Have you ever wished you could browse all the Haskell packages
together in your IDE, with full navigation using go-to-definition
and find-references? Here’s a demo of something I hacked together
while at ZuriHac 2025 over the weekend:
In the previous post I talked about
how to index all of Hackage (actually Stackage, strictly speaking,
because it’s not in general possible to build all of Hackage together)
using Glean. Since that post I made some
more progress on the indexer:
The indexer now indexes
types. You can
see type-on-hover working in the demo. The types are similar to what
you see in the Haddock-generated hyperlinked source, except that
here it’s always using the type of the definition and not the type
at the usage site, which might be more specific. That’s a TODO for
later.
Fixed a bunch of things, enriched the index with details about
constructors, fields and class methods, and made indexing more
efficient.
The DB size including types is now about 1GB, and it takes just
under 10 minutes on my 9-year-old laptop to index the nearly 3000
packages in my stackage LTS 21.21 snapshot.
Hooking it up to VS Code
The architecture looks like this:
The LSP server is a modified version of
static-ls, which is
already designed to provide an LSP service based on static
information. I just reimplemented a few of its handlers to make calls
to Glass instead of the existing hie/hiedb implementations. You can
see the changes on my fork of
static-ls. Of
course, these changes are still quite hacky and not suitable for
upstreaming.
Glass
is a “Language-agnostic Symbol Server”. Essentially it provides an API
abstraction over Glean with operations that are useful for code
navigation and search.
Where to next?
There remain a few issues to solve before this can be useful.
Make Glean more easily installable. There’s a general concensus that
cabal install glean would lower the barrier to entry
significantly; in order to do this we need to build the folly
dependency using Cabal.
Clean up and ship the LSP server, somehow. Once Glean is
cabal-installable, we can depend on it from an LSP server package.
Think about continuous integration to build the Glean
DB. Perhaps this can piggyback off the stackage CI infra? If we
can already build a complete stackage snapshot, and Glean is
easily installable, then indexing would be fairly
straightforward. I’d love to hear suggestions on how best to do
this.
And looking forwards a bit further:
Think about how to handle multiple packages versions. There’s no
fundamental problem with indexing multiple package versions, except
that Glass’s SymbolID format currently doesn’t include the package
version but that’s easily fixable. We could for example build
multiple stackage LTS instances and index them all in a single Glean
DB. There would be advantages to doing this, if for instance there
were packages in common between two Stackage instances then the
Glean DB would only contain a single copy. A lot of the type
structure would be shared too.
Provide search functionality in the LSP. Glean can provide
simple textual search for names, and with some work could also
provide Hoogle-like type search.
Think about how to index local projects and local changes. Glean
supports stacked and
incremental DBs, so we
could build a DB for a local project stacked on top of the full
Stackage DB. You would be able to go-to-definition directly from
a file in your project to the packages it depends on in
Stackage. We could re-index new .hie files as they are
generated, rather like how static-ls currently handles changes.
Integrate with HLS? Perhaps Glean could be used to handle
references outside of the current project, switching seamlessly
from GHC-based navigation to Glean-based navigation if you jump
into a non-local package.
More use cases?
I talked with a few people at ZuriHac about potential use cases for
Glean within the Haskell ecosystem. Using it in haskell.org came up
a few times, as a way to power search, navigation and analysis. Also
mentioned was the possibility of using it as a Hoogle
backend. Potentially we could replace the Haddock-generated
hyperlinked sources on haskell.org with a Glean-based browser, which
would allow navigating links between packages and find-references.
Another use cases that came up was the possibility of doing impact
analysis for core library changes (or any API changes really). Some of
this is already possible using find-references, but more complex cases
such as finding instances that override certain methods aren’t
possible yet until we extend the indexer to capture richer
information.
If you’re interested in using Glean for something, why not jump on the
Glean discord server and tell us about it!
Today we’re continuing our study of Rust and Haskell solutions to basic coding problems. This algorithm is going to be a little harder than the last few we’ve done in this series, and it will get trickier from here!
For a complete study of problem solving techniques in Haskell, make sure to check out Solve.hs. This course runs the gamut from basic solving techniques to advanced data structures and algorithms, so you’ll learn a lot!
The Problem
Today’s problem is Zigzag Conversion. This is an odd problem that stretches your ability to think iteratively and spatially. The idea is that you’re given an input string and a number of “rows”. You need to then imagine the input word written as a zig-zag pattern, where you write the letters in order first going down, and then diagonally up to the right until you get back to the first row. Then it goes down again. Your output must be characters re-ordered in “row-order” after this zig-zag rearrangement.
This makes the most sense looking at examples. Let’s go through several variations with the string MONDAYMORNINGHASKELL. Here’s what it looks like with 3 rows.
M A R G K
O D Y O N N H S E L
N M I A L
So to get the answer, we read along the top line first (MARGK), then the second (ODYONNHSEL), and then the third (NMIAL). So the final answer is MARGKODYONNHSELNMIAL.
Now let’s look at the same string in 4 rows:
M M G L
O Y O N H E L
N A R I A K
D N S
The answer here is MMGLOYONHELNARIAKDNS.
Here’s 5 rows:
M R K
O O N S E
N M I A L
D Y N H L
A G
The answer here is MRKOONSENMIALDYNHLAG.
And now that we have the pattern, we can also consider 2 rows, which doesn’t visually look like a zig-zag as much:
M N A M R I G A K L
O D Y O N N H S E L
This gives the answer MNAMRIGAKLODYONNHSEL.
Finally, if there’s only 1 row, you can simply return the original string.
The Algorithm
So how do we go about solving this? The algorithm here is a bit more involved than the last few weeks!
Our output order is row-by-row, so for our solution we should think in a row-by-row fashion. If we can devise a function that will determine the indices of the original string that belong in each row, then we can simply loop over the rows and append these results!
In order to create this function, we have to think about the zig-zag in terms of “cycles”. Each cycle begins at the top row, goes down to the bottom row, and then up diagonally to the second row. The next element to go at the top row starts a new cycle. By thinking about cycles, we’ll discover a few key facts:
With n rows (n >= 2), a complete cycle has 2n - 2 letters.
The top and bottom row get one letter per cycle.
All other rows get two letters per cycle.
Now we can start to think mathematically about the indices that belong in each row. It’s easiest to think about the top and bottom rows, since they only get one letter each cycle. Each of these has a starting index (0 and n - 1, respectively), and then we add the cycle length 2n - 2 to these starting indices until it exceeds the length.
The middle rows have this same pattern, only now they have 2 starting indices. They have the starting index from the “down” direction and then their first index going up and to the right. The first index for row i is obviously i - 1, but the second index is harder to see.
The easiest way to find the second index is backwards! The next cycle starts at 2n - 2. So row index 1 has its second index at 2n - 2 - 1, and row index 2 has its second index at 2n - 2 - 2, and so on! The pattern of adding the “cycle number” will work for all starting indices.
Once we have the indices for each row, our task is simple. We build a string for each row and combine them together in order.
So suppose we have our 4-row example.
M M G L
O Y O N H E L
N A R I A K
D N S
The “cycle num” is 6 (2 * 4 - 2). So the first row has indices [0, 6, 12, 18]. The fourth row starts with index 3, and so its indices also go up by 6 each time: [3, 9, 15].
The second row (index 1) has starting indices 1 and 5 (6 - 1). So its indices are [1, 5, 7, 11, 13, 17, 19]. Then the third row has indices [2, 4, 8, 10, 14, 16].
A vector input will allow us to efficiently use and combine these indices.
As a final note, the “cycle num” logic doesn’t end up working with only 1 row. The cycle length using our calculation would be 0, not 1 as it should. The discrepancy is because our “cycle num” logic really depends on having a “first” and “last” row. So if we only have 1 row, we’ll hardcode that case and return the input string.
Rust Solution
In our rust solution, we’ll accumulate our result string in place. To accomplish this we’ll do a few setup steps:
Handle our base case (1 row)
Get the string length and cycle number
Make a vector of the input chars for easy indexing (Rust doesn’t allow string indexing)
Initialize our mutable result string
pub fn convert(s: String, num_rows: i32) -> String {
if (num_rows == 1) {
return s;
}
let n = s.len();
let nr = num_rows as usize; // Convenience for comparison
let cycleLen: usize = (2 * nr - 2);
let sChars: Vec<char> = s.chars().collect();
let mut result = String::new();
...
}
Now we have to add the rows in order. Since the logic differs for the first and last rows, we have 3 sections: first row, middle rows, and last row. The first and last row are straightforward using our algorithm. Each is a simple while loop.
pub fn convert(s: String, num_rows: i32) -> String {
if (num_rows == 1) {
return s;
}
let n = s.len();
let nr = num_rows as usize; // Convenience for comparison
let cycleLen: usize = (2 * nr - 2);
let sChars: Vec<char> = s.chars().collect();
let mut result = String::new();
// First Row
let mut i = 0;
while i < n {
result.push(sChars[i]);
i += cycleLen;
}
// Middle Rows
...
// Last Row
i = (nr - 1);
while i < n {
result.push(sChars[i]);
i += cycleLen;
}
return result;
}
Now the middle rows section is similar. We loop through each of the possible rows in the middle. For each of these, we’ll do a while loop similar to the first and last row. These loops are different though, because we have to track two possible values, the “first” and “second” of each cycle.
If the “first” is already past the end of the vector, then we’re already done and can skip the loop. But even if not, we still need an “if check” on the “second” value as well. Each time through the loop, we increase both values by cycleLen.
pub fn convert(s: String, num_rows: i32) -> String {
if (num_rows == 1) {
return s;
}
let n = s.len();
let nr = num_rows as usize; // Convenience for comparison
let cycleLen: usize = (2 * nr - 2);
let sChars: Vec<char> = s.chars().collect();
let mut result = String::new();
// First Row
let mut i = 0;
while i < n {
result.push(sChars[i]);
i += cycleLen;
}
// Middle Rows
for row in 1..(nr - 1) {
let mut first = row;
let mut second = cycleLen - row;
while first < n {
result.push(sChars[first]);
if second < n {
result.push(sChars[second]);
}
first += cycleLen;
second += cycleLen;
}
}
// Last Row
i = (nr - 1);
while i < n {
result.push(sChars[i]);
i += cycleLen;
}
return result;
}
And that’s our complete solution!
Haskell Solution
The Haskell solution follows the same algorithm, but we’ll make a few stylistic changes compared to Rust. In Haskell, we’ll go ahead and define specific lists of indices for each row. That way, we can combine these lists and make our final string all at once using concatMap. This approach will let us demonstrate the power of ranges in Haskell.
We start our defining our base case and core parameters:
zigzagConversion :: String -> Int -> String
zigzagConversion input numRows = if numRows == 1 then input
else ...
where
n = length input
cycleLen = 2 * numRows - 2
...
Now we can define index-lists for the first and last rows. These are just ranges! We have the starting element, and we know to increment it by cycleLen. The range should go no higher than n - 1. Funny enough, the range can figure out that it should be empty in the edge case that our input is too small to fill all the rows!
In Rust, we used a while-loop with two state values to calculate the middle rows. Hopefully you know from this series now that this while loop translates into a recursive function in Haskell. We’ll accumulate our list of indices as a tail argument, and keep the two stateful values as our other input parameters. We’ll combine all our lists together into one big list of int-lists, allRows.
zigzagConversion :: String -> Int -> String
zigzagConversion input numRows = if numRows == 1 then input
else ...
where
n = length input
cycleLen = 2 * numRows - 2
firstRow :: [Int]
firstRow = [0,cycleLen..n - 1]
lastRow :: [Int]
lastRow = [numRows - 1, numRows - 1 + cycleLen..n - 1]
middleRow :: Int -> Int -> [Int] -> [Int]
middleRow first second acc = if first >= n then reverse acc
else if second >= n then reverse (first : acc)
else middleRow (first + cycleLen) (second + cycleLen) (second : first : acc)
middleRows :: [[Int]]
middleRows = map (\i -> middleRow i (cycleLen - i) []) [1..numRows-2]
allRows :: [[Int]]
allRows = firstRow : middleRows <> [lastRow]
...
Now we bring it all together with one final step. We make a vector from our input, and define a function to turn a single int-list into a single String. Then at the top level of our function (the original else branch), we use concatMap to bring these together into our final result String.
zigzagConversion :: String -> Int -> String
zigzagConversion input numRows = if numRows == 1 then input
else concatMap rowIndicesToString allRows
where
n = length input
cycleLen = 2 * numRows - 2
firstRow :: [Int]
firstRow = [0,cycleLen..n - 1]
lastRow :: [Int]
lastRow = [numRows - 1, numRows - 1 + cycleLen..n - 1]
middleRow :: Int -> Int -> [Int] -> [Int]
middleRow first second acc = if first >= n then reverse acc
else if second >= n then reverse (first : acc)
else middleRow (first + cycleLen) (second + cycleLen) (second : first : acc)
middleRows :: [[Int]]
middleRows = map (\i -> middleRow i (cycleLen - i) []) [1..numRows-2]
allRows :: [[Int]]
allRows = firstRow : middleRows <> [lastRow]
inputV :: V.Vector Char
inputV = V.fromList input
rowIndicesToString :: [Int] -> String
rowIndicesToString = map (inputV V.!)
Conclusion
This comparison once again showed how while loops in Rust track with recursive functions in Haskell. We also saw some nifty Haskell features like ranges and tail recursion. Most of all, we saw that even with a trickier algorithm, we can still keep the same basic shape of our algorithm in a functional or imperative style.
To learn more about these problem solving concepts, take a look at Solve.hs, our comprehensive course on problem solving in Haskell. You’ll learn about recursion, list manipulation, data structures, graph algorithms, and so much more!
They, along with our other clients who contribute to funding our open-source
work, are making a real difference to Haskell’s sustainability. If your company
uses Haskell and is not already funding its development, why not join them?
Introduction
The Haskell language has just turned 35 years old, and the vibrant Haskell
community continues to thrive thanks to the efforts of dedicated volunteers
across many different roles and open source projects. Haskell has always been a
collective, community-driven project without a controlling corporate sponsor,
which has many benefits. But it leads to challenges funding some activities that
are crucial to the sustainability of the ecosystem:
Maintenance of core parts of the Haskell toolchain: large, complex
open-source projects need full-time paid maintainers to function effectively
and avoid the “Nebraska problem”.
Community-building and infrastructure support: from running servers for
crucial services, to coordinating volunteers and assisting open-source
projects, to organizing events.
Both Well-Typed and the Haskell Foundation rely on funding from commercial users
of Haskell to make this possible. Thus we are happy to be working in partnership
to make it easy for companies to contribute back and support all this important
work, while receiving tangible benefits themselves. Haskell Ecosystem Supporter
companies make a single contract and in return get support with their specific
needs, work to sustain the Haskell tools on which they rely, and sponsor the
Haskell Foundation. You can find out more about the approach we’ve agreed in
the Foundation’s announcement of Ecosystem
Partnerships.
Well-Typed have been keen supporters and sponsors of the Haskell
Foundation since it was established five
years ago, and Andres Löh from Well-Typed is currently Chair
of the Foundation Board. We would be delighted to see
other companies supporting the Haskell Foundation and the wider Haskell
community through similar partnerships.
Our offer
Well-Typed offer multiple tiers of support, to meet the needs of clients of
different sizes:
Tier
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Next steps
If this sounds like something you or your company would be interested in, and
you would like to help support the Haskell ecosystem, then:
Today we continue our series exploring LeetCode problems and comparing Haskell and Rust solutions. We’re staying in the realm of list/vector manipulation, but the problems are going to start getting more challenging!
If you want to learn more about problem solving in Haskell, you should take a closer look at Solve.hs! You’ll particularly learn how to translate common ideas from loop-based into Haskell’s recursive ideas!
The Problem
Today’s problem is Product of Array Except Self. The idea is that we are given a vector of n integers. We are supposed to return another vector of n integers, where output[i] is equivalent to the product of all the input integers except for input[i].
The key constraint here is that we are not allowed to use division. If we could use division, the answer would be simple! We would find the product of the input numbers and then divide this product by each input number to find the corresponding value. But division is more expensive than most other numeric operations, so we want to avoid it if possible!
The Algorithm
The approach we’ll use in this article relies on “prefix products” and “suffix products”. We’ll make two separate vectors called prefixes and suffixes, where prefixes[i] is the product of all numbers strictly before index i, and suffixes[i] is the product of all numbers strictly after index i.
Then, we can easily produce our results. The value output[i] is simply the product of prefixes[i] and suffixes[i].
As an example, our input might be [3, 4, 5]. The prefixes vector should be [1, 3, 12], and the suffixes vector should be [20, 5, 1]. Then our final output should be [20, 15, 12].
impl Solution {
pub fn product_except_self(nums: Vec<i32>) -> Vec<i32> {
let n = nums.len();
let mut prefixes = vec![0; n];
let mut suffixes = vec![0; n];
let mut totalPrefix = 1;
let mut totalSuffix = 1;
// Loop 1: Populate prefixes & suffixes
for i in 0..n {
prefixes[i] = totalPrefix;
totalPrefix *= nums[i];
suffixes[n - i - 1] = totalSuffix;
totalSuffix *= nums[n - i - 1];
}
let mut results = vec![0; n];
// Loop 2: Populate results
for i in 0..n {
results[i] = prefixes[i] * suffixes[i];
}
return results;
}
}
The two for-loops provide this solution with its shape. The first loop generates our vectors prefixes and suffixes. We keep track of a running tally of the totalPrefix and the totalSuffix. Each of these is initially 1.
let n = nums.len();
let mut prefixes = vec![0; n];
let mut suffixes = vec![0; n];
let mut totalPrefix = 1;
let mut totalSuffix = 1;
On each iteration, we assign the current “total prefix” to the prefixes vector in the front index i, and then the “total suffix” to the suffixes vector in the back index n - i - 1. Then we multiply each total value by the input value (nums) from that index so it’s ready for the next iteration.
// Loop 1: Populate prefixes & suffixes
for i in 0..n {
prefixes[i] = totalPrefix;
totalPrefix *= nums[i];
suffixes[n - i - 1] = totalSuffix;
totalSuffix *= nums[n - i - 1];
}
And now we calculate the result, by taking the product of prefixes and suffixes at each index.
let mut results = vec![0; n];
// Loop 2: Populate results
for i in 0..n {
results[i] = prefixes[i] * suffixes[i];
}
return results;
Haskell Solution
In Haskell, we can follow this same template. However, a couple differences stand out. First, we don’t use for-loops. We have to use recursion or recursive helpers to accomplish these loops. Second, when constructing prefixes and suffixes, we want to use lists instead of modifying mutable vectors.
When performing recursion and accumulating linked lists, it can be tricky to reason about which lists need to be reversed at which points in our algorithm. For this reason, it’s often very helpful in Haskell to start from the end of our algorithm.
Let’s write out a template of our solution that leaves prefixes and suffixes as undefined stubs. Then the first step we’ll work through is how to get the solution from that:
productOfArrayExceptSelf :: V.Vector Int -> V.Vector Int
productOfArrayExceptSelf inputs = solution ???
where
n = V.length inputs
solution :: ??? -> V.Vector Int
prefixes :: [Int]
prefixes = undefined
suffixes :: [Int]
suffixes = undefined
So given prefixes and suffixes, how do we find our solution? The ideal case is that both these lists are already in reverse-index order with respect to the input vector (i.e. n - 1 to 0). Then we don’t need to do an additional reverse to get our solution.
We can then implement solution as a simple tail recursive helper function that peels one element off each input and multiplies them together. When we’re out of inputs, it returns its result:
productOfArrayExceptSelf :: V.Vector Int -> V.Vector Int
productOfArrayExceptSelf inputs = solution (prefixes, suffixes, [])
where
n = V.length inputs
-- Loop 2: Populate Results
solution :: ([Int], [Int], [Int]) -> V.Vector Int
solution ([], [], acc) = V.fromList acc
solution (p : ps, s : ss, acc) = solution (ps, ss, p * s : acc)
solution _ = error “Prefixes and suffixes must be the same size!”
prefixes :: [Int]
suffixes :: [Int]
So now we’ve done “Loop 2” already, and we just have to implement “Loop 1” so that it produces the right results. Again, we’ll make a tail recursive helper, and this will produce both prefixes and suffixes at once. It will take the index, as well as the “total” prefix and suffix so far, and then two accumulator lists. At the end of this, we want both lists in reverse index order.
Now we fill in mkPrefixSuffix as we would any tail recursive helper. First we satisfy the base case. This occurs once i is at least n. We’ll return the accumulated lists.
mkPrefixSuffix :: (Int, Int, [Int], Int, [Int]) -> ([Int], [Int])
mkPrefixSuffix (i, totalPre, pres, totalSuff, suffs) = if i >= n then (pres, reverse suffs)
else ...
But observe we’ll need to reverse suffixes! This becomes clear when we map out what each iteration of the loop looks like for a simple input. Doing this kind of “loop tracking” is a very helpful problem solving skill for walking through your code!
Our prefixes are [12, 3, 1], which is properly reversed, but the suffixes are [20, 5, 1]. We don’t want both lists ending in 1! So we reverse the suffixes.
Now that we’ve figured this out, it’s simple enough to fill in the recursive case using what we already know from “Loop 1” in the Rust solution. We get the “front” index of input with i, and the “back” index with n - i - 1, use these to get the new products, and then save the old products in our list.
mkPrefixSuffix :: (Int, Int, [Int], Int, [Int]) -> ([Int], [Int])
mkPrefixSuffix (i, totalPre, pres, totalSuff, suffs) = if i >= n then (pres, reverse suffs)
else
let nextPre = nums V.! i
nextSuff = nums V.! (n - i - 1)
in mkPrefixSuffix (i + 1, totalPre * nextPre, totalPre : pres, totalSuff * nextSuff, totalSuff : suffs)
Here’s our complete Haskell solution!
productOfArrayExceptSelf :: V.Vector Int -> V.Vector Int
productOfArrayExceptSelf inputs = solution (prefixes, suffixes, [])
where
n = V.length inputs
solution :: ([Int], [Int], [Int]) -> V.Vector Int
solution ([], [], acc) = V.fromList acc
solution (p : ps, s : ss, acc) = solution (ps, ss, p * s : acc)
solution _ = error "Invalid solution!"
prefixes :: [Int]
suffixes :: [Int]
(prefixes, suffixes) = mkPrefixSuffix (0, 1, [], 1, [])
mkPrefixSuffix:: (Int, Int, [Int], Int, [Int]) -> ([Int], [Int])
mkPrefixSuffix (i, totalPre, pres, totalSuff, suffs) = if i >= n then (pres, reverse suffs)
else
let nextPre = inputs V.! i
nextSuff = inputs V.! (n - i - 1)
in mkPrefixSuffix (i + 1, totalPre * nextPre, totalPre : pres, totalSuff * nextSuff, totalSuff : suffs)
Conclusion
In this comparison, we saw a couple important differences in problem solving with a loop-based language like Rust compared to Haskell.
For-loops have to become recursion in Haskell
We want to use lists in Haskell, not mutable vectors
It takes a bit of planning to figure out when to reverse lists!
This led us to a couple important insights when solving problems in Haskell.
“Starting from the end” can be very helpful in plotting out our solution
“Loop tracking” is a very helpful skill to guide our solutions
For an in-depth look at these sorts of comparisons, check out our Solve.hs course. You’ll learn all the most important tips and tricks for solving coding problems in Haskell! In particular you’ll get an in-depth look at tail recursion, a vital concept for solving problems in Haskell.
A lot of strong engineers that I know haven't really taken a serious look at AI coding; they've used LLMs to ask questions or write simple scripts and appreciate that it is a useful tool, but haven't actually tried building a nontrivial application entirely from scratch in vibe coding style (here, I use the term in its original meaning: when you do AI coding without carefully reviewing the output). This is understandable: if you're not working on a green field project, there aren't that many opportunities to write code in this style--standard practice for established projects is that someone else needs to review all of the code you write: this is a bad match for vibe coding! So in this post, I want to give a concrete case study of a nontrivial system that was entirely vibe coded (ScubaDuck), to argue the following claims:
AI coding can be done on a manager's schedule: you don't need continuous blocks of coding time and context-switching is considerably less harmful. ScubaDuck was implemented in three days of part time work, where all of the work happened when the baby was napping.
AI coding substantially lowers the cost of doing projects in tech stacks you are less familiar with. ScubaDuck is mostly JavaScript UI code, which is not something I write on a day-to-day basis.
AI coding is an unlock for "sidequests": support software that's ancillary to your main task that is nice to have, but not essential. If previously you would have decided the cost outweighed the benefit, AI coding reducing the cost means you should redo these calculations.
Vibe coding works and can produce working software. ScubaDuck is an existence proof that vibe coding is a viable strategy for generating JavaScript UI code (NB: I don't claim vibe coding will work for all domains, nor do I claim this is the only domain for it works. Hopefully you can also build some intuition for where it is more or less likely to work). You will not one shot it (ScubaDuck was 150 prompts in the end) but if you are prompting the LLM to also generate tests, you can reliably fix issues without causing regressions to existing code.
Vibe coding is good for situations where buggy software is low impact; be on the lookout for ways to engineer this sort of situation. ScubaDuck is a read-only interface, where the only downside to being buggy is you can't issue the queries you want to issue.
Update: You can see all of my prompts and the resulting agent trajectories at scubaduck-prompts.
What is ScubaDuck?
ScubaDuck is a discount implementation of Meta's internal Scuba realtime database system. You can read more about what exactly this is on GitHub, but it's not so important for the purposes of this post: the key details you need to know about ScubaDuck is that it consists of a Python server that exposes an API to perform queries against a DuckDB database, and an HTML and JavaScript frontend application which implements the forms for building these queries and rendering of the output data. Both the forms and output data rendering have nontrivial JavaScript enhancements: some form inputs are chip inputs and support autocomplete, and the time series view is an SVG chart. All of these components were coded from scratch, so the project has no third-party JavaScript dependencies.
So on the one hand, this project is pretty simple. There are no stringent performance or uptime requirements, it's a pretty standard server-client program that the LLM has seen millions of times before (this is good!) On the other hand, the exact behavior of the frontend UI is quite intricate and would be very difficult to one-shot in a single prompt. Indeed, as I was coding and testing the application, I frequently ran into situations that I didn't anticipate in my original specification, and that I had to ask Codex to refine. Another way to put it is that ScubaDuck is a relatively simple functional specification (although this too was not one shot), but I did a lot of polishing of small behaviors so that the interface behaved in the way that I expected Scuba to behave. Here, it was helpful that I had a very clear idea of what I wanted (since I've used Scuba quite a lot at work).
Going into ScubaDuck, I had a pretty good sense that this project should be a good fit for LLMs. HTML, JavaScript and Python are all extremely high resource languages, and I'd heard lots of people raving about how good LLMs were at transforming wireframes and mockups into fully functional websites. It is also fully self contained and straightforward-ish to test (only "ish" because you do have to use something like Playwright to actually test the frontend UI, which honestly is a slog. But fortunately, the LLM can write the tests for you!) One design decision I made, which I didn't originally anticipate but worked out in the end, was the decision to not use any third-party JavaScript libraries. This was by accident: Python has no native of bundling third party JavaScript, but I wanted the tool to work offline. I wasn't sure if you could vibe code an SVG charting library from scratch, but apparently you can and it's quite easy!
Agent setup
ScubaDuck was implemented with OpenAI Codex in the cloud (not the CLI tool). Codex's cloud offering requires you to initialize a hermetic environment which the coding agent can execute commands in. It's pretty well known now that AI coding agents work much better if they are able to run the code they write and see if it worked or not, so this is quite an important part of the process. Unfortunately, this was somewhat time consuming trial and error to setup. I had a fairly detailed initial prompt, and what I would do was submit it to Codex, watch it fail, read over the trajectory (the agent logs) to see what happened (Codex wanted to use npm! Codex couldn't download something from the internet! Codex tried to use a package that wasn't available!) and then fixed whatever environment misconfiguration had caused it to fail, or edited AGENTS.md to instruct it to not do some behavior. According to my history, the first day of the project was spent unsuccessfully trying to get the project setup, and my first successful Codex PR only happened on May 19.
At the end of setup, I had the following:
A pyproject.toml with exactly the dependencies I wanted to be used (duckdb, flask and python-dateutil), a lockfile for it (since I was using uv) and my preferred configuration for various tools (pytest, ruff). I'm a big fan of pytest-xdist for vibe coded projects, since you can prompt the LLM to write tests that will work when run in parallel and it does a pretty good job at this. Later I'd also add a pyright configuration, though initially I left it out because I saw Codex doing some strange things on account of duckdb being untyped, and I didn't want to debug it at the time (the fix, by the way, is instructing the LLM to define stubs as necessary in this case.)
An AGENTS.md file with some basic instructions to try to get Codex to stop doing things I saw it doing in the initial trajectories that I didn't want it to do. Nothing fancy, just if you see Codex do something bad, tell it not to do it in AGENTS.md. A good example of this is the "There are no nested AGENTS.md files, this is the only agents file": Codex is post-trained to look for nested AGENTS.md files, but you can save a few tool calls if you tell it there aren't any. (Note: folklore for Claude 3.7 is that instruction following for this sort of rules following was not great. Word on the street is that both Codex and Claude 4 are substantially better at this. Extra note: For uv users, another notable instruction in AGENTS.md is how to activate the venv, since at time of writing I couldn't get Codex to make this happen automatically.)
A setup script for the environment. This took the most debugging, because Codex runs all Internet access through a proxy and sometimes it works imperfectly.
Interact with the application and find things that are broken
Prompt the LLM to fix them
Repeat
For example, after the very first PR, some very mild poking around immediately revealed the bugs fixed in #2:
There's a race condition in the current test logic for matching against table contents in run_query. Specifically, if there were previously valid results in lastResults, and for some reason Dive doesn't do anything, then we will still see the old results. The testing framework should explicitly clear lastResults before attempting an interaction.
Filter functionality does not work. We will first add a failing test, and then fix it. The failing test should click "Add Filter", then select "user" as the field, and then add an "alice" chip (by typing alice in the text box and pressing ENTER). Then when we dive, we should see two alice rows. Right now, NO request is issued at all when we click Dive. Diagnose and then fix the problem.
Prompt the agent to write tests. It's very helpful to prompt the agent to generate tests for whatever bugs its fixing. For frontend code, I decided to use playwright to write these tests. An example in #11:
This is kind of a very manual and somewhat brittle test. I doubt I would have had the patience to manually write and debug this. But the LLM will happily do it. There is also a fear that the test might not actually be testing anything for real. But because these tests are all generated from bug reports, you can prompt the model to write the test first, check that it's failing, and then fix the problem. I didn't instruct this in the above prompt, but I did instruct it for #155, for example. The trajectory shows that the LLM demonstrates it can repro the bug before fixing the issue:
I’ll add a case-insensitive test for "samples" to check server behavior with lowercase input.
I’ve found the bug happens when "Samples" is lowercase. I’ll adjust the query to handle case insensitivity and add a test for it. Time to modify server.py to fix the issue!
Beware BC-breaking changes. The LLM generated tests worked well when I didn't need to change behavior in a BC-breaking way. Example prompt:
Change time column to default to none unconditionally. (I want to default it to a time column if we find a good candidate but this threshold needs tuning and I don't have the tuning right now.)
This hit a Codex timeout:
This attempt to update the default time column led to extensive frontend test failures that require deeper changes. The environment made it difficult to fully diagnose the Playwright test behaviors within the time available, so the work could not be completed.
In standard software engineering practice, when this happens, decouple the BC compatible and BC breaking changes!
Make it so that Time Column can be set to (none). When it is set this way, the Start/End fields are hidden and we don't apply a filter on time range. (#115)
and then later instead of defaulting the time column to none, I added a heuristic to pick a column that looked like time, which picked the same column that all of the existing tests had also expected to be called with.
Refactors have to be split up. Codex's timeout means that you can't ask it to do too much in one go. Here's a prompt that timed out:
scubaduck/index.html has gotten a bit long. Let's split out some of the JS code into dedicated JS files for their functionality. Also setup the necessary Flask scaffolding to serve these JS files. I think splitting out these specific components would be good:
Dropdown implementation
Sidebar resizing
JS controlling the View Settings (e.g., updateDisplayTypeUI, as well as one off interactions on form elements, columns handling, filter handling, the actual Dive implementation (including query updating), reading in defaults from query string)
Table rendering (e.g., formatNumber, sorting)
Chip input implementation
Chart rendering (showTimeSeries)
Make changes to AGENTS.md or README.md describing the structure so you can quickly find where the components you need are
I eventually did manage the refactor by prompting Codex to individually move out the pieces I wanted to extract one-by-one. This is a place where I think Claude Code probably would have performed better.
Parallelizing tasks. As you can see from the lengths of my prompts, it does take a while to write a good prompt; you're basically writing a bug report with enough detail that the LLM can repro it and then fix it. So sometimes I would be bottlenecked on prompt writing. However, sometimes the prompts were quite short. In those cases, Codex encourages you to submit more tasks that can run in parallel. I found this worked well, and I'd sometimes have as many as five instances going (once again, rate limited by discovering problems, making designs and typing prompts!) One irritation is when the tasks end up conflicting with each other. Sometimes the conflicts are easy to fix, but if it feels nontrivial, it's often better to just ask Codex to redo one of the PRs on latest main after the other has landed. To avoid merge conflicts, it helps to have only one "main feature" agent going at any time, and then ask the agent to do random bugfixes in parallel with it. Once you have no more tasks to get running, you can go do something else while you wait for the agents to finish (manager schedule!)
Prompting
As a reminder, I've posted all of my prompts (including the ones that failed) at scubaduck-prompts, and I think it's helpful to skim through them to get a flavor of what I was asking the LLM. But to summarize, what did I spend most of my time on prompting Codex to do? My general vibe (ahem) is that I spent most of my time doing minor enhancements, where I instructed Codex to make some part of the program work slightly differently, in a way that was previously unspecified from the previous prompt. The metaphor I had in my head while I was working on the project was like that of a sculptor chiseling away marble: in the beginning, anything is possible, but as I kept prompting, I continuously narrowed down the space of possible programs I had until I had exactly the one I wanted. One big thing I want to note is that Codex rarely needed to make updates to my tests; for the most part, tests that were added never got taken away, because I never "changed my mind". I suspect that the vibe coding process would have been rockier if I was having to change behavior frequently.
One of the things that surprised me the most about the process was how easy it was to implement a line chart in SVG with Codex. My first prompt resulted in a chart that looked broken on the test data:
We're going to add a new View type, to go along with Samples and Table: Time Series. Time Series supports all the fields that Table supports, and a few more:
X-axis: Main group by dimension, e.g., the x-axis on time series view. This is our custom dropdown selector, but only time columns are populated here. It should prefer a default setting from the following list, most preferred first: "time", "timestamp"
Granularity: Choose the time interval between data points on the chart.
For example, a granularity of 1 hour means there will be a data point every 60 minutes that is aggregated with the chosen Aggregate function over the data for the granularity period before point. This is a plain drop down. The valid values are: Auto, Fine, 1 second, 5 seconds, 10 seconds, 30 seconds, 1 minute, 4 minutes, 5 minutes, 10 minutes, 15 minutes, 30 minutes, 1 hour, 3 hours, 6 hours, 1 day, 1 week, 30 days. The semantics of the Auto setting is that it sets the interval to whatever would result in maximum 100 buckets (if there are not enough data points for that many buckets, it just picks the finest time interval that makes sense), and Fine which sets the interval to 500 buckets.
Fill Missing Buckets: This is a dropdown. For now, it has the settings "Fill with 0 (Per Series)" (default), "Connect (Per Series)" and "Leave blank".
Additionally, the default setting of Limit is 7, as it controls how many elements from group by will be plotted (the actual number of lines plotted could be a multiple of this, as we will plot every selected Column).
Unlike Samples and Table, we will instead display a line chart in the right panel. To plot the line chart, we will implement it by hand with JS and SVG, similar to how highcharts implements it. We will not use any third party dependencies. Lines will be plotted as paths, no smoothing, no dots for individual data points. Each series (as generated by group by) should be plotted with a different color, assigned using a best practices color palette for graph design. There should be a rendering of x-axis and y-axis; the x-axis should have slanted labels to aid readability. When we mouse over the chart, a vertical line should snap to the center of the time bucket that we are closest to. We should also display a crosshair on all of the series showing us their values at that data point, and highlight the closest point we are on, and increase the thickness of the series that point is on. To the left of the graph (still in the right panel), there should be a legend. The legend looks like this:
[GROUP BY VALUE] [AGGREGATE]
[First Column name, with series color]
[Number of samples for the first column]
[Second Column name, with series color]
[Number of samples for the second column]
... for all columns
----
... for all group by values (up to the limit)
So for example, if I group by user, I might see:
Alice AVG
value
4 (samples)
The highlighted series (which has a thicker line) should also be highlighted in the legend).
This was kind of terrifying, because I initially thought I didn't have a good way to test the SVG outputs. But after doing some regular old-fashioned debugging and reading the code (yes, this part not vibe coded), I figured out the problem, and also realized that Playwright can test that an SVG path is not just entirely straight. After the initial bugs were fixed, I mostly had to add missing features like x-axis/y-axis and interactivity features (amusingly, Codex ignored most of the instructions in the latter half of the prompt, giving only the barest bones legend. I suspect this was because I had some files which were too long). My general take after this was that JS chart libraries are going to become obsolete: it's much easier to vibe code a bespoke implementation and then customize the heck out of it.
Conclusion
ScubaDuck was implemented in about 150 Codex prompts. As you can see from the sample prompts above, the prompts are recognizably programming, they just happen to be in plain English language. This is a big help, because I never had to keep track of the nest of callbacks and state machines for implementing complex UI elements in JavaScript. I had to be fluent in what I wanted my program to do, and a good QA tester for the application to discover new problems that needed to be fixed, but I did not have to worry at all about the vagaries of SVG DOM elements or pixel position computation minutiae. It's hard to say how long it would have taken to code this by hand, but I think reproducing a UI that's been in production for years at Meta in three (part-time) days is pretty good!
Despite having done a bit of AI coding before, I also learned a bit from working on Codex. Codex made it blindingly clear that the parallel modality (and subsequent conflict resolution) is important. It made me adjust up my estimation of the capability of LLMs to write raw HTML/JS and evoked a future where people vibe code components in place of taking on a third party dependency. I was very appreciative of no rate limit Codex (though I doubt it's going to last.) It also reminded me how difficult it will be to setup agent environments for "real" projects (like PyTorch).
Hopefully, this case study has given you some ideas for things to try. Go forth and vibe code, responsibly!
I don't know about you, but testing isn't my favourite part of
software development.
It's usually the last thing standing between me and shipping a shiny
new feature, and writing tests is often an annoying process with a lot
of boilerplate and fighting against your system to get your app into a
good start starting for the test or mocking out whichever services your
app depends on.
Much ink has been spilled about how to organize your code in order to
make this easier, but the fact that so many blog posts and frameworks
exist for this express purpose suggests to me that we as a community of
software developers haven't quite solved this issue yet.
Keep reading to see how I've solved this problem for myself by simply
avoiding unit testing altogether.
An alternative testing
method
When I first started at Unison Computing I was submitting my first
feature when I learned there were precious few unit tests. I found it
rather surprising for a codebase for a compiler for a programming
language! How do you prevent regressions without unit tests?
The answer is what the Unison team has dubbed transcript
tests. These are a variation on the concept of golden-file
tests.
A Unison transcript is a markdown file which explains in
standard what behaviour it is going to test, then intersperses
code-blocks which outline the steps involved in testing that feature
using a mix of Unison code and UCM commands (UCM is Unison's CLI tool).
After that comes the magic trick; UCM itself can understand and run
these transcript files directly and record the results of each
block.
When running a transcript file with the ucm transcript
command UCM produces a deterministic output file containing the result
of processing each code block. Unless the behaviour of UCM has changed
since the last time it was run the resulting file will always be the
same.
Each block in the markdown file is either a command, which is sent to
the UCM shell tool, or it represents an update to a file on the
(virtual) file-system, in which case it will be typechecked against the
state of the codebase.
Here's a quick example of a transcript for testing UCM's view command
so you can get a feel for it.
# Testing the `view` command
First, let's write a simple definition to view:
``` unison
isZero = cases
0 -> true
_ -> false
```
Now we add the definition to the codebase, and view it.
``` ucm
scratch/main> update
scratch/main> view isZero
```
We run this transcript file with
ucm transcript my-transcript.md which produces the
my-transcript.output.md file.
Notice how compiler output is added inline, ignore the hashed names,
It's because I'm skipping the step which adds names for Unison's
builtins.
# Testing the `view` command
First, let's write a simple definition to view:
``` unison
isZero = cases
0 -> true
_ -> false
```
``` ucm :added-by-ucm
Loading changes detected in scratch.u.
I found and typechecked these definitions in scratch.u. If you
do an `add` or `update`, here's how your codebase would
change:
� These new definitions are ok to `add`:
isZero : ##Nat -> ##Boolean
```
Now we add the definition to the codebase, and view it.
``` ucm
scratch/main> update
Done.
scratch/main> view isZero
isZero : ##Nat -> ##Boolean
isZero = cases
0 -> true
_ -> false
```
Feel free to browse through the collection
of transcripts we test in CI to keep UCM working as expected.
Testing in CI
Running transcript tests in CI is pretty trivial; we discover all
markdown files within our transcript directory and run them all. After
the outputs have been written we can use
git diff --exit-code which will then fail with a non-zero
code if anything of the outputs have changed from what was committed.
Conveniently, git will also report exactly what changed, and
what the old output was.
This failure method allows the developer to know exactly which file
has unexpected behaviour so they can easily re-run that file or recreate
the state in their own codebase if they desire.
Transcript tests in other
domains
I liked the transcript tests in UCM so much that when I was tasked
with building out the Unison Share webapp I decided to use
transcript-style testing for that too. Fast forward a few years and
Unison Share is now a fully-featured package repository and code
collaboration platform running in production without a
single unit test.
If you're interested in how I've adapted transcript tests to work
well for a webapp, I'll leave a few notes at the end of the post.
Benefits of transcript tests
Here's a shortlist of benefits I've found working with transcript
tests over alternatives like unit tests.
You write a transcript using the same syntax as you'd
interact with UCM itself.
This allows all your users to codify any buggy behaviour they've
encountered into a deterministic transcript. Knowing exactly how to
reproduce the behaviour your users are seeing is a huge boon, and having
a single standardized format for accepting bug reports helps reduce a
lot of the mental work that usually goes into reproducing bug reports
from a variety of sources. This also means that the bug report itself
can go directly into the test suite if we so desire.
All tests are written against the tool's external
interface.
The tests use the same interface that the users of your software will
employ, which means that internal refactors won't ever break
tests unless there's a change in behaviour that's externally
observable.
This has been a huge benefit for me personally. I'd often find myself
hesitant to re-work code because I knew that at the end I'd be rewriting
thousands of lines of tests. If you always have to rewrite your tests at
the same time you've rewritten your code, how do you have any confidence
that the tests still work as intended?
Updating tests is trivial
In the common case where transcripts are mismatched because some help
message was altered, or perhaps the behaviour has changed but the change
is intended, you don't need to rewrite any complex assertions, or mock
out any new dependencies. You can simply look at the new output, and if
it's reasonable you commit the changed transcript output files.
It can't be understated how convenient this is when making sweeping
changes; e.g. making changes to Unison's pretty printer. We don't need
to manually update test-cases, we just run the transcripts locally and
commit the output if it all looks good!
Transcript changes appear in PR reviews
Since all transcript outputs are committed, any change in behaviour
will show up in the PR diff in an easy-to-read form. This allows
reviewers to trivially see the old and new behaviour for each relevant
feature.
Transcript tests are documentation
Each transcript shows how a feature is intended to be used by
end-users.
Transcripts as a collaboration tool
When I'm implementing new features in Unison Share I need to
communicate the shape of a JSON API with our Frontend designer Simon.
Typically I'll just write a transcript test which exercises all possible
variants of the new feature, then I can just point at the transcript
output as the interface for those APIs.
It's beneficial for both of us since I don't need to keep an example
up-to-date for him, and he knows that the output is actually accurate
since it's generated from an execution of the service itself.
Transcript testing for
Webapps
I've adapted transcript testing a bit for the Unison Share webapp. I
run the standard Share executable locally with its dependencies mocked
out via docker-compose. I've got a SQL file which resets the database
with a known set of test fixtures, then use a zsh script to reset my
application state in between running each transcript.
Each transcript file is just a zsh script that interacts with the
running server using a few bash functions which wrap curl commands, but
save the output to json files, which serve as the transcript output.
I've also got helpers for capturing specific fields from an API call
into local variables which I can then interpolate into future queries,
this is handy if you need to, for example, create a project then switch
it from private to public, then fetch that project via API.
Here's a small snippet from one of my transcripts for testing Unison
Share's project APIs:
#!/usr/bin/env zsh# Fail the transcript if any command failsset-e# Load utility functions and variables for user credentialssource"../../transcript_helpers.sh"# Run a UCM transcript to upload some code to load in projects.transcript_ucm transcript prelude.md# I should be able to see the fixture project as an unauthenticated user.fetch"$unauthenticated_user" GET project-get-simple '/users/test/projects/publictestproject'# I should be able to create a new project as an authenticated user.fetch"$transcripts_user" POST project-create '/users/transcripts/projects/containers''{ "summary": "This is my project", "visibility": "private", "tags": []}'fetch"$transcripts_user" GET project-list '/users/transcripts/projects'
You can see the output files generated by the full transcript in
this directory.
Requirements of
a good transcript testing tool
After working with two different transcript testing tools across two
different apps I've got a few criteria for what makes a good transcript
testing tool, if you're thinking of adding transcript tests to your app
consider the following:
Transcripts should be deterministic
This is critical. Transcripts are only useful if they produce the
same result on every run, on every operating system, at every time of
day.
You may need to make a few changes in your app to adapt or remove
randomness, at least when in the context of a transcript test.
In Share there were a lot of timestamps, random IDs, and JWTs (which
contain a timestamp). The actual values of these weren't important for
the tests themselves, so I solved the issue by piping the curl output
through a sed script before writing to disk. The script
matches timestamps, UUIDs, and JWTs and replaces them with placeholders
like <TIMESTAMP>, <UUID>, and
<JWT> accordingly.
A special mode in your app for transcript testing which avoids
randomness can be useful, but use custom modes sparingly lest your app's
behaviour differ too much during transcripts and you can't test the real
thing.
I also make sure that the data returned by APIs is always sorted by
something other than randomized IDs, it's a small price to pay, and
reduces randomness and heisenbugs in the app as a helpful byproduct.
Transcripts should be isolated
Each individual transcript should be run in its own pristine
environment. Databases should be reset to known state, if the
file-system is used, it should be cleared or even better, a virtual
file-system should be used.
Transcripts should be self-contained
Everything that pertains to a given test-case's state or
configuration should be evident from within the transcript file itself.
I've found that changes in behaviour from the file's location or name
can just end up being confusing.
Difficulties working with
Transcripts
Transcripts often require custom tooling
In UCM's case the transcript tooling has evolved slowly over many
years, it has it's own parser, and you can even test UCM's API server by
using special code blocks for that.
Share has a variety of zsh utility scripts which provide
helpers for fetching endpoints using curl, and filtering output to
capture data for future calls. It also has a few tools for making
database calls and assertions.
Don't shy away from investing a bit of time into making transcript
testing sustainable and pleasant, it will pay dividends down the
road.
Intensive Setup**
As opposed to unit tests which are generally pretty lightweight;
transcript tests are full integration tests, and require setting up
data, and sometimes executing entire flows so that we can get the system
into a good state for testing each feature.
You can mitigate the setup time by testing multiple features with
each transcript.
I haven't personally found transcript tests to take too much time in
CI, largely because I think transcript testing tends to produce fewer
tests, but of higher value than unit testing. I've seen many unit test
suites bogged down by particular unit tests which generate hundreds of
test cases that aren't actually providing real value. Also, any
setup/teardown is going to be more costly on thousands of unit-tests as
compared to dozens or hundreds of transcript tests.
Service Mocking
Since transcript tests run against the system-under-test's external
interface, you won't have traditional mocking/stubbing frameworks
available to you. Instead, you'll mock out the system's dependencies by
specifying custom services using environment variables, or wiring things
up in docker-compose.
Most systems have a setup for local development anyways, so
integrating transcript tests against it has the added benefit that
they'll ensure your local development setup is tested in CI, is
consistent for all members of your team, and continues to work as
expected.
In Summary
Hopefully this post has helped you to consider your relationship with
unit tests and perhaps think about whether other testing techniques may
work better for your app.
Transcript tests surely aren't ideal for all
possible apps or teams, but my last few years at Unison have proven to
me that tests can be more helpful, efficient, and readable than I'd
previously thought possible.
Let me know how it works out for you!
Hopefully you learned something �! Did you know I'm currently writing a book? It's all about Lenses and Optics! It takes you all the way from beginner to optics-wizard and it's currently in early access! Consider supporting it, and more posts like this one by pledging on my Patreon page! It takes quite a bit of work to put
these things together, if I managed to teach your something or even just entertain you for a minute or two
maybe send a few bucks my way for a coffee? Cheers! �
Andy Gordon from Cogna is interviewed by Sam and Matti. We learn about Andy’s influential work including the origins of the bind symbol in haskell, and the introduction of lambdas in Excel. We go onto discuss his current work at Cogna on using AI to allow non-programmers to write apps using natural language. We delve deeper into the ethics of AI and consider the most likely AI apocalypse.
As software engineers at Modus Create,
we are always on the lookout for tools that can enhance our productivity and code quality.
The advent of AI-powered coding assistants such as GitHub Copilot
has sparked excitement in the development community.
Copilot code completions propose snippets at the current cursor that the user can quickly insert,
while Copilot Chat allows users to discuss their code with an AI.
These tools promise to revolutionize software development,
allowing engineers to focus on higher-level tasks while delegating implementation details to machines.
However, their adoption also raises questions:
Do they genuinely improve developer productivity?
How do they affect code quality and maintainability?
Which users and tasks benefit the most from these AI-driven coding assistants?
This blog post explores the challenges of measuring the impact of AI tools in
our software engineering practices, with a focus on GitHub Copilot. Note that
the data discussed in the post was collected in Q2 2024. We expect that GitHub
Copilot has improved since then; we have also not yet had the opportunity to
quantitatively investigate newer interfaces to AI development, like
Cursor or Windsurf.
“Developer Productivity”
At Modus Create, we’re passionate about improving the experience of developers,
both for our own teams and those at clients.
We have been working for years on tools that we think improve developer productivity,
for instance with
Nix,
Bazel,
Python,
and many more.
But measuring developer productivity is a notoriously difficult task.
At the heart of this question lies the nature of software development itself.
Is it a productive activity that can fit scientific management,
be objectively measured, and be optimized?
Part of the research on developer productivity goes down this path, trying to measure things like the time it takes to complete standardized tasks.
Another trend suggests that developers themselves can be their own assessors of productivity, where
frameworks like
SPACE are used to guide self-assessment.
Each of these angles has strengths and weaknesses. To get as broad a picture as possible, we tried to use
a bit of both. We found, though, that data collection issues made our task timings unusable (more on this below). Therefore,
all our conclusions are drawn from self-assessments.
Our in-house experiment
To gain a deeper understanding of the impact of GitHub Copilot at Modus Create,
we designed and conducted an in-house experiment.
We managed to recruit 22 participants in total, ranging from Junior to Principal software engineers.
They had a wide range of programming experience.
The experiment consisted of four coding tasks that participants
needed to complete using Python within an existing
codebase. The tasks were designed
to evaluate different aspects of software development:
Data ingestion: Loading and parsing data from a file into a Pandas DataFrame
Data analysis: Performing statistical computations and aggregations using Pandas’ groupby operations
Test development: Writing tests using Python’s unittest framework
Data visualization: Creating interactive plots using the Streamlit library
Participants had varied levels of experience with the required tools. Most participants
had at least a little bit of Python experience, but Pandas experience was less common
and hardly anyone had used Streamlit before.1
Upon completion of the assigned tasks, all participants completed a comprehensive survey to provide
detailed feedback on their experience.
The survey consisted of approximately 50 questions designed to assess multiple dimensions of the
development process, including:
Assessment of participant expertise levels regarding task requirements, AI tooling and GitHub Copilot proficiency
Evaluation of task-specific perceived productivity
Analysis of the impact on learning and knowledge acquisition
Insights into potential future GitHub Copilot adoption
Perceived productivity gains
We asked participants the following questions.
Question
Choices
If you didn't have Copilot, reaching the answer for task X would have taken...
Less time
About the same time
More time
This question was core to our study,
as it allowed us to directly measure the perceived productivity gain of using Copilot versus not using it.
The result was clear:
almost every Copilot user felt more productive using Copilot on every task.
We also broke out the same data by Python experience level, and found that more experienced Python
users found less productivity gain than less experienced users. In this plot, we grouped the “no Python
experience” and “beginner” users into the “less experienced” group, with the rest of the users in the “more experienced group”.
To better understand how participants tackled these tasks, we collected information by asking for each task:
Question
Choices
Which of the following have you used to complete task X?
Copilot code completions
Copilot Chat
Google search
Library documentation
My knowledge
We were also interested in comparing these usages across profiles of developers,
so we asked this question as well:
Question
Choices
How would you describe your Python level?
No Python experience
Beginner
Intermediate
Advanced
We could then visualize how participants who felt more productive with Copilot solved each problem,
and see if there were variations depending on their profile. Since each participant could choose multiple
options, sometimes there are more responses than participants.
Apparently, people don’t like library documentation. Also, we thought it was
strange that the most experienced Python users never reported using their own
knowledge. It would be interesting to dig more into this, but we don’t have
the data available. One theory is that when reviewing AI suggestions everyone
relied on their own Python knowledge, but experienced users took that knowledge
for granted and so didn’t report using it.
Among people who felt more productive on tasks “Write unit tests” and “Plot with Streamlit”,
we really see more usage of Copilot Chat than other sources.
Our hypothesis is that these tasks typically require making more global changes to the code or adding code in places that are not obvious at first.
In these scenarios, Copilot Chat is more useful because it will tell you where and what code to add.
In other tasks, it was clearer where to add code, so participants could likely place their cursor and prompt Copilot for a suggestion.
This is supported by the questions we asked:
Question
Choices
Which of the following do you think is true?
Copilot is better with acceleration: it helps accelerate work that I already know how to do.
Copilot is better with exploration: it helps me explore the problem and how to solve it when I am not sure how to proceed.
This question uses checkboxes, so respondents were not restricted to a single answer.
On average, participants thought Copilot was suited for both acceleration and exploration, but with some notable
differences depending on experience level: experienced Pythonistas strongly favored Copilot for acceleration,
while less experienced users thought it was better for exploration.
We also found that the participants’ perspective on acceleration versus exploration seems related to the usage of Copilot Chat.
The most interesting part of this chart is that
participants who think Copilot is good for exploration or bad for acceleration relied most heavily on Copilot Chat.
This suggests that users find the autocomplete features more useful for acceleration, while the chat features —
which allow general questions, divorced from a specific code location — are useful for exploration.
But it is interesting to note how
usage of Copilot Chat versus autocomplete is correlated with how users perceive Copilot as a whole.
For more on acceleration versus exploration with Copilot, this OOPSLA23 talk which inspired to ask this question
is worth watching.
Copilot will make code flow
The SPACE framework mentions “flow”
as an important aspect of productivity.
Some research associates productivity with the ability to get complex tasks done with minimal distractions or interruptions.
This conceptualization of productivity is echoed by many developers when they talk about “getting into the flow” […].
This concept of flow is really interesting,
because it is a way to measure productivity that is not based on outputs,
but rather on the experience of the developers themselves. And although
“flow” might be subjective and perceptual,
studies have linked it to higher productivity and reduced stress; see
this open-access book chapter for a readable overview of the research.
To get an idea of Copilot’s impact on flow, we asked the following questions:
Question
Choices
Did Copilot decrease your need to switch out of your IDE (for example to search for answers or check the documentation)?
Significantly
A bit
No
Did Copilot enhance your capacity to stay in your development flow?
Significantly
A bit
No
The results were unambiguous: most users found that Copilot helped significantly, and a strong majority
found that it helped at least a little.
Learnings from organizing the experiment
Although the experiment went well overall, we noted a few challenges worth sharing.
First, ensuring active participation in the experiment required a collective effort within the company.
Spreading the word internally about the experiment and looking for participants is an effort not to be underestimated.
In our case, we benefited from great support from internal leaders and managers who helped communicate with and recruit participants.
Even so, we would have liked to have more participants. It turns out that engineers are sometimes just too busy!
Second, keeping participants focused on the experiment was harder than expected.
We had asked participants to make a git commit at the end of each task,
thinking that we could use this data to quantify the time it took for each participant to complete their tasks.
When looking at the data, we were surprised to see that the time between commits varied widely and was often much longer than expected.
When asked, several participants reported that they had to interrupt our experiment to deal with higher-priority tasks.
In the end, we discarded the timing data: they were too limited and too heavily influenced by external factors to provide useful conclusions.
For the same reason, we haven’t even mentioned yet that our study had a control group: since the timing data
wasn’t useful, we’ve omitted the control group entirely from the data presented here.
The ideal scenario of securing dedicated, uninterrupted time from a large pool
of engineers proved impractical within our organizational context. Nevertheless,
despite these limitations, we successfully gathered a meaningful dataset
that contributes valuable perspectives to the existing body of research on
AI-assisted development.
Further references
Speaking of other work out there, there’s a lot of it! It turns out that many people are excited by the potential
of code assistants and want to understand them better. Who knew? Here is some further reading that we found
particularly interesting:
Experiments at Microsoft and Accenture introduced
Copilot into engineers’ day-to-day workflow and measured the impact on various productivity metrics,
like the number of opened pull requests; they found that Copilot usage significantly increased the
number of successful builds. They had a much larger sample size than we did — Microsoft
and Accenture have a lot of engineers — but unlike us they didn’t specifically consider
the uptake of unfamiliar tools and libraries.
A research team from Microsoft and MIT recruited developers from Upwork, gave them a task,
and measured the time it took with and without Copilot’s help; they
found that Copilot users were about 50% faster. They
did a better job than we did at measuring completion time (they used GitHub Classroom), but we think
our exit survey asked more interesting questions.
The Pragmatic Engineer ran a survey
about how engineers are using AI tooling, covering popular tools and their perceived impact
on development.
Conclusion
Our experiment provided valuable insights into the impact of GitHub Copilot on
developer experiences at Modus Create. Overall, developers reported increased
productivity and a more seamless workflow. Participants used Copilot extensively
in specific coding scenarios, such as automated testing and modifying code that
used libraries they were unfamiliar with, and they felt more productive in those cases.
It was particularly interesting to see how the interface to the AI assistant
(chat vs. completion) affected participants’ opinions on what the assistant was useful
for, with chat-heavy users prioritizing exploration over acceleration and completion-heavy
users the other way around. As interfaces and tooling continue to evolve — faster than we can design
and run experiments to test them — we expect them to play a huge
role in the success of AI-powered code assistants.
We made a small mistake with the wording in Pandas and Streamlit questions:
we gave them the options “I have never used it”, “I have heard of it”, “I
have used it before in a limited way”, “I am comfortable with it”, and “I am
an advanced user”. The problem, of course, is that these responses
aren’t mutually exclusive. Given the order the responses were presented in, we
think it’s reasonable to interpret “I have never used it” responses to mean that they’d
heard of it but never used it. For the plot, we’ve combined “I have never used it”
and “I have heard of it” into “Never used it”.↩
How to turn polling insight into an optimal ballot — and why anything else is wasted.
“approve of�? What does that mean anyway?
I have written previously about how approval and range voting methods are intrinsically tactical. This doesn’t mean that they are more tactical than other election systems (nearly all of which are shown to sometimes be tactical by Gibbard’s Theorem when there are three or more options). Rather, it means that tactical voting is unavoidable. Voting in such a system requires answering the question of where to set your approval threshold or how to map your preferences to a ranged voting scale. These questions don’t have more or less “honest� answers. They are always tactical choices.
But I haven’t dug deeper into what these tactics look like. Here, I’ll do the mathematical analysis to show what effective voting looks like in these systems, and make some surprising observations along the way.
Mathematical formalism for approval voting
We’ll start by assuming an approval election, so the question is where to put your threshold. At what level of approval do you switch from voting not to approve a candidate to approving them?
We’ll keep the notation minimal:
As is standard in probability, I’ll write ℙ[X] for the probability of an event X, and �[X] for the expected value of a (numerical) random variable X.
I will use B to refer to a random collection (multiset) of ballots, drawn from some probability distribution reflecting what we know from polling and other information sources on other voters. B will usually not include the approval vote that you’re considering casting, and to include that approval, we’ll write B ∪ {c}, where c is the candidate you contemplate approving.
I’ll write W(·) to indicate the winner of an election with a given set of ballots. This is the candidate with the most approvals. We’ll assume some tiebreaker is in place that’s independent of individual voting decisions; for instance, candidates could be shuffled into a random order before votes are cast, in in the event of a tie for number of approvals, we’ll pick the candidate who comes first in that shuffled order.
U(·) will be your utility function, so U(c) is the utility (i.e., happiness, satisfaction, or perceived social welfare) that you personally will get from candidate c winning the election. This doesn’t mean you have to be selfish, per se, as accomplishing some altruistic goal is still a form of utility, but we evaluate that utility from your point of view even though other voters may disagree.
With this notation established, we can clearly state, almost tautologically, when you should approve of a candidate c. You should approve of c whenever:
�[U(W(B ∪ {c}))] > �[U(W(B))]
That’s just saying you should approve of c if your expected utility from the election with your approval of c is more than your utility without it.
The role of pivotal votes and exact strategy
This inequality can be made more useful by isolating the circumstances in which your vote makes a difference in the outcome. That is, W(B ∪ {c}) ≠W(B). Non-pivotal votes contribute zero to the net expectation, and can be ignored.
In approval voting, approving a candidate can only change the outcome by making that candidate the winner. This means a pivotal vote is equivalent to both of:
W(B ∪ {c}) = c
W(B) ≠ c
It’s useful to have notation for this, so we’ll define V(B, c) to mean that W(B ∪ {c}) ≠W(B), or equivalently, that W(B ∪ {c}) = c and W(B) ≠c. To remember this notation, recall that V is the pivotal letter in the word “pivot�, and also visually resembles a pivot.
With this in mind, the expected gain in utility from approving c is:
�[U(W(B ∪ {c}))] - �[U(W(B))]. But since the utility gain is zero except for pivotal votes, this is the same as
ℙ[V(B,c)] · (�[U(W(B ∪ {c})) | V(B,c)] - �[U(W(B)) | V(B,c)]). But since V(B,c) implies that W(B ∪ {c}) = c, so this simplifies to
ℙ[V(B,c)] · (U(c) - �[U(W(B)) | V(B, c)])
Therefore, you ought to approve of a candidate c whenever
U(c) > �[U(W(B)) | V(B, c)]
This is much easier to interpret. You should approve of a candidate c precisely when the utility you obtain from c winning is greater than the expected utility in cases where c is right on the verge of winning (but someone else wins instead).
There are a few observations worth making about this:
The expectation clarifies why the threshold setting part of approval voting is intrinsically tactical. It involves evaluating how likely each other candidate is to win, and using that information to compute an expectation. That means advice to vote only based on internal feelings like whether you consider a candidate acceptable is always wrong. An effective vote takes into account external information about how others are likely to vote, including polling and understanding of public opinion and mood.
The conditional expectation, assuming V(B, c), tells us that the optimal strategy for whether to approve of some candidate c depends on the very specific situation where c is right on the verge of winning the election. If c is a frontrunner in the election, this scenario isn’t likely to be too different from the general case, and the conditional probability doesn’t change much. However, if c is a long-shot candidate from some minor party, but somehow nearly ties for a win, we’re in a strange situation indeed: perhaps a major last-minute scandal, a drastic polling error, or a fundamental misunderstanding of the public mood. Here, the conditonal expected utility of an alternate winner might be quite different from your unconditional expectation. If, say, voters prove to have an unexpected appetite for extremism, this can affect the runner-ups, as well.
Counter-intuitively, an optimal strategy might even involve approving some candidates that you like less than some that you don’t approve! This can happen because different candidates are evaluated against different thresholds. Therefore, a single voter’s best approval ballot isn’t necessarily monotonic in their utility rankings. This adds a level of strategic complexity I hadn’t anticipated in my earlier writings on strategy in approval voting.
Approximate strategy
The strategy described above is rigorously optimal, but not at all easy to apply. Imagining the bizarre scenarios in which each candidate, no matter how minor, might tie for a win, is challenging to do well. We’re fortunate, then, that there’s a good approximation. Remember that the utility gain from approving a candidate was equal to
ℙ[V(B,c)] · (U(c) - �[U(W(B)) | V(B, c)])
In precisely the cases where V(B, c) is a bizarre assumption that’s difficult to imagine, we’re also multiplying by ℙ[V(B,c)], which is vanishingly small, so this vote is very unlikely to make a difference in the outcome. For front-runners, who are relatively much more likely to be in a tie for the win, the conditional probability changes a lot less: scenarios that end in a near-tie are not too different from the baseline expectation.
This happens because ℙ[V(B,c)] falls off quite quickly indeed as the popularity of c decreases, especially for large numbers of voters. For a national scale election (say, about 10 million voters), if c expects around 45% of approvals, then ℙ[V(B,c)] is around one in a million. That’s a small number, telling us that very large elections aren’t likely to be decided by a one-vote margin anyway. But it’s gargantuan compared to the number if c expects only 5% of approvals. Then ℙ[V(B,c)] is around one in 10^70. That’s about one in a quadrillion-vigintillion, if you want to know, and near the scale of possibly picking one atom at random from the entire universe! The probability of casting a pivotal vote drops off exponentially, and by this point it’s effectively zero.
With that in mind, we can drop the condition on the probability in the second term, giving us a new rule: Approve of a candidate c any time that:
U(c) > �[U(W(B))]
That is, approve of any candidate whose win you would like better than you expect to like the outcome of the election. In other words, imagine you have no other information on election night, and hear that this candidate has won. If this would be good news, approve of the candidate on your ballot. If it would be bad news, don’t.
This rule is still tactical. To determine how much you expect to like the outcome of the election, you need to have beliefs about who else is likely to win, which still requires an understanding of polling and public opinion and mood.
However, there is one threshold, derived from real polling data in realistic scenarios, and you can cast your approval ballot monotonically based on that single threshold.
This is no longer a true optimal strategy, but with enough voters, the exponential falloff in ℙ[V(B,c)] as c becomes less popular is a pretty good assurance that the incorrect votes you might cast by using this strategy instead of the optimal ones are extremely unlikely to matter. In practice, this is probably the best rule to communicate to voters in an approval election with moderate to large numbers of voters.
We can get closer with the following hypothetical: Imagine that on election night, you have no information on the results except for a headline that proclaims: Election Too Close To Call. With that as your prior, you ask of each candidate, is it good or bad news to hear now that this candidate has won. If it would be good news, then you approve of them. This still leaves one threshold, but we’re no longer making the leap that the pivotal condition for front-runners is unnecessary; we’re imagining a world in which at least some candidates, almost surely the front-runners, are tied. If this changes your decision (which it likely would only in very marginal cases), you can use this more accurate approximation.
Reducing range to approval voting
I promised to look at strategy for range voting, as well. Armed with an appreciation of approval strategy, it’s easy to extend this to an optimal range strategy, as well, for large-scale elections.
The key is to recognize that a range voting election with options 0, 1, 2, …, n is mathematically equivalent to an approval election where everyone is just allowed to vote n times. The number you mark on the range ballot can be interpreted as saying how many of your approval ballots you want to mark as approving that candidate.
Looking at it this way presents the obvious question: why would you vote differently on some ballots than others? In what situation could that possibly be the right choice?
For small elections, say if you’re voting on places to go out and eat with your friends or coworkers, it’s possible that adding in a handful of approvals materially changes the election so that the optimal vote is different. Then it may well be optimal to cast a range ballot using some intermediate number.
For large elections, though, you’re presented with pretty much exactly the same question each time, and you may as well give the same answer. Therefore, in large-scale elections, the optimal way to vote with a range ballot is always to rate everyone either the minimum or maximum possible score. This reduces a range election exactly to an approval election. The additional expressiveness of a range ballot is a siren call: by using it, you always vote less effectively than you would have by ignoring it and using only the two extreme choices.
Since we’re discussing political elections, which have relatively large numbers of voters, this answers the question for range elections, as well: Rate a candidate the maximum score if you like them better than you expect to like the outcome of the election. Otherwise, rate them the minimum score.
Summing it up
What we’ve learned, then, is that optimal voting in approval or range systems boils down to two nested rules.
Exact rule (for the mathematically fearless): approve c iff U(c) > �[ U(W(B)) | your extra vote for c is pivotal ]. This Bayesian test weighs each candidate against the expected utility in the razor-thin worlds where they tie for first.
Large-electorate shortcut (for everyone else): because those pivotal worlds become astronomically rare as the field grows, the condition shrinks to a single cutoff: approve (or give a maximum score) to every candidate whose victory you expect to enjoy more than you expected to like the result. (If you can, imagine only cases where you know the election is close.)
We’ve seen why the first rule is the gold standard; but the second captures virtually all of its benefit when millions are voting. Either way, strategy is inseparable from sincerity: you must translate beliefs about polling into a utility threshold, and then measure every candidate against it. We’ve also seen by a clear mathematical equivalence why range ballots add no real leverage in large-scale elections, instead only offering false choices that are always wrong.
The entire playbook fits on a sticky note: compute the threshold, vote all-or-nothing, and let the math do the rest.
Shows are under the banner of The Provocateurs (formerly Cabaret of Dangerous Ideas). Tickets go on sale Wednesday 7 May, around noon. The official blurb is brief:
Professor Philip Wadler (The University of Edinburgh) separates the hopes and threats of AI from the chatbot bullshit.
Welcome to the second article in our Rust vs. Haskell problem solving series. Last week we saw some basic differences between Rust loops and Haskell recursion. We also saw how to use the concept of “folding” to simplify a recursive loop function.
This week, we’ll look at another simple problem and consider multiple solutions in each language. We’ll consider what a “basic” solution looks like, using relatively few library functions. Then we’ll consider more “advanced” solutions that make use of library functionality, and greatly simplify the structure of our solutions.
To learn more about problem solving in Haskell, including the importance of list library functions, take a look at our course Solve.hs! You’ll write most of Haskell’s list API from scratch so you get an in-depth understanding of the functions that are available!
The Problem
This week’s problem is Reverse Words in a String. The idea is simple. Our input is a string, which naturally has “words” separated by whitespace. We want to return a string that has all the words reversed! So if the input is ”A quick brown fox”, the result should be ”fox brown quick A”.
Notice that all whitespace is truncated in our output. We should only have a single space between words in our answer, with no leading or trailing whitespace.
The Algorithm
The algorithmic idea is simple and hardly needs explanation. We want to gather letters from the input word until we encounter whitespace. Then we append this buffered word to a growing result string, and keep following this process until we run out of input.
There is one wrinkle, which is whether we want to accumulate our answer in the forward or reverse direction. This changes across languages!
In Haskell, it’s actually more efficient to accumulate the “back” of our resulting string first, meaning we should start by iterating from the front of the input. This is more consistent with linked list construction.
In Rust, we’ll iterate from the back of the input so that we can accumulate our result from the “front”.
Basic Rust Solution
In our basic solution, we’re going to consider a character-by-character approach. As outlined in our algorithm, we can accomplish this task with a single loop, with two stateful values. First, we have the “current” word we’re accumulating of non-whitespace characters. Second, we have the final “result” we’re accumulating.
It’s efficient to append to the end of strings, meaning we want to construct our result from front-to-back. This means we’ll loop through the characters of our string in reverse, as shown with .rev() here:
pub fn reverse_words(s: String) -> String {
let mut current = String::new();
let mut result = String::new();
for c in s.chars().rev() {
...
}
}
Within the loop, we now just have to consider what to do with each character. If the character is not whitespace, the answer is simple. We just append this character to our “current” word. Because we’re looping through the input in reverse, our “current” word will also be in reverse!
pub fn reverse_words(s: String) -> String {
let mut current = String::new();
let mut result = String::new();
for c in s.chars().rev() {
if !c.is_whitespace() {
current.push(c);
} else {
...
}
}
}
So what happens when we encounter whitespace? There’s a few conditions to consider:
If “current” is empty, do nothing.
If “result” is empty, append “current” (in reverse order) to result.
If “result” is not empty, add a space and then append “current” in reverse.
Regardless, clear “current” and prepare to gather a new string.
Here’s what the code looks like:
pub fn reverse_words(s: String) -> String {
let mut current = String::new();
let mut result = String::new();
for c in s.chars().rev() {
if !c.is_whitespace() {
current.push(c);
} else {
// Step 1: Skip if empty
if !current.is_empty() {
// Step 2/3 Only push an empty space is result is not empty
if !result.is_empty() {
result.push(' ');
}
// Step 2/3 Reverse current and append
for b in current.chars().rev() {
result.push(b);
}
// Step 4: Clear “current”
current.clear();
}
}
}
}
There’s one final trick. Unless the word begins with whitespace, we’ll still have non-empty current at the end and we will not have appended it. So we do one final check, and once again append “current” in reverse order.
Here’s our final basic solution:
pub fn reverse_words(s: String) -> String {
let mut current = String::new();
let mut result = String::new();
for c in s.chars().rev() {
if !c.is_whitespace() {
current.push(c);
} else {
// Step 1: Skip if empty
if !current.is_empty() {
// Step 2/3 Only push an empty space is result is not empty
if !result.is_empty() {
result.push(' ');
}
// Step 2/3 Reverse current and append
for b in current.chars().rev() {
result.push(b);
}
// Step 4: Clear “current”
current.clear();
}
}
}
if !current.is_empty() {
if !result.is_empty() {
result.push(' ');
}
for b in current.chars().rev() {
result.push(b);
}
}
return result;
}
Advanced Rust Solution
Looping character-by-character is a bit cumbersome. However, since basic whitespace related operations are so common, there are some useful library functions for dealing with them.
Rust also prioritizes the ability to chain iterative operations together. This gives us the following one-line solution!
Join them together with one space in between them.
What is interesting about this structure is that each stage of the process has a separate type. Step 1 creates a SplitWhitespace struct. Step 2 creates a Reverse struct. Step 3 then creates a normal vector, and step 4 concludes by producing a string.
The two preliminary structures are essentially wrappers with iterators to help chain the operations together. As we’ll see, the comparable Haskell solution only uses basic lists, and this is a noteworthy difference between the languages.
Basic Haskell Solution
Our “basic” Haskell solution will follow the same outline as the basic Rust solution, but we’ll work in the opposite direction! We’ll loop through the input in forward order, and accumulate our output in reverse order.
Before we even get started though, we can make an observation from our basic Rust solution that we duplicated some code! The concept of combining the “current” word and the “result” had several edge cases to handle, so let’s write a combine function to handle these.
-- “current” is reversed and then goes in *front* of result
-- (Rust version put “current” at the back)
combine :: (String, String) -> String
combine (current, res) = if null current then res
else reverse current <> if null res then "" else (' ' : res)
Now let’s think about our loop structure. We are going through the input, character-by-character. This means we should be able to use a fold, like we did last week! Whenever we’re using a fold, we want to think about the “state” we’re passing through each iteration. In our case, the state is the “current” word and the “result” string. This means our folding function should look like this:
Now we just have to distinguish between the “whitespace” case and the non-whitespace case. If we encounter a space, we just combine the current word with the accumulated result. If we encounter a normal character, we append this to our current word (again, accumulating “current” in reverse).
loop :: (String, String) -> Char -> (String, String)
loop (currentWord, result) c = if isSpace c
then ("", combine (currentWord, result))
else (c : currentWord, result)
Now to complete the solution, we just call ‘foldl’ with our ‘loop’ and the input, and we just have to remember to combine the final “current” word with the output! Here’s our complete “basic” solution.
reverseWords :: String -> String
reverseWords input = combine $ foldl loop ("", "") input
where
combine :: (String, String) -> String
combine (current, res) = if null current then res
else reverse current <> if null res then "" else (' ' : res)
loop :: (String, String) -> Char -> (String, String)
loop (currentWord, result) c = if isSpace c
then ("", combine (currentWord, result))
else (c : currentWord, result)
Advanced Haskell Solutions
Now that we’ve seen a basic, character-by-character solution in Haskell, we can also consider more advanced solutions that incorporate library functions. The first improvement we can make is to lean on list functions like break and dropWhile.
Using break splits off the first part of a list that does not satisfy a predicate. We’ll use this to gather non-space characters. Then dropWhile allows us to drop the first series of characters in a list that satisfy a predicate. We’ll use this to get rid of whitespace as we move along!
So we’ll define this solution using a basic recursive loop rather than a fold, because each iteration will consume a variable number of characters. The “state” of this loop will be two strings: the remaining part of the input, and the accumulated result.
Since there’s no “current” word, our base case is easy. If the remaining input is empty, we return the accumulated result.
Combine this word with the output (if it’s not null)
Recurse with the new output, dropping the initial whitespace from the remainder.
Here’s what it looks like:
loop :: (String, String) -> String
loop ([], output) = output
loop (cs, output) =
-- Step 1: Separate next word from rest
let (nextWord, rest) = L.break isSpace cs
-- Step 2: Make new output (account for edge cases)
-- (Can’t use ‘combine’ from above because we aren’t reversing!)
newOutput = if null output then nextWord
else if null nextWord then output
else nextWord <> (' ' : output)
-- Drop spaces from remainder and recurse
in loop (L.dropWhile isSpace rest, newOutput)
And completing the function is as simple as calling this loop with the base inputs:
The final (and recommended) Haskell solution uses the library functions words and unwords. These do exactly what we want for this problem! We separate words based on whitespace using words, and then join them with a single space with unwords. All we have to do in between is reverse.
This has a similar elegance to the advanced Rust solution, but is much simpler to understand since there are no complex structs or iterators involved. The types of all functions involved simply relate to lists. Here are the signatures, specialized to String for this problem.
A simple problem will often have many solutions, but in this case, each of these solutions teaches us something new about the language we’re working with. Working character-by-character helps us understand some of the core mechanics of the language, showing us how it works under the hood. But using library functions helps us see the breadth of available options we have for simplifying future code we write.
In our Solve.hs course, you’ll go through all of these steps with Haskell. You’ll implement list library functions, data structures, and algorithms from scratch so you understand how they work under the hood. Then, you’ll know they exist and be able to apply them to efficiently solve harder problems. Take a look at the course today!
A quincunx is the X-shaped pattern of pips on the #5 face of a die.
It's so-called because the Romans had a common copper coin called an
as, and it was divided (monetarily, not physically) into twelve
uncia. There was a bronze coin worth five uncia called a quīncunx, which
is a contraction of quīnque (“five”) + uncia, and the coin had
that pattern of dots on it to indicate its value.
Uncia generally meant a twelfth of something. It was not just a
twelfth of an as, but also a twelfth of a pound , which is where we
get the word “ounce”, and a twelfth of a foot, which is where we get
the word “inch”.
The story I always heard about the connection between the coin and the
X-shaped pattern of dots was the one that is told by Wikipedia:
Its value was sometimes
represented by a pattern of five dots arranged at the corners and the
center of a square, like the pips of a die. So, this pattern also came
to be called quincunx.
Or the Big Dictionary:
… [from a] coin of this value (occasionally marked with a pattern
resembling the five spots on a dice cube),…
But today I did Google image search for qunicunxes. And while most
had five dots, I found not even one that had the dots arranged in an
X pattern.
(I believe the heads here are Minerva, goddess of wisdom. The owl is
also associated with Minerva.)
Where's the quincunx that actually has a quincuncial arrangement of
dots? Nowhere to be found, it seems. But everyone says it, so it must be true.
Addenda
The first common use of “quincunx” as an English word was to refer
to trees that were planted in a quincuncial pattern, although not
necessarily in groups of exactly five, in which each square of four
trees had a fifth at its center.
Similarly, the
Galton Box,
has a quincuncial arrangement of little pegs. Galton himself called
it a “quincunx”.
The OED also offers this fascinating aside:
Latinquincunx occurs earlier in an English context. Compare
the following use apparently with reference to a v-shaped figure:
1545Decusis, tenne hole partes or ten Asses...It is also a
fourme in any thynge representyng the letter, X, whiche parted in
the middel, maketh an other figure called Quincunx, V.
which shows that for someone, a quincuncial shape was a V and not
an X, presumably because V is the Roman numeral for five.
A decussis was a coin worth not ten uncia but ten asses, and
it did indeed have an X on the front. A five-as coin was a
quincussis and it had a V. I wonder if the author was confused?
The source is
Bibliotheca Eliotæ.
The OED does not provide a page number.
It wasn't until after I published this that I realized that
today's date was the extremely quincuncial 2025-05-25. I thank the
gods of chance and fortune for this little gift.
The quince is so-named because, like other fruits in the apple family,
it has a natural fivefold symmetry:
This is because their fruits develop from five-petaled flowers, and
the symmetry persists through development. These are pear blossoms:
You can see this in most apples if you cut them into equatorial slices:
The fivefold symmetry isn't usually apparent from the outside once the
structure leaves the flowering stage. But perfect Red Delicious
specimens do have five little feet:
P.S.: I was just kidding about the name of the quince, which actually
has nothing to do with any of this. It is a coincidence.
I thought it might be fun to try to use Glean to index as much of
Hackage as I could, and then do some rough comparisons against hiedb and also play around to see what interesting queries
we could run against a database of all the code in Hackage.
This project was mostly just for fun: Glean is not going to replace
hiedb any time soon, for reasons that will become clear. Neither are
we ready (yet) to build an HLS plugin that can use Glean, but
hopefully this at least demonstrates that such a thing should be
possible, and Glean might offer some advantages over hiedb in
performance and flexibility.
A bit of background:
Glean is a code-indexing system
that we developed at Meta. It’s used internally at Meta for a wide
range of use cases, including code browsing, documentation
generation and code analysis. You can read about the ways in which
Glean is used at Meta in Indexing
Code At Scale with Glean.
hiedb is a code-indexing system for Haskell. It takes
the .hie files that GHC produces when given the option
-fwrite-ide-info and writes the information to a SQLite database
in various tables. The idea is that putting the information in a DB
allows certain operations that an IDE needs to do, such as
go-to-definition, to be fast.
You can think of Glean as a general-purpose system that does the same
job as hiedb, but for multiple languages and with a more flexible
data model. The open-source version of Glean comes with indexers for
ten languages or
so, and moreover Glean supports SCIP which has
indexers for various languages available from SourceGraph.
Since a hiedb is just a SQLite DB with a few tables, if you want you
can query it directly using SQL. However, most users will access the
data through either the command-line hiedb tool or through the API,
which provide the higher-level operations such as go-to-definition and
find-references. Glean has a similar setup: you can make raw queries
using Glean’s query language (Angle) using the
Glean shell or the command-line tool, while the higher-level
operations that know about symbols and references are provided by a
separate system called Glass which also has a command-line tool and
API. In Glean the raw data is language-specific, while the Glass
interface provides a language-agnostic view of the data in a way
that’s useful for tools that need to navigate or search code.
An ulterior motive
In part all of this was an excuse to rewrite Glean’s Haskell
indexer. We built a Haskell indexer a while ago but it’s pretty
limited in what information it stores, only capturing enough
information to do go-to-definition and find-references and only for a
subset of identifiers. Furthermore the old indexer works by first
producing a hiedb and consuming that, which is both unnecessary and
limits the information we can collect. By processing the .hie files
directly we have access to richer information, and we don’t have the
intermediate step of creating the hiedb which can be slow.
The rest of this post
The rest of the post is organised as follows, feel free to jump
around:
Performance: a few results comparing hiedb with Glean on an
index of all of Hackage
Queries: A couple of examples of queries we can do with
a Glean index of Hackage: searching by name, and finding dead code.
Apparatus: more details on how I set
everything up and how it all works.
What’s next: some thoughts on what we still need to add to
the indexer.
Performance
All of this was perfomed on a build of 2900+ packages from Hackage,
for more details see Building all of Hackage
below.
Indexing performance
I used this hiedb command:
hiedb index -D /tmp/hiedb . --skip-types
I’m using --skip-types because at the time of writing I haven’t
implemented type indexing in Glean’s Haskell indexer, so this should
hopefully give a more realistic comparison.
I should note that in the case of Glean the only parallelism is
between the indexer and the server that is writing to the DB. We
didn’t try to index multiple .hie files in parallel, although that
would be fairly trivial to do. I suspect hiedb is also
single-threaded just going by the CPU load during indexing.
Size of the resulting DB
hiedb: 5.2GB
Glean: 0.8GB
It’s quite possible that hiedb is simply storing more information, but
Glean does have a rather efficient storage system based on RocksDB.
Performance of find-references
Let’s look up all the references of Data.Aeson.encode:
hiedb -D /tmp/hiedb name-refs encode Data.Aeson
This is the query using Glass:
cabal run glass-democlient -- --service localhost:12345 \
references stackage/hs/aeson/Data/Aeson/var/encode
(side note: hiedb found 416 references while Glean found 415. I
haven’t yet checked where this discrepancy comes from.)
But these results don’t really tell the whole story.
In the case of hiedb, name-refs does a full table scan so it’s
going to take time proportional to the number of refs in the DB. Glean
meanwhile has indexed the references by name, so it can serve this
query very efficiently. The actual query takes a few milliseconds, the
main overhead is encoding and decoding the results.
The reason the Glass query takes longer than the raw Glean query is
because Glass also fetches additional information about each
reference, so it performs a lot more queries.
We can also do the raw hiedb query using the sqlite shell:
sqlite> select count(*) from refs where occ = "v:encode" AND mod = "Data.Aeson";
417
Run Time: real 2.038 user 1.213905 sys 0.823001
Of course hiedb could index the refs table to make this query much
faster, but it’s interesting to note that Glean has already done that
and it was still quicker to index and produced a smaller DB.
Performance of find-definition
Let’s find the definition of Data.Aeson.encode, first with hiedb:
(worth noting that hiedb is giving the span of the identifier only,
while Glass is giving the span of the whole definition. This is just a
different choice; the .hie file contains both.)
Again, the issue with hiedb is that its data is not indexed in a way
that makes this query efficient: the defs table is indexed by the
pair (hieFile,occ) not occ alone. Interestingly, when the module
is known it ought to be possible to do a more efficient query with
hiedb by first looking up the hieFile and then using that to query
defs.
What other queries can we do with Glean?
I’ll look at a couple of examples here, but really the possibilities
are endless. We can collect whatever data we like from the .hie
file, and design the schema around whatever efficient queries we want
to support.
Search by case-insensitive prefix
Let’s search for all identifiers that start with the case-insensitive
prefix "withasync":
In less than 0.1 seconds we find 55 such identifiers in Hackage. (the
output isn’t very readable so I didn’t include it here, but for
example this finds results not just in async but in a bunch of
packages that wrap async too).
Case-insensitive prefix search is supported by an index that Glean
produces when the DB is created. It works in the same way as efficient
find-references, more details on that below.
Why only prefix and not suffix or infix? What about fuzzy search? We
could certainly provide a suffix search too; infix gets more tricky
and it’s not clear that Glean is the best tool to use for infix or
fuzzy text search: there are better data representations for that kind
of thing. Still, case-insensitive prefix search is a useful thing to
have.
Could we support Hoogle using Glean? Absolutely. That said, Hoogle
doesn’t seem too slow. Also we need to index types in Glean before it
could be used for type search.
Identify dead code
Dead code is, by definition, code that isn’t used anywhere. We have a
handy way to find that: any identifier with no references isn’t
used. But it’s not quite that simple: we want to ignore references
in imports and exports, and from the type signature.
Admittedly finding unreferenced code within Hackage isn’t all that
useful, because the libraries in Hackage are consumed by end-user code
that we haven’t indexed so we can’t see all the references. But you
could index your own project using Glean and use it to find dead
code. In fact, I did that for Glean itself and identified one entire
module that was dead, amongst a handful of other dead things.
Here’s a query to find dead code:
N where
N = hs.Name _;
N.sort.external?;
hs.ModuleSource { mod = N.mod, file = F };
!(
hs.NameRefs { target = N, file = RefFile, uses = R };
RefFile != F;
coderef = (R[..]).kind
)
Without going into all the details, here’s roughly how it works:
N = hs.Name _; declares N to be a fact of hs.Name
N.sort.external?; requires N to be external (i.e. exported), as
opposed to a local variable
hs.ModuleSource { mod = N.mod, file = F }; finds the file F
corresponding to this name’s module
The last part is checking to see that there are no references to
this name that are (a) in a different file and (b) are in code,
i.e. not import/export references. Restricting to other files isn’t
exactly what we want, but it’s enough to exclude references from
the type signature. Ideally we would be able to identify those more
precisely (that’s on the TODO list).
You can try this on Hackage and it will find a lot of stuff. It might
be useful to focus on particular modules to find things that aren’t
used anywhere, for example I was interested in which identifiers in
Control.Concurrent.Async aren’t used:
N where
N = hs.Name _;
N.mod.name = "Control.Concurrent.Async";
N.mod.unit = "async-2.2.4-inplace";
N.sort.external?;
hs.ModuleSource { mod = N.mod, file = F };
!(
hs.NameRefs { target = N, file = RefFile, uses = R };
RefFile != F;
coderef = (R[..]).kind
)
This finds 21 identifiers, which I can use to decide what to deprecate!
Apparatus
Building all of Hackage
The goal was to build as much of Hackage as possible and then to index
it using both hiedb and Glean, and see how they differ.
To avoid problems with dependency resolution, I used a Stackage LTS
snapshot of package versions. Using LTS-21.21 and GHC 9.4.7, I was
able to build 2922 packages. About 50 failed for some reason or other.
And did a large cabal get to fetch all the packages in LTS-21.21.
Then
cabal build all --keep-going
After a few retries to install any required RPMs to get the dependency
resolution phase to pass, and to delete a few packages that weren’t
going to configure successfully, I went away for a few hours to let
the build complete.
It’s entirely possible there’s a better way to do this that I don’t
know about - please let me know!
Building Glean
The Haskell indexer I’m using is in this pull
request which at the time of writing isn’t merged yet. (Since I’ve
left Meta I’m just a regular open-source contributor and have to wait
for my PRs to be merged just like everyone else!).
Admittedly Glean is not the easiest thing in the world to build,
mainly because it has a couple of troublesome dependencies:
folly (Meta’s library of
highly-optimised C++ utilities) and RocksDB.
Glean depends on a very up to date version of these libraries so we
can’t use any distro packaged versions.
Full instructions for building Glean are
here but roughly it goes like
this on Linux:
Install a bunch of dependencies with apt or yum
Build the C++ dependencies with ./install-deps.sh and set some env vars
make
The Makefile is needed because there are some codegen steps that
would be awkward to incorporate into the Cabal setup. After the first
make you can usually just switch to cabal for rebuilding stuff
unless you change something (e.g. a schema) that requires re-running
the codegen.
Running Glean
I’ve done everything here with a running Glean server, which was
started like this:
While it’s possible to run Glean queries directly on the DB without a
server, running a server is the normal way because it avoids the
latency from opening the DB each time, and it keeps an in-memory cache
which significantly speeds up repeated queries.
The examples that use Glass were done using a running Glass server,
started like this:
cabal run glass-server -- --service localhost:1234 --port 12345
How does it work?
The interesting part of the Haskell indexer is the schema in hs.angle. Every
language that Glean indexes needs a schema, which describes the data
that the indexer will store in the DB. Unlike an SQL schema, a Glean
schema looks more like a set of datatype declarations, and it really
does correspond to a set of (code-generated) types that you can work
with when programmatically writing data, making queries, or inspecting
results. For more about Glean schemas, see the
documentation.
Being able to design your own schema means that you can design
something that is a close match for the requirements of the language
you’re indexing. In our Glean schema for Haskell, we use a Name,
OccName, and Module structure that’s similar to the one GHC uses
internally and is stored in the .hie files.
The indexer
itself
just reads the .hie files and produces Glean data using datatypes
that are generated from the schema. For example, here’s a fragment of
the indexer that produces Module facts, which contain a ModuleName
and a UnitName:
here NameRefs is a predicate—which you can think of as a datatype,
or a table in SQL—defined in terms of another predicate,
FileXRefs. The facts of the predicate NameRefs (rows of the table)
are derived automatically using this definition when the DB is
created. If you’re familiar with SQL, a stored derived predicate in
Glean is rather like a materialized view in SQL.
What’s next?
As I mentioned earlier, the indexer doesn’t yet index types, so that
would be an obvious next step. There are a handful of weird corner
cases that aren’t handled correctly, particularly around record
selectors, and it would be good to iron those out.
Longer term ideally the Glean data would be rich enough to produce the
Haddock docs. In fact Meta’s internal code browser does produce
documentation on the fly from Glean data for some languages - Hack and
C++ in particular. Doing it for Haskell is a bit tricky because while
I believe the .hie file does contain enough information to do this,
it’s not easy to reconstruct the full ASTs for declarations. Doing it
by running the compiler—perhaps using the Haddock API—would be
an option, but that involves a deeper integration with Cabal so it’s
somewhat more awkward to go that route.
Could HLS use Glean? Perhaps it would be useful to have a full Hackage
index to be able to go-to-definition from library references? As a
plugin this might make sense, but there are a lot of things to fix and
polish before it’s really practical.
Longer term should we be thinking about replacing hiedb with Glean?
Again, we’re some way off from that. The issue of incremental updates
is an interesting one - Glean does support incremental
indexing
but so far it’s been aimed at speeding up whole-repository indexing
rather than supporting IDE features.
Today will be the first in a series where we’ll be exploring some LeetCode problems and comparing different solutions from Haskell and Rust. The main idea is to demonstrate how you might translate ideas between the recursive core of Haskell, and the loop-based framing of most other languages.
If you want to learn more about problem solving in Haskell, you should take a closer look at Solve.hs! This course will give you an in-depth walkthrough of problem solving ideas in Haskell, including how concepts compare to more typical languages.
The Problem
The first problem we’ll consider is called H-Index. In academia, a person has an “H-Index” of n if they have published at least n papers that have n or more citations. So the input to our problem is a list of integers, where each integer is the number of citations of a particular paper the author wrote. Our job is to calculate the author’s H-Index.
The Algorithm
This problem is fairly straightforward if you sort the input list. Once we do this, we can look at any index i, and consider the number of remaining entries (e.g. n - i), and we’ll know that the number of papers with at least that many citations is n - i.
So we can accomplish this task with a single loop over the sorted list. Throughout this loop, we’ll be tracking the maximum “H-Index” we’ve seen so far (maxH). At each iteration, we take the following steps:
Get the number of remaining papers (rem) and the citations at this index (next)
If the rem is greater than next, then update maxH to next if next is larger.
Otherwise, update maxH to rem if rem is greater.
The last step is a key edge case! If we have the list [1, 1, 1, 9, 9], we’ll get to index 3, with next being 9 and rem being 2. The remainder is smaller than the index, but we would still update maxH to 2, because there are at least 2 citations remaining that are 2 or greater.
Rust Solution
Here’s our Rust solution:
pub fn h_index(citations: Vec<i32>) -> i32 {
let mut cp = citations.clone();
cp.sort();
let n = cp.len();
let mut maxH: i32 = 0;
for i in 0..n {
let next = cp[i];
let rem: i32 = (n - i) as i32;
if (rem >= next) {
maxH = std::cmp::max(next, maxH);
} else {
maxH = std::cmp::max(rem, maxH);
}
}
return maxH;
}
We have the first part, where we clone the input, sort it, and set up our loop variables:
pub fn h_index(citations: Vec<i32>) -> i32 {
let mut cp = citations.clone();
cp.sort();
let n = cp.len();
let mut maxH: i32 = 0;
...
}
Then we have the loop itself, where we have our two cases to consider:
for i in 0..n {
let next = cp[i];
let rem: i32 = (n - i) as i32;
if (rem >= next) {
// There are at least ‘next’ papers >= ‘next’
maxH = std::cmp::max(next, maxH);
} else {
// ‘next’ > ‘rem’, so there are at least ‘rem’ papers >= ‘rem’
maxH = std::cmp::max(rem, maxH);
}
}
So this is pretty straightforward. Now how do we approach this kind of problem in Haskell?
Haskell Solution
Our Haskell solution will have the same structure, but instead of running a loop and indexing into a vector, we’ll use a linked list and call a recursive function. Let’s begin by getting the length and sorting our input:
import qualified Data.List as L
hIndex :: [Int] -> Int
hIndex inputs = ...
where
n = length inputs
sorted = L.sort inputs
...
Now we need to think about our recursive loop function. At each iteration, we need access to the remaining number of values, the next citation value, and we need to pass along maxH. As with many list-based recursive functions, we’ll peel off one element of the input list each time. Ultimately we’ll return maxH from this loop when we hit our base case of an empty input list. So its type signature should look like this:
loop :: (Int, [Int], Int) -> Int
When writing a recursive function, we always handle the base case first:
Now in the recursive case, we can apply our algorithm, updating maxH if necessary:
loop :: (Int, [Int], Int) -> Int
loop (_, [], maxH) = maxH
loop (remaining, next : rest, maxH) = if remaining >= next
then loop (remaining - 1, rest, max next maxH)
else loop (remaining - 1, rest, max remaining maxH)
To finish up, all we need to do is call our loop function with the appropriate initial inputs (n, sorted, 0). Here’s our complete Haskell solution:
import qualified Data.List as L
hIndex :: [Int] -> Int
hIndex inputs = loop (n, sorted, 0)
where
n = length inputs
sorted = L.sort inputs
loop :: (Int, [Int], Int) -> Int
loop (_, [], maxH) = maxH
loop (remaining, next : rest, maxH) = if remaining >= next
then loop (remaining - 1, rest, max next maxH)
else loop (remaining - 1, rest, max remaining maxH)
Using a Fold
Now we can notice that our loop has a particular structure. We have one piece of accumulated state (maxH), and this changes based on each value in our list (combined with the remaining values). We can easily re-imagine this kind of loop using a fold. We just have to think of the folding function like this:
loop :: Int -> (Int, Int) -> Int
loop maxH (remaining, next) = if remaining >= next
then max next maxH
else max remaining maxH
This has the a -> b -> a structure of a left-fold function, where a is our accumulated maxH value, and the other values come from our list. The main benefit here is that our loop function no longer has to deal with the burden of calling a base case or passing the “shrinking” list as an argument to the next recursive call.
We can invoke this loop at the top level like so:
hIndex :: [Int] -> Int
hIndex inputs = foldl loop 0 (zip [n,n-1..1] sorted)
where
n = length inputs
sorted = L.sort inputs
loop :: Int -> (Int, Int) -> Int
loop maxH (remaining, next) = if remaining >= next
then max next maxH
else max remaining maxH
We just have to zip the decreasing indices together with our sorted list. Now our recursive “loop” is more like a typical for-loop. We’re only considering one element at a time, and we’re updating the important state each time.
Conclusion
In this comparison, we saw a good comparison between a normal for-loop in Rust, and a recursive solution in Haskell. We also saw how we could simplify this recursive formulation into a “fold” structure.
If you're interested in learning more about writing recursive functions in Haskell, check out our Solve.hs course. You’ll learn how to start thinking about problems in a functional way, and you’ll learn the step-by-step processes for tackling problems with basic recursion and folds like we saw in this example.
I was the chief editor for this blog for the past 8 years or so,
and I’ve just recently passed the mantle to Chris Harrison. I thought
I’d take the opportunity to write a little bit about this blog, how
it’s operated and what it means to us. Besides, we do like when things
get meta here, so this is a blog post about the blog.
A little bit of history
One of the tenets under which Mathieu Boespflug founded
Tweag was that software engineers naturally don’t write enough.
Writing is an essential part of the engineering job. We write issues,
pull requests, code comments, documentation of various sorts. We
discuss and debate online, we have to arbitrate trade-offs. Most of
this is common, in fact, to every engineering profession. And
although we typically become engineers because of our taste for the
technical part of our job (I certainly did, I don’t know for sure how
ubiquitous it is); writing is still a big part of our job.
So Mathieu reasoned, if writing is to be such a big part of our job,
but we’re technical people at heart, not writers, for the company to
be at its best it needs to make it clear that precise and clear
writing is important to our job. To that effect, there ought to be
venues for us to write, where it was made an expectation, an actual
requirement, for us to write. The blog is one of them.
At first, the blog was just there. After all, our blog was, and still
is, just (part of) a Github repository. So we’d make pull requests and
merge blog posts. But, of course, people seek review. When you’re
posting in the name of a company, you tend to be a little more careful
about what you write; besides, we’re all so used to having our PRs
reviewed. This was all very ad hoc, there was no process for it.
Whatever my reason at the time (I honestly don’t remember after
all this time), I ended up participating in the review of most blog
posts. It’s a poorly kept, but surprisingly little discussed secret
that the most common way to get a responsibility in a company is to
just assume said responsibility. Do the thing, and it will become your
charge. Anyway, I became editor in chief.
Why we blog
Besides being a venue to exercise our technical writing, this blog has
been tremendously useful to us. Keeping a high-quality, serious,
technical blog helped establish trust in our work, build bridges with
community, and attract clients and employees.
It’s very rewarding for us, as individuals, to be recognised by our
peers, and to work with a company which is likewise recognised. This
is certainly a great motivation for us to write.
Besides, it fits very well with our open-source values, both as a
company and as individuals. We strongly believe in open-source, and
the value it has to the world. So we participate. And open-source
software isn’t just about putting software’s sources out there.
Open-source is also about sharing knowledge. Which is another important
element of the company’s ethos. We do have formal and informal venues
for sharing knowledge internally, all very important stuff, but it’s a
topic for another time. The blog is one of our main venues for sharing
knowledge with the rest of the world.
So we do share, we talk about our open-source journey, the software we
build, or the software we use. But we also share what we learn through
working with clients, not necessarily on open-source projects.
So this blog is something that operates at the sweet spot of any
company’s activities: it’s something we do because we believe that
it’s the right thing to do, and it’s also something which is valuable
to the company. In fact the blog brought so many clients that it
allowed Tweag to operate without a marketing team for quite a few
years. Nowadays, the company’s bigger and we can’t rely on a blog
alone, but it’s still a great way to connect.
How we blog
Writing a blog post is making a pull request against the Github
repository for the tweag.io website. A post is a
Markdown file, possibly accompanied by resources (such as
images). It’s a workflow which is very familiar to software
engineers. All the review process is, in fact, a Github pull request
review.
The author of a blog post is first tasked with obtaining
a “technical review” from their peers: people close to the topic
review the blog post for accuracy and relevance. The blog post editors
are all engineers (the entire process, in fact, is owned by
engineers), but aren’t necessarily very familiar with the blog post’s
topic, so this first round of review couldn’t be performed by the
editing team, it’s also much better for scalability as technical
review uses a little time from a lot of people, rather than a lot of
time from a few.
When the author and technical reviewers are satisfied with the blog
post, they send it to the editing team, which will do two rounds of
review (which we cleverly call “first editorial review” and “second
editorial review”). The editing team reviews for clarity and writing
quality. Something that isn’t a concern of reviewers (or really of
the process as a whole), on the other hand, is SEO. Certainly our blog
posts make our website easier to find, and of course this is important
to us. But this is the result, we hope, of making blog posts that you
like.
We can propose edits in three different ways: we can ask
question in the review thread, propose “suggestions” in the Github
interface which the author is free to accept or not, or we can push
directly to the blog post (in which case we try our best to preserve
the author’s voice, and always leave time for the author to check that
the edits are correct). The choice between the three is primarily
driven by our degree of confidence in the edit. Really we do what makes
sense to reduce the amount of back-and-forth.
Reviewing blog posts on Github with a team distributed all around the
world is essentially a distributed process. As with any distributed process,
we try to minimise the amount of synchronisation. Our target is to be
able to do a review in two weeks or less (one week for each
round). Sometimes it takes more though, when the blog post presents
more difficult editing challenges, or when the author doesn’t have
enough availability outside of their client work to respond to our
reviews quickly enough. But we hit the target more often than not.
The two rounds of review are functionally identical. We just found
that, in practice, having two rounds of review helps raise the quality
a lot. The first reviewer often gets involved deeply in the writing of
the blog post, to the point that they stop seeing the blog post as a
reader. The second reviewer receives a blog post in almost publishable
condition and brings fresh eyes to conduct the finishing touch.
What blogging means to us
When Tweag was a standalone company, blogging was almost our entire
marketing strategy. Now that we’re part of the substantially larger
Modus Create, we aren’t in a position to rely solely on technical
blogging for marketing. Yet, this blog is still an important part of
our strategy.
All of what I wrote above still applies, but I’d like to bring up
something else. See, a company is a little bit faceless. It doesn’t
have a drive, it doesn’t have a personality: people do. This is truer
the larger a company is. We don’t aspire to be a faceless abstract
entity, though. Modus Create is made of people, people that we believe
to be interesting, and that we hope you’ll find interesting too. In
fact our business is largely for clients (maybe you!) to find our
people interesting and hire us. This blog is one of the ways we use to
promote individuals. Who they are, what they like, what they know,
what they can do. This is why our writing recommendations encourage
authors to use the pronoun “I” where many of us would be tempted to say “we”.
A more collectivist consideration is communities. We take part in a
number of communities (Haskell, Nix, Bazel, Typescript, …), and it’s
all too easy for a company to say it’s part of a community but really
meaning that they just use Haskell, Nix, Bazel, Typescript. But it’s
not really what being part of a community means, does it? It also
entails taking part in community life. There’s a bunch of things
we do: we write libraries,
contribute
upstream,
help with
governance,
sponsor and
sometimes organise events, …. And we blog, which also plays its part in
community life. This is why we don’t shy away from highly
specialised blog posts. Sure they are addressed to a pretty narrow,
sometimes quite niche audience, but they are relevant to one of our
communities. Of course it helps establish our technical
chops. Hopefully it also builds trust.
I should note that what I’m writing about is what works for us. It’s
not a recipe that will automatically work for you. There are many
excellent technical blogs out there, which, I assume, probably have
different approaches to us, such as
CockroachDB’s,
Netflix’s,
fly.io’s and Trail of
Bits’s (the latter two I wasn’t aware
of, by the way, they were brought to my attention by this blog post’s
technical review). If you like this blog, you’ll probably like theirs,
go check them out! Dan Luu shares some more general
considerations. But at the end of
the day, what makes a strong communication strategy is to build
around and empower the people you already have. Capitalise on your
strengths, don’t go against the grain just to imitate what someone
else does, however successful they appear to be.
Sign-off
A paradox of the work of software engineers (and probably all
engineering disciplines really) is that writing is such an integral
part of our job, but most of us receive virtually no training in our
studies.
I’ve never had technical writing classes, myself. The way I learned
writing was in part by gleaning some of the popular wisdom taught
among my peers, but mostly by co-authoring scientific articles with
better technical writers than me and seeing them do their
magic. Honestly, I found no better learning experience than seeing an
entire paragraph of mine rewritten in a mere few words that were also
more precise.
I don’t know how to teach technical writing formally. So a lot of the
process we’ve converged on reflects my attempt to replicate what
worked on me for the company: teaching technical writing by
example. By showing how a blog post can be improved, hopefully the
authors will learn to write better next time. Because the blog is this
as well: a tool to teach ourselves, collectively, how to be technical
writers.
This is a very soft target, it’s hard to measure the degree to which we’ve
succeeded. But, because I don’t really know how to close this post
without getting a touch emotional (it is, after all, no small moment
for me), I’ve witnessed many of our engineers mature as writers, and I
can’t help but feeling some parent-like pride at their growth.
Today, 2025-05-14, at 1830 UTC (11:30 am PDT, 2:30 pm EDT, 7:30 pm GMT, 20:30 CET, …)
we are streaming the 44th episode of the Haskell Unfolder live on YouTube.
Many Haskell programmers will be familiar with property based testing of pure functions (for those who are not, various episodes of the Haskell Unfolder have discussed this: #4, #21, #38 and #40). Property based testing for stateful systems (“IO code”) is however much less well-known, which is a pity as it is just as useful! In this episode we will demonstrate how we can use quickcheck-lockstep to verify the responses we get from a simple stateful API; as we will see, all of the lessons from property based testing for pure functions can be applied in this stateful setting also.
About the Haskell Unfolder
The Haskell Unfolder is a YouTube series about all things Haskell hosted by
Edsko de Vries and Andres Löh, with episodes appearing approximately every two
weeks. All episodes are live-streamed, and we try to respond to audience
questions. All episodes are also available as recordings afterwards.
In last week’s article, I discussed how Monday Morning Haskell courses compare to other Haskell courses that I’ve seen out there online. Obviously I’m wildly biased, but I think MMH courses have some serious advantages.
But there’s still the elephant in the room…how do my courses compare to the possibility of using generative AI (e.g. ChatGPT) to learn Haskell instead? While AI is a great tool that has opened up a lot of doors in terms of learning complex concepts, human-developed courses still have some important advantages over the current way you would learn from a chatbot.
Analogy: Going to the Library
I’ll start my case by drawing an (imperfect) analogy. Suppose you are enrolled in a college and want to learn a particular subject, like physical chemistry. You could enroll in your school’s physical chemistry course. Or you could spend the same amount of time going to the library. After all, the library has tons of books on physical chemistry. So you could read all these books and gain the same level of insight, right?
In this example, most people would recognize the shortcomings of just going to the library. For example, you are now responsible for determining the curriculum and course of study.
You could, of course, look at the table of contents of an introductory book and just run with that way of organizing the material. But how much of that do you need to learn? Most college courses aren’t going all the way through a textbook, because the professor already has a good idea of what material is the most important, and has organized the course around that.
A professor will also know when and how to introduce supplemental material from other sources. If you’re just “learning from the library”, you’d be responsible for selecting which materials are the most important, and you probably aren’t qualified!
Also, while textbooks may have practice problems, and they may even have answers to those problems, you still have to do the work of figuring out which problems to study, and how many you need to study before you know the material. Taking a full course with assignments would solve this for you.
Finally, textbooks will rarely tell you about the human process of learning a particular subject. You probably aren’t going to read a sentence like “lots of students struggle to understand this, here’s a way of thinking about the problem that has helped a lot of them.” These are insights you’ll gain from working with a professor (who has taught real students) or other students in the class.
So let’s sum up the shortcomings of “learning from the library”:
Direction - You must take on the cognitive overhead of determining which areas of the subject to study.
Filtering - You must figure out how much detail is necessary, and how much practice you need to learn it.
Human Learning Insight - Textbooks are generally lacking in the actual insights and breakthroughs that help students understand particularly challenging ideas.
From Physical to Online Learning
Now let’s consider what changes about the analogy if instead of comparing physical learning environments, we think about the current online learning environment. Entering an online course is significantly easier than enrolling in a university course. You don’t have to wait for the start of the semester, or go to a physical location.
But using ChatGPT as your “library” is vastly easier than studying from textbooks. In a matter of minutes, you can get tons of information on your screen that would have taken hours or days of effort at the library. And best of all, you can get information on virtually any topic, rather than just those that have pre-existing textbooks.
But I would still claim that using Chatbots for learning shares some of the drawbacks of “learning from the library”. And for these reasons, it’s still worthwhile to consider online courses where they exist instead of relying solely on ChatGPT. Some of these drawbacks might seem counterintuitive, but let’s think about it.
Direction of Study
You might think, “I don’t need to set my own direction, ChatGPT will do that for me!” And yes, you can ask it to lay out a syllabus for you (I did this myself in one of the examples below). This will give you a list of topics to study.
But it won’t just write out the whole course for you based on this initial syllabus in one go. You have to keep prompting it to provide you with the information you want. And it will get sidetracked, consistently asking you to go deeper and deeper down particular rabbit holes.
So it’s still up to you to determine how much you really want to study about particular topics, and you need to maintain the discipline to pull it back out and shift gears. A human-designed course puts these limits in there for you, so that you don’t need to carry that cognitive load.
Filtering
This brings us to the next issue of “filtering”. ChatGPT will provide you with a lot of information, all at once. You’ll ask a simple question, and get a very complicated answer with lots of tables comparing various different ways of looking at the question.
Sometimes, this is nice. It will expose you to ideas you wouldn’t have thought of otherwise. Sometimes though, it’s very distracting. It takes you away from the core of what you’re trying to learn. You have to make sure you aren’t getting dragged into an infinite loop of concepts.
The “practice” problem also exists. ChatGPT can keep coming up with practice problems, but it’s up to you to know how many you really need to study. In our case study below, we’ll also consider that it’s not necessarily the best tool for coming up with practice problems.
Again, a human-designed course does the filtering and measuring for you.
Human Insight
Once at my job, I was reviewing a teammate’s code that implemented a complicated algorithm. I told him, “after I looked closely at this one particular line, my understanding of this algorithm went from like 30% to 70%”, so adding an explanatory comment here would be very helpful!”.
This experience helped me understand the idea of “knowledge inflection points”. These are the key insights that really help you understand a topic. I’ve had several of these with various Haskell concepts, from monads, to folds, to data structures and certain algorithms. I’ve done my best to incorporate these insights into my course content.
An example from Solve.hs might be my understanding of “the common API” of Haskell data structures. This made it much easier for me to reason about using different structures in Haskell.
An AI probably wouldn’t frame the issue in the way I did, unless you already have the knowledge to prompt it. AI’s don’t have the experience of “learning” a concept piece-by-piece, and knowing when things finally “clicked”. You could try asking the chatbot what insights help people learn a topic, but it will only be able to try piecing that information together from what other people have written. On the whole, it still doesn’t beat the experience of someone who’s been there.
Human insights around learning are always going to get baked into a human-designed course, whereas AI is not generally going to be thinking in these terms.
Case Study: Learning Concurrency
I wanted to share a couple case studies that highlight some of the promise but also some of the frustrations with using AI for learning. Here’s a link to an extensive, multi-day study I did with ChatGPT to learn about concurrency topics. It helped me review a lot of topics I had learned in college (10 years ago), and also learn many new things. But there were still some pain points.
The “filtering” problem should be very evident. For each prompt I gave, ChatGPT provided tons of information. It was entirely up to me to figure out how much of this I really needed to know in order to be satisfied.
The “direction” problem is also clear. I started by asking for an organizational outline, and the chatbot duly obliged. But as I dug into certain topics, its preference was to ask me to keep going deeper down certain knowledge paths. I had to consistently drag it back to the syllabus it originally designed.
There were also no clear insights on what the key knowledge was. Over the course of the study, I figured some of these out for myself. But again, I had to filter through a lot of data to get there.
Another drawback I haven’t mentioned yet is the “memory” issue. Chatbots have limited, token-based memory, so they’ll forget what you’ve already learned over even a medium length study. My concurrency study introduced the idea of a “lock-free queue” using compare-and-swap operations early on. ChatGPT reintroduces this idea later as if I had never heard of it. Human-designed courses will avoid this sort of behavior.
I didn’t ask for practice problems in this study, so let’s consider another case study where I was specifically looking to do this in Haskell.
Case Study: Dijkstra’s Algorithm
In this quick study, I asked ChatGPT to come up with a practice problem for learning Dijkstra’s algorithm. Some things were good about its response, but some things weren’t.
On the positive side, the code works, the tests work, and some of the follow-up suggestions are also pretty good. For example, putting a bound on the number of nodes your path can have, or allowing multiple starts are simple extensions that didn’t occur to me when I was writing problems.
My main gripe is that the problems are a bit too obvious as graph problems. It started essentially with “implement Dijkstra’s algorithm” rather than giving me a practice problem using Dijkstra’s algorithm. And when I asked for a “disguised graph problem”, it gave me the delivery problem which wasn’t much of a disguise.
Also, the code used PSQueue, rather than the more beginner-friendly Data.Heap. This package may be better for certain things, but the type operator it uses would be a bit more confusing for a novice.
The line-by-line explanations were pretty good on the whole, but I don’t know that they’re a perfect substitute for really good visual/slide-based instructions like you would find in one of my courses.
With enough prompt engineering, you could get around these issues. But that’s exactly my point. It’s nice to not have to keep coming up with new prompts to get what you’re looking for, especially when you get a long explanation after every question.
Conclusion
Generative AI is a massive innovation for learning, especially on subjects that don’t have a lot of good guide material. But extensive, well-thought-out, human-designed content still has some significant advantages. The content is informed by the personal experience of someone who has actually been in your shoes and has had to learn something the same way you’ll learn it. This is not something an AI can relate to.
Prompt engineering involves a lot of cognitive effort. You have to constantly be directing the flow of what you’re supposed to learn, filter out the unnecessary parts, and then you have to learn it! While the freedom of being able to learn almost anything can be desirable, it can also be exhausting to always be directing the flow. It can be much easier and more helpful to just follow the lead of what another person has done.
I’ve used generative AI for learning and will continue to do so. But when human-designed content is available, I’ll look there first, and consider using AI as a supplement where I feel there are gaps.
When it comes to generating content, I don’t like AI as much, certainly not as a general purpose content producer. But it certainly has its uses. Looking back on course creation, I wish I had used it for writing test cases, for example. Another idea might be translating my work into other languages.
I’ll continue to experiment with AI going forward. But a solid guiding principle is that you should be using AI to enhance yourself, and not replace yourself. I still believe that human content has an edge over AI content for the same subject matter, so I encourage you to take another look at our courses at Monday Morning Haskell Academy, and to subscribe to our mailing list for future updates and discounts!
[ I started thinking about this about twenty years ago, and then writing it down in 2019, but it seems to be obsolete. I am publishing it anyway. ]
The canonical division of the year into seasons in the northern
temperate zone goes something like this:
Spring: March 21 – June 21
Summer: June 21 – September 21
Autumn: September 21 – December 21
Winter: December 21 – March 21
Living in the mid-Atlantic region of the northeast U.S., I have never
been happy with this. It is just not a good description of the
climate.
I begin by observing that the year is not equally partitioned between
the four seasons. The summer and winter are longer, and spring and
autumn are brief and happy interludes in between.
I have no problem with spring beginning in the middle of March. I
think that is just right. March famously comes in like a lion and
goes out like a lamb. The beginning of March is crappy, like
February, and frequently has snowstorms and freezes. By the end of
March, spring is usually skipping along, with singing birds and not just the early
flowers (snowdrops, crocuses, daffodil) but many of the later ones also.
By the middle of May the spring flowers are over and the weather is
getting warm, often uncomfortably so. Summer continues through the
beginning of September, which is still good for swimming and
lightweight clothes. In late September it finally gives way to
autumn.
Autumn is jacket weather but not overcoat weather. Its last gasp is
in the middle of November. By this time all the leaves have changed,
and the ones that are going to fall off the trees have done so. The
cool autumn mist has become a chilly winter mist. The cold winter
rains begin at the end of November.
So my first cut would look something like this:
Months
Seasons
January
February
March
April
May
June
July
August
September
October
November
December
Winter
Spring
Summer
Autumn
Winter
Note that this puts Thanksgiving where it belongs at the boundary
between autumn (harvest season) and winter (did we harvest enough to
survive?). Also, it puts the winter solstice (December 21) about one
quarter of the way through the winter. This is correct. By the
solstice the days have gotten short, and after that the cold starts to
kick in. (“As the days begin to lengthen, the cold begins to
strengthen”.) The conventional division takes the solstice as the
beginning of winter, which I just find perplexing. December 1 is
not the very coldest part of winter, but it certainly isn't autumn.
There is something to be said for it though. I think I can
distinguish several subseasons — ten in fact:
Dominus Seasonal Calendar
Months
Seasons
Sub-seasons
January
February
March
April
May
June
July
August
September
October
November
December
Winter
Spring
Summer
Autumn
Winter
Midwinter
Late Winter
Early spring
Late spring
Early Summer
Midsummer
Late Summer
Early autumn
Late autumn
Early winter
Midwinter
Midwinter, beginning around the solstice, is when the really crappy
weather arrives, day after day of bitter cold. In contrast, early and
late winter are typically much milder. By late February the snow is
usually starting to melt. (March, of course, is always unpredictable,
and usually has one nasty practical joke hiding up its sleeve. Often,
March is pleasant and springy in the second week, and then mocks you
by turning back into January for the third week. This takes people by
surprise almost every year and I wonder why they never seem to catch
on.)
Similarly, the really hot weather is mostly confined to
midsummer. Early and late summer may be warm but you do not get
blazing sun and you have to fry your eggs indoors, not on the
pavement.
Why the seasons seem to turn in the middle of each month, and not at
the beginning, I can't say. Someone messed up, but who? Probably the
Romans. I hear that the Persians and the Baha’i start their year on
the vernal equinox. Smart!
Weather in other places is very different, even in the temperate
zones. For example, in southern California they don't have any of the
traditional seasons. They have a period of cooler damp weather in the
winter months, and then instead of summer they have a period of gloomy
haze from June through August.
However
I may have waited too long to publish this article, as climate change
seems to have rendered it obsolete. In recent years, we have barely
had midwinter, and instead of the usual two to three annual snows we
have zero. Midsummer has grown from two to four months, and summer
now lasts into October.
Today, 2025-05-07, at 1830 UTC (11:30 am PDT, 2:30 pm EDT, 7:30 pm GMT, 20:30 CET, …)
we are streaming the 43th episode of the Haskell Unfolder live on YouTube.
In this episode, we are going to look at two interacting “features” of the Haskell language (the monomorphism restriction and defaulting) that can be somewhat surprising, in particular to newcomers: there are situations where Haskell’s type inference algorithm deliberately refuses to infer the most general type. We are going to look at a number of examples, explain what exactly is going on, and why.
About the Haskell Unfolder
The Haskell Unfolder is a YouTube series about all things Haskell hosted by
Edsko de Vries and Andres Löh, with episodes appearing approximately every two
weeks. All episodes are live-streamed, and we try to respond to audience
questions. All episodes are also available as recordings afterwards.
In the UK, it’s very common that your employer pays you once a
month. When this happens, they give you a document called a payslip,
that has some numbers on it, such as how much your salary is, how much
they paid you this month, how much went to HMRC in tax, how much went
to your pension, and a few other numbers. But they never show any
workings, so you really have no way to check whether any of these
numbers are correct. There are plenty of online take-home-pay
calculators, but these all focus on the full year; they have no
facility to calculate your next payslip.
About half way through April 2024, I stopped working for one
company. Everything was wrapped up – I received my final payslip from
them, along with my P45. I then had a few months off, and started a
new job in July 2024. When you start a new job it always takes a while
for money things to get sorted out, for example pension enrolment and
sorting out pension contributions, so it’s really worthwhile to keep a
close eye on your payslips particularly for these first few
months. Mine were arriving and some numbers looked right, but other
numbers, such as the amount of tax I was paying, were changing
dramatically, month to month. I had no idea why; whether they should
be changing like that; whether they were going to keep changing or
would eventually settle down. I had no way to check any of these
numbers. Was I going to get in trouble with HMRC and get investigated?
I was also a little on edge because this was the first job where my
pension contributions were using a thing called Qualifying
Earnings. In
all my previous jobs, if I chose for 10% of my salary to go into my
pension, then that’s what would happen. But now there was this thing
called Qualifying Earnings, which is (numbers correct at time of
writing) a band from £6240 to £50,270. If you’re earning, say £30k,
then your x% contribution is actually x% of £30,000-£6240. If
you’re earning above £50,270, then any further increase to your salary
will not result in any extra contributions to your pension because
you’re above the band. The 2008 Pensions Act, which created the legal
requirement for all employees to have workplace pensions and for
automatic enrolment (with a minimum 8% combined contribution from the
employer and employee), also created this concept of Qualifying
Earnings. I consider this is a pretty scummy way of reducing employer
pension contributions for large firms. It complicates the maths and no
doubt adds confusion for people trying to check their own
payslips. Given that 74% of the population have pensions that are too
small to retire
on,
this whole concept of Qualifying Earnings seems amoral at best.
These days, a lot of smaller companies outsource their payroll
processing. In my case, I was officially working for an international
Employer of Record
and they were then outsourcing payroll processing to local firms with
country-specific expertise. So when I started asking questions, there
was no ability to go and sit with someone and work through it. Or have
a call. It was all messages passed across multiple different systems,
and partial answers at best would come back several days later. Even
if your payroll is done in-house, I strongly suspect that a lot of the
time, some software package will be being used that does all the
calculations and quite likely no one will actually understand or be
able to explain the maths that’s going on.
After a while of getting no-where, and after uncovering some
substantial mistakes that had been made that affected me, I decided to
spend some weekends actually figuring out how
PAYE
works, and writing some code that can calculate my next payslip. This
library is available for
anyone to use. There’s a
README that
hopefully explains the basic principles of how the calculations are
done. This only works if your
tax-code ends
in an L, and it only works if you’re in National Insurance
categoryA. All the code can do is use some details you provide to predict
your next payslips. Also, I’m not a trained accountant or financial
adviser, and even for my own payslips, every month, the numbers don’t
quite match up (but they’re within £1). So please treat this as a toy,
rather than the basis for building a payroll processor!
Getting started
The library is written in Go so you’ll need Go
installed. Then, in a terminal do:
$ mkdir payslips
$ cd payslips
$ go mod init mypayslips
$ go get wellquite.org/tax@latest
Now we need to write a tiny amount of code. In your new payslips
directory, create a main.go file, and open it in your editor. You
want something like this:
We create a list of
Payslips. The first
payslip must specify a year, and your tax-code. These details are
automatically applied to the payslips that follow, if not explicitly
provided. Many of the calculations rely on year-to-date totals, and so
we must have a complete record of your payslips from the start of the
tax year. So that means the first payslip is month 1 (in this example,
April 2024), then month 2 (May 2024) and so on. If you have no income
for a month then you can just put in an empty payslip ({}). The
above example describes being paid in April and May 2024, then nothing
in June, and then being paid (with a higher salary) in July, August
and September.
Save this main.go file. Then, back in your terminal, in your
payslips directory, just do:
go run main.go
You should get some output showing all sorts of calculations,
including income tax, and personal
allowance. With a little luck,
if you change the numbers to match your own salary and other details,
the numbers produced should match quite closely your own payslips,
provided nothing you’re doing is too exotic.
There is documentation for all the different
fields that you can
provide in each payslip. In general, the code will try to fill in
missing values. It should be able to cope with things like
salary-sacrifice, or, if you change job within a month and have
several payslips for the same month, this should work too. Everything
is run locally on your computer: please feel free to check the
source – there are no 3rd
party libraries at all, and nothing imports the net package. It’ll
work just the same if you yank out your network cable or disable your
WiFi.
Note however, this code is lightly tested. Whilst it works for me
(and one or two friends), I make no claims that it correctly models
the entirety of PAYE, so it may very well not work for you. Feedback,
contributions, corrections, and patches are all very welcome!
Why doesn’t [the Data.Map function] unionWith :: (a -> a -> a) -> Map k a -> Map k a -> Map k a allow for different value types the way intersectionWith :: (a -> b -> c) -> Map k a -> Map k b -> Map k c does?
This is a very reasonable question, and it lead down an interesting rabbit hole of at the intersection of API design and efficient implementation.
To answer the original question, what would the type of a different value type of unionWith look like? It would be something in the flavor of:
unionWith :: (Maybe a ->Maybe b -> c) ->Map k a ->Map k b ->Map k c
But this new Maybe a -> Maybe b -> c parameter is somewhat lossy, in that it gives the impression that it could be called with Nothing Nothing as parameters, which doesn’t fit into the vibe of being a “union.”
So instead we could restrict that possibility by using These a b:
dataThese a b =This a |That b |These a bunionWith :: (These a b -> c) ->Map k a ->Map k b ->Map k c
which seems reasonable enough.
But let’s take reasonableness out of the picture and start again from first principles. Instead let’s ask ourselves the deep philsophical question of what even IS a map?
A Map k v is a particularly efficient implementation of functions with type k -> Maybe v. But why is this Maybe here? It’s really only to encode the “default” value of performing a lookup. Nothing goes wrong if we generalize this to be Monoid v => k -> v. In fact, it helps us make sense of the right bias present in Data.Map, where we see:
lookup k (singleton k v1 <> singleton k v2) =Just v2
This equality is hard to justify under the normal understanding of Map k v being an encoding of a function k -> Maybe v. But under the general monoid interpretation, we get a nice semigroup homomorphism:
lookup k (m1 <> m2) =lookup k m1 <>lookup k m2
where the monoid in question has been specialized to be Last.
Of course, we also have a monoid homomorphism:
lookup k mempty=mempty
Let’s re-evaluate the original question in terms of this newly-generalized Map. Now that we’ve removed all of the unnecessary baggage of Maybe, we can again think about the desired type of unionWith:
unionWith :: (a -> b -> c)->Map k a->Map k b->Map k c
which looks awfully familiar. This new type signature automatically resolves our original concerns about “what should we do if the key isn’t present?”—just call the function with mempty as a parameter!
We can give some semantics as to what unionWith ought to do again by relating it to the observation lookup. The relevant law here seems like it ought to be:
lookup k (unionWith f m n) = f (lookup k m) (lookup k n)
By choosing a degenerate function f, say, \_ _ -> nontrivial, where nontrivial is some value that is notmempty, we can see the beginnings of a problem:
lookup k (unionWith f m n)= f (lookup k m) (lookup k n)=<let f = \_ _ -> nontrivial> nontrivial
Regardless of the key we lookup in our unionWithed Map, we need to get back nontrivial. How can we implement such a thing? I see only two ways:
explicitly associate every key in the map with nontrivial, or
keep nontrivial around as a default value in the map
#1 is clearly a non-starter, given that we want our Maps to be efficient encodings of functions, which leaves us with only #2. This is actually a pretty common construction, which stems immediately from the fact that a pair of monoids is itself a monoid. The construction would look something like this:
dataMap k v =Map { defaultValue :: v , implementation ::Data.Map.Map k v }deriving stock Genericderiving (Semigroup, Monoid) via (Generically (Map k v))unionWith :: (a -> b -> c)->Map k a->Map k b->Map k cunionWith f (Map def1 imp1) (Map def2 imp2) =Map (f def1 def2) (liftA2 f imp1 imp2)
Seems fine, right? The nail in the coffin comes from when we reintroduce our semigroup homomorphism:
lookup k (m1 <> m2) =lookup k m1 <>lookup k m2
Without loss of generalization, take m2 = pure nontrivial (where pure is just unionWith with a constant function.) This gives us:
lookup k (m1 <>pure nontrivial) =lookup k m1 <> nontrivial
Making this thing efficient is a further complication! We again have two options:
modify the value at every key by multiplying in nontrivial, or
finding a way of suspending this computation
#1 clearly requires \(O(n)\) work, which again forces us to look at #2. But #2 seems very challenging, because the monoidal values we need to suspend need not span the entire Map. For example, consider a Map constructed a la:
Representing this thing efficiently certainly isn’t impossible, but you’re not going to be able to do it on the balanced binary search trees that underlie the implementation of Data.Map.Map.
I find this quite an interesting result. I always assumed that Data.Map.Map (or at least, Data.Map.Monoidal.MonoidalMap) didn’t have an Applicative instance because it would require a Monoid constraint on its output—but that’s not the sort of thing we can express in Haskell.
But the analysis above says that’s not actually the reason! It’s that there can be no efficient implementation of Applicative, even if we could constrain the result.
What I find so cool about this style of analysis is that we didn’t actually write any code, nor did we peek into the implementation of Data.Map (except to know that it’s implemented as a balanced BST.) All we did was look at the obvious laws, instantiate them with degenerate inputs, and think about what would be required to to efficiently get the right answer.
Usually I write about solutions to problems I’ve worked out, but I’ve found myself increasingly becoming interesting in where solutions come from. Maybe it’s because I’ve been reading Boorstin’s excellent The Discoverers, which I’d strongly recommend.
Regardless of why, I thought I’d switch up the usual dance step today, and discuss what solving my most-recent-big-problem actually looked like, in terms of what I tried, where I looked, and what the timeline was.
The Problem
The problem is to serialize a program graph into a series of let-bindings. For example, given the following graph:
+
/ \
f ---> g
| / \
a \ /
expensive
which represents the program:
f a (g expensive expensive) + g expensive expensive
Unfortunately, this is a naive representation of the program, since it duplicates the work required to compute expensive four times, and g expensive expensive twice. Instead, we would prefer to generate the equivalent-but-more-efficient program:
let$0= expensive$1= g $0$0in f a $1+$1
This transformation is affectionately known as sharing, since it shares the computed answer whenever there is repeated work to be done.
So this is what we’re trying to do. Given the original graph, determine the best place to insert these let-bindings, for some reasonable definition of “best.” We can assume there are no side effects involved, so any place that an expression is well-scoped is an acceptable solution.
In order to understand some of my attempted solutions, it’s worth noting that our final solution should build something of type Expr, and the original graph is represented as a IntMap (ExprF Int). ExprF is the Base functor of Expr, with all of its self-references replaced by some type variable, in this case Int. Thus, the graph above looks much more like:
I spent over a year trying to solve this problem, with various mostly-working solutions during that time. My strategy here was to think really hard, write up some algorithm that seemed plausible, and then run it against our (small) battery of integration tests to make sure it got the same answer as before.
Why not property test it? I tried, but found it very challenging to implement well-typed generators that would reliably introduce shared thunks. But maybe there’s a different lesson to be learned here about writing good generators.
Anyway. For eight months, one of these think-really-hard algorithms fit the bill and didn’t give us any problems. It was a weird, bespoke solution to the problem that independetly kept track of all of the free variables in every graph fragment, and tried to let-bind a fragment as soon as we landed in a context where all of the free variables were in scope. It seemed to work, but it was extremely messy and unmaintainable.
At the time of writing, this sharing algorithm was the only source of let-binds in our entire language, which meant that it didn’t need to account for let-binds in the program.
Of course, that invariant eventually changed. We added a way in the source langauge to introduce lets, which meant my algorithm was wrong. And I had written it sufficiently long ago that I no longer remembered exactly why it worked. Which meant the theory of my program was lost, and thus that we ought to rewrite it.
Unfolding a Solution
I went back to the problem statement, and stared at it for a long time (back to the think-really-hard algorithm!) Upon staring at the problem, I realized that what I was really trying to do was determine where diamond patterns arose in the propgram graph.
Recall our original graph:
+
/ \
f ---> g
| / \
a \ /
expensive
If we redraw it such that g is on a different rank than f, then the two diamond patterns become much clearer:
+
/ \
f |
| \ |
a \ /
g
/ \
\ /
expensive
The insight I came up with is that if a node n is the source of a diamond, then we must let-bind the sink of the diamond immediately before inlining the definition of n.
This gives rise to the question of “how do we identify a diamond?” What we can do is give a mapping from each node to its reachable set of nodes. For example, in the above, we’d compute the map:
+ -> {+, f, a, g, expensive}
f -> {f, a, g, expensive}
a -> {a}
g -> {g, expensive}
expensive -> {expensive}
Then when we go to inline a node, say, +, we can look for any nodes that are reachable via more than one of its immediate subterms. Since the immediate subterms of + are f and g, we can take the intersections of their reachable sets:
{f, a, g, expensive} union {g, expensive}
giving us
{g, expensive}
which is exactly the set of nodes that we need to perform sharing on. If you topologically sort this set, it gives you the order that you should perform your let bindings.
EXCEPT there’s a kink in the whole thing. What happens if one of the terms in this diamond contains free variables? In particular, we might have something like this:
+
/ \
f |
| \ |
a \ /
λx
/ \
\ /
expensive
|
x
This gives us an analogous set of reachable nodes when we look at +, but we obviously can’t lift expensive x above the lambda.
Resolving this problem required giving up on the notion of memoizing the entire reachable set of nodes, and to instead crawl the graph ensuring that everything is well-scoped.
Performance Woes
My algorithm looked fine, and, importantly, got the right answer in a reasonable amount of time on our (small) battery of integration tests. So I shipped it, commended myself on a job well done, and thought nothing more about it. For about a week, until a bug report came in saying that our compiler now seemed to hang on big programs.
Which was something I hadn’t noticed, since we didn’t have any big programs in our integration tests.
Damn!
Upon digging in to what exactly was so slow, I noticed that my algorithm was accidentally quadratic. I needed to fold over every node in the graph, and that required looking at the entire reachable set underneath it. I had put in some of the obvious safeguards, hoping that they would prune the search tree early, but it wasn’t enough sacrifice for the Great God of Asymptotes.
Did I mention that at this point in the story, having this algorithm working fast was on the critical path of the company? Everybody else was blocked on me figuring this out. Talk about pressure!
Anyway. You’ll notice above that in my description of the algorithm, everything sounds fine. But the juice is in the details, as the common saying goes. Computing reachability isn’t quite the right thing to be using here, as it gave us the wrong answer for the lambda example above. Which is unfortunate because reachability is something we can do in linear time.
And then when reachability didn’t work, I just threw away the fast performance and hoped my bespoke algorithm would do the job. My only redemption comes from the fact that at least it got the right answer, even if it did so very slowly.
Finding the Kernel
Back to the drawing board.
Whenever I have graph theory problems, I call up my boy Vikrem. He’s good at nerd stuff like this.
We rubberducked the problem, and tried to reframe the problem in the language of graph theory. We had a Merkiv–Maguire moment where we indepdently realized that the goal was somehow related to finding the lowest common ancestor (LCA) of a node.
Which is to say, roughly, that we are looking for forks in the diamond diagram. Which we already knew, but it was nice to have some language for.
Our new problem is that LCA is defined only over trees. There are some extensions to DAGs, but none of them seem to be particularly well founded. However, searching for exactly that brought me to this stackoverflow question, where nestled in the comments is someone suggesting that the poster isn’t looking for LCA, but instead for a related notion the lowest single common ancestor. LSCA is defined in a 2010 paper New common ancestor problems in trees and directed acyclic graphs.
The standard definition of LCA(x, y) = l is that “l is an ancestor of x and of y, and that no descendent of l has this property.”
But the definition of LSCA(x, y) = l is that “l lies on all root-to-x paths, and that l lies on all root-to-y paths, and that no descendent of l has this property.”
The distinction between the two is easily seen in the following graph:
0
/ \
1 2
| X |
3 4
Under the standard definition, LCA is not uniquely defined for DAGs. That is, LCA(3, 4) = {1, 2}. But neither 1 nor 2 lies on all paths from the root. Under LSCA therefore we get LSCA(3, 4) = 0, which is the obviously-correct place to let-bind 3 and 4.
The paper gives a preprocessing scheme for computing LSCA by building a “lowest single ancestor” (LSA) tree. The LSA of a node is the LSCA of all of its in-edges. This definition cashes out to mean “the most immediate diamond above any node.” Finally! This is exactly what we’re looking for, since this is where we must insert our let-bindings! Even better, the paper gives us an algorithm for computing the LSA tree in linear time!
The First Implementer
Of course, I’m lazy and would prefer not to implement this thing. So instead I searched on hackage for lsca, and found nothing. But then I searched for lca and found that, like always, Ed Kmett was 13 years ahead of me.
The lca package implements an \(O(log n)\) algorithm for computing the LCA of any two nodes in a graph. Which is very convenient for me, since the LSCA algorithm requires being able to do this.
Time to roll up the sleeves and get cracking I suppose.
The paper was surprisingly straightforward, and my first attempt implemented the (imperative) algorithms as given (imperatively.) The first step is to do a topological sort on the DAG in order to know in which order one ought to unfold the LSA tree.
But as is so often the case, this topological sort isn’t actually relevant to the algorithm; it’s just an encoding detail of expressing the algorithm imperatively. But you don’t need that when you’ve got laziness on your side! Instead you can just tie the know and do something cool like this:
lsaTree ::Ord v =>Map v (Set v) ->Map v (Path v)lsaTree input = fix $ \result -> M.fromList $do (node, parents) <- M.toList inputlet parentResults =fmap (result M.!) parents...
Notice how we use fix to bind the eventual result of the final computation. Then we can chase pointers by looking them up in result—even though it’s not yet “computed.” Who cares what order the computer does it in. Why is that a thing I should need to specify?
Anyway. The exact details of implementing LSA are not particularly important for the remainder of this blog post. If you’re interested, you can peep the PR, which is delightfully small.
Tying It All Back Together
Equipped with my LSA tree, I was now ready to go back and solve the original problem of figuring out where to stick let-bindings. It’s easy now. Given the original program graph, find the LSA for each node. The LSA is the place you should insert the let binding.
So given the map of nodes to their LSAs, invert that map and get back a map of nodes to descendents who have this node as an LSA. Now when you go to inline a node, just look up everything in this map and inline it first.
It turns out to be a very elegant solution. It’s one third of the length of my horrible ad-hoc implementations, and it runs in linear time of the number of nodes in the graph. All in all, very good.
More often than I’m comfortable about, people will ask me how I can have so many good ideas. And what I like about this story is that it’s pretty typical of how I actually “have” “good” ideas. I’m reminded of the fact that luck favors the prepared mind. Attentive readers will notice that none of this process was due to brilliance on my part. I happened to know Vikrem who’s a genius. Together we pulled at some ancient graph theory strings and remembered a fact that someone else had thought important to teach us. That wasn’t actually the right path, but it lead us to stackoverflow where someone had linked to a relevant paper. I implemented the paper using a library that someone else had done the heavy lifting on, and simplified the implementation using this knot-tying trick I picked up somewhere along the way.
Also, I’m just really pleased that the solution came from trying to reverse engineer the relevant graph-theory search terms. Maybe that’s the actual takeaway here.
A few days ago I got angry at xargs for the hundredth time, because
for me xargs is one of those "then he had two problems" technologies.
It never does what I want by default and I can never remember how to
use it. This time what I wanted wasn't complicated: I had a bunch of
PDF documents in /tmp and I wanted to use GPG to encrypt some of
them, something like this:
gpg -ac $(ls *.pdf | menupick)
menupick
is a lovely little utility that reads lines from standard input,
presents a menu, prompts on the terminal for a selection from the
items, and then prints the selection to standard output. Anyway, this
didn't work because some of the filenames I wanted had spaces in them,
and the shell sucks. Also because
gpg probably only does one file at a time.
I could have done it this way:
ls *.pdf | menupick | while read f; do gpg -ac "$f"; done
but that's a lot to type. I thought “aha, I'll use xargs.” Then I
had two problems.
ls *.pdf | menupick | xargs gpg -ac
This doesn't work because xargs wants to batch up the inputs to run
as few instances of gpg as possible, and gpg only does one file at
a time. I glanced at the xargs manual looking for the "one at a
time please" option (which should have been the default) but I didn't
see it amongst the forest of other options.
I think now that I needed -n 1 but I didn't find it immediately, and
I was tired of looking it up every time when it was what I wanted
every time. After many years of not remembering how to get xargs to
do what I wanted, I decided the time had come to write a stripped-down
replacement that just did what I wanted and nothing else.
(In hindsight I should perhaps have looked to see if gpg's
--multifile option did what I wanted, but it's okay that I didn't,
this solution is more general and I will use it over and over in
coming years.)
xar is a worse version of xargs, but worse is better (for me)
First I wrote a comment that specified the scope of the project:
# Version of xargs that will be easier to use
#
# 1. Replace each % with the filename, if there are any
# 2. Otherwise put the filename at the end of the line
# 3. Run one command per argument unless there is (some flag)
# 4. On error, continue anyway
# 5. Need -0 flag to allow NUL-termination
There! It will do one thing well, as Brian and Rob commanded us in
the Beginning Times.
I wrote a draft implementation that did not even do all those things,
just items 2 and 4, then I fleshed it out with item 1. I decided that
I would postpone 3 and 5 until I needed them. (5 at least isn't a
YAGNI, because I know I have needed it in the past.)
The result was this:
import subprocess
import sys
def command_has_percent(command):
for word in command:
if "%" in word:
return True
return False
def substitute_percents(target, replacement):
return [ s.replace("%", replacement) for s in target ]
def run_command_with_filename(command_template, filename):
command = command_template.copy()
if not command_has_percent(command):
command.append("%")
res = subprocess.run(substitute_percents(command, filename), check=False)
return res.returncode == 0
if __name__ == '__main__':
template = sys.argv[1:]
ok = True
for line in sys.stdin:
if line.endswith("\n"):
line = line[:-1]
if not run_command_with_filename(template, line):
ok = False
exit(0 if ok else 1)
Short, clean, simple, easy to use. I called it xar, ran
ls *.pdf | menupick | xar gpg -ac
and was content.
Now again, with Claude
The following day I thought this would be the perfect opportunity to
try getting some LLM help with programming. I already had a baseline
version of xar working, and had thought through the problem
specification. Now I could do it over with the LLM and compare the
two results. The program I wanted was small and self-contained. If
the LLM sabotaged me somehow, I would probably notice, and if I
didn't, it wouldn't matter, because I had a working version
already.
So I asked Claude about it. I knew Claude was not the right choice,
because Claude is not really designed for this task, and has no
integration with the coding editor, but Claude is what I am familiar
with so I thought I'd give it a shot.
On the whole it went extremely well.
The complete transcript is here. I
imagine this was pretty much a maximally good experience, that all the
terrible problems of LLM coding arise in larger, more complicated
systems. But this program is small and self-contained, with no
subtleties, so the LLM could be at its best.
Phase 1
First I just told Claude what I wanted and let it write the program
from scratch.
Let's write a program in Python called xar. It will be a
stripped-down version of the standard Unix xargs utility, but it
will do less and be simpler to use.
It will get command-line
arguments that specify a command template. Then it will read lines
from stdin. Each line will be appended to the command template to
produce a command. The program will run the resulting command, one
command for each line of input.
Any questions?
Default Claude is very chatty and I am going to omit most of its
chatty comments, which make me impatient.
The program it wrote it was not what I would have written, but it was
good enough. If I had just used it right off the bat, instead of
writing my own, it would have been enough, and it would have taken
somewhere between 2% and 10% as long to produce.
So the one-line summary of this article is: I should stop writing
simple command-line utilities, and just have Claude write them
instead, because Claude's are good enough, and definitely better than
yak-shaving.
Here's Claude's initial version:
#!/usr/bin/env python3
import sys
import subprocess
import shlex
def main():
if len(sys.argv) < 2:
print("Usage: xar command [args...]", file=sys.stderr)
sys.exit(1)
command_template = sys.argv[1:]
for line in sys.stdin:
line = line.rstrip('\n')
full_command = command_template + [line]
try:
result = subprocess.run(full_command, check=True)
except subprocess.CalledProcessError as e:
print(f"Command failed with exit code {e.returncode}: {' '.join(full_command)}",
file=sys.stderr)
except FileNotFoundError:
print(f"Command not found: {full_command[0]}", file=sys.stderr)
sys.exit(1)
if __name__ == "__main__":
main()
Claude's version had numerous comments, which I have omitted. I later
told it to stop putting in comments, which it did.
Claude's use of check here was not what I wanted, because that makes
subprocess.run raise an exception when the subcommand fails, and
then the exception has to be immediately caught and handled. My
original control flow had been simpler:
res = subprocess.run(substitute_percents(command, filename), check=False)
return res.returncode == 0
…
if not run_command_with_filename(template, line):
ok = False
Claude's program pulled in shlex without noticing that it was
unused. But Claude did teach me about str.rstrip('\n') which I had
not known about before (or maybe had forgotten), so that was one small
win already.
Argument parsing
The next step was a big win. Python's library for command-line
argument handling is called argparse and it is really nice. If I
were still writing programs in Perl, I would implement a Perl version
of argparse because Perl has 29 available argument parsing libraries
and they are all loathsome. The one problem with argparse is I never
remember off the top of my head how to use it. I think the module is
called argparse but it provides a class called Argparser but I
often get these backward and try to use argparser and Argparse.
Instead of figuring it out every time I usually dig up some previous
Python program and then copy-paste the argument parser from there,
amending it to suit the purpose.
But this time I didn't have to do that. Instead, I just said to
Claude:
This is good, now please add code at the top to handle argument
parsing with the standard Argparse library, even though there are
no options yet.
Claude handed me back pretty much the same program, but with the
argument parser at the top:
def parse_arguments():
"""Parse command-line arguments using argparse."""
parser = argparse.ArgumentParser(
description="A simplified version of xargs that appends each line from stdin to a command template.",
prog="xar"
)
# Store all positional arguments as the command to execute
parser.add_argument(
"command",
nargs="+",
help="The command template to execute for each input line"
)
return parser.parse_args()
Let's pause for a moment. Maybe you kids are unimpressed by this.
But if someone had sent this interaction back in time and showed it to
me even as late as five years ago, I would have been stunned. It
would have been completely science-fictional. The code that it
produced is not so impressive, maybe, but that I was able to get it in
response to an unstructured request in plain English is like seeing a
nuclear-powered jetpack in action.
I make some changes
It bothered me that the command-line arguments were being taken
implicitly from sys.argv; that makes the program hard to test. A
better way to do this is to have the list of command-line arguments be a
function to the main function, and then for the runtime to call
main(sys.argv[1:]) instead of just main(). That way you can test
the program without having to fake up sys.argv. So I changed that,
and while I was making changes I got rid of Claude's comments and the
unnecessary exception handling. I pasted the result back at Claude,
saying:
Good so far. I have adjusted your source code in several ways:
I removed the comments.
I removed the exception handling for the subcommand. If one of the subcommands fails, that no longer terminates the main loop
Command-line arguments are passed to the main() function rather than being accessed directly
Here is the code. Tell me if you see any problems or if you have any questions about it
I had also sabotaged the code with at least one likely error.
Python's subprocess.run returns a “result” structure that includes
the subcommand's output and exit status, and I can never remember the
name of the component that records the exit status. I either look it
up or get it by trial-and-error. This time I didn't bother to either,
I just guessed that it was result.exitcode, which I handed off to
Claude even though I thought was probably wrong.
Claude's reply said:
Here are a couple of issues I noticed:
There's an error in the error handling code - you're using result.exitcode but in the subprocess.run() function, the attribute is called returncode not exitcode
Awesome, now I don't have to remember that name of that component.
Claude continued:
The parse_arguments() function definition is missing in the pasted code, but it would need to be updated to accept the argv parameter
I had passed sys.argv[1:] to main, and then had main call parse_arguments(argv).
But I hadn't changed parse_arguments to expect an argument list. (I
hadn't included it in my paste at all, in fact.) Claude was exactly
right, and continued with the corrected version:
#!/usr/bin/env python3
import sys
import subprocess
import argparse
def parse_arguments(argv):
…
return parser.parse_args(argv)
def main(argv):
# Parse command-line arguments
args = parse_arguments(argv)
…
if __name__ == "__main__":
ok = main(sys.argv[1:])
exit(0 if ok else 1)
Handling NUL-separated values
I said
It's great, thanks for catching those errors. Now let's add a -0 command-line option that tells the program to split the input on "\0" characters instead of at line breaks.
Recall that my handwritten xar did not do this yet; I had left that
feature for another day. Claude just added it for me, adding
parser.add_argument(
"-0", "--null",
action="store_true",
help="Input items are terminated by a null character instead of by newline"
)
to the argument parser and changing the main loop to:
# Split input based on the delimiter option
if args.null:
# Split on null characters
items = stdin_data.split('\0')
else:
# Split on newlines
items = stdin_data.splitlines()
for item in items:
if not item:
continue
…
I was tired of Claude's comments, so I said
Give me the same thing, with no comments
which it did, so I said
From now on leave out the comments. I'm a Python expert and I don't need them. If there is something I don't understand I will ask you.
Claude complied. If I were going to do use Claude again in the future
I would include that in the canned instructions that Claude is given
up front. Instead I will probably use a tool better-suited to
programming, and do whatever the analogous thing is.
Template filling
Now I told Claude to add the % feature:
I changed if not item to if item == "". Now let's make the following change:
If the command template includes any % characters, each of these should be replaced with the input item.
Otherwise, if there were no % characters, the input item should be appended to the end of the command as in the current version
Claude did this. It used an explicit loop instead of the list
comprehension that I had used (and preferred), but it did do it correctly:
for arg in command_template:
if '%' in arg:
has_placeholder = True
full_command.append(arg.replace('%', item))
else:
full_command.append(arg)
if not has_placeholder:
full_command.append(item)
Even without the list comprehension, I would have factored
out the common code:
for arg in command_template:
if '%' in arg:
has_placeholder = True
full_command.append(arg.replace('%', item))
if not has_placeholder:
full_command.append(item)
But I am not going to complain, my code is simpler but is
doing unnecessary work.
Claude also took my hint to change item == "" even though I didn't
explicitly tell it to change that.
At this point the main loop of the main function was 15 lines long,
because Claude had stuck all the %-processing inline. So I said:
Good, let's extract the command template processing into a subroutine.
It did this right, understanding correctly what code I was referring
to and extracting it into a subroutine called
process_command_template. More science fiction: I can say "command
template processing" and it guesses what I had in mind!
This cut the main loop to 7 lines. That worked so well I tried it
again:
Good, now let's extract the part of main that processes stdin into a subroutine that returns the items array
It pulled the correct code into a function called process_stdin. It
did not make the novice mistake of passing the entire args structure
to this function. In the caller it had process_stdin(args.null) and
inside of process_stdin this parameter was named
use_null_delimiter,
YAGNI?
At this point I was satisfied but I thought I might as well ask if it
should do something else before we concluded:
Can you think of any features I left out that would be useful enough
to warrant inclusion? Remember this program is supposed to be small
and easy to use, in contrast to the existing xargs which is very
complicated.
Claude had four suggestions:
A -p or --parallel option to run commands in parallel
A -n or --max-args option to specify the maximum number of items to pass per command
A simple -v or --verbose flag to show commands as they're executed
A way to replace the command's standard input with the item instead of adding it as an argument
All reasonable suggestions, nothing stupid. (It also supplied code for #3,
which I had not asked for and did not want, but as I said before,
default Claude is very chatty.)
Parallelization
I didn't want any of these, and I knew that #2–4 would be easy to add if I
did want any of them later. But #1 was harder. I've done code like
this in the past, where the program has a worker pool and runs a new
process whenever the worker pool isn't at capacity. It's not even that
hard. In Perl you can play a cute trick and use something like
$workers{spawn()} = 1 while delete $workers{wait()};
where the workers hash maps process IDs to dummy values. A child
exits, wait() awakens and returns the process ID of the completed
child, which is then deleted from the map, and the loop starts another
worker.
I wanted to see how Claude would do it, and the result was an even
bigger win than I had had previously, because Claude wrote this:
with concurrent.futures.ProcessPoolExecutor(max_workers=args.parallel) as executor:
futures = [executor.submit(execute_command, cmd, args.verbose) for cmd in commands]
for future in concurrent.futures.as_completed(futures):
success = future.result()
if not success:
ok = False
What's so great about this? What's great is that I hadn't known about
concurrent.futures or ProcessPoolExecutor. And while I might have
suspected that something like them existed, I didn't know what they
were called. But now I do know about them.
If someone had asked me to write the --parallel option, I would have
had to have this conversation with myself:
Python probably has something like this already. But how long will
it take me to track it down? And once I do, will the API
documentation be any good, or will it be spotty and incorrect? And
will there be only one module, or will there be three and I will
have to pick the right one? And having picked module F6, will I
find out an hour later that F6 is old and unmaintained and that
people will tell me “Oh, you should have used A1, it is the new
hotness, everyone knows that.”
When I put all that uncertainty on a balance, and weigh it
against the known costs of doing it myself, which one wins?
The right choice is: I should do the research, find the good module (A1, not
F6), and figure out how to use it.
But one of my biggest weaknesses as a programmer is that I too often
make the wrong choice in this situation. I think “oh, I've done this
before, it will be quicker to just do it myself”, and then I do and it
is.
Let me repeat, it is quicker to do it myself. But that is still
the wrong choice.
Maybe the thing I wrote would be sooner or smaller or faster or more technically
suitable to the project than the canned module would have been. But it
would only have been more be technically suitable today. If it
needed a new feature in the future it might have to be changed by
someone who had never seen it before, whereas the canned module could well
already have the needed feature ready to go, already documented, and
perhaps already familiar to whoever had to make the change. My
bespoke version would certainly be unfamiliar to every other
programmer on the project — including perhaps myself six months later — so would be the wrong thing to use.
I'm really good at hacking this stuff up. Which is a problem. It
makes me want to hack stuff up, even when I shouldn't.
Claude tips the balance strongly toward the correct side, which is
that I should use the prepackaged module that someone else wrote and
not hack something up.
And now I know about concurrent.futures.ProcessPoolExecutor! The
world is full of manuals, how can I decide which ones I should read?
One way is to read them all, which used to work back when I was
younger, but now I have more responsibilities and I don't have time to
read the entire Python library reference including all the useless
bits I will never use. But here's Claude pointing out to me that this
is something I need to know about, now, today, and I should read this one. That is
valuable knowledge.
Claude flubs
While testing the program I did
xar --parallel=2 bash -c '...'
which didn't work because the argparse library tried to parse the
-c as a flag to xar, when I wanted it to be a flag to bash.
This was my fault, and there is a standard way around it, which is
that I should have said
xar --parallel=2 -- bash -c '...'
which tells argparse to drop the -- and take everything after it
literally with no interpretation.
I thought I'd see what Claude had to say about this, and it dropped
the ball in a very LLM sort of way.
I asked:
When I did xar --parallel=2 bash -c '...' it misunderstood the
arguments. It thought I was trying to pass -c to xar and failed
with unrecognized arguments. What should I do about that?
Claude replied:
There are two main approaches to handle this:
Use the -- separator to indicate the end of arguments for xar: …
Modify your code to handle this case specifically by looking for a command terminator in the argument parser.
Here's how you could modify your code to implement the first
approach:
And then it followed with the same, unmodified code…
Which was correct, because -- is already handled by argparse, so
no change was actually required to implement the first approach! But
Claude's characterization of it as a modification was misleading,
wrong, and potentially very confusing.
I said:
I don't see what the difference is in that argument parsing code as compared to what I had before.
and Claude got into the LLM tail-chasing mode where it got more and
more confused:
You're right, I apologize for the confusion. There's no difference
in the argument parsing code I provided compared to what you had
before. The issue is with how argparse handles arguments by
default.
To fix the issue with xar --parallel=2 bash -c '...', you need to
actually implement support for the -- separator in your code…
# Find the index of '--' if it exists
try:
separator_index = argv.index('--')
xar_args = argv[:separator_index]
command_args = argv[separator_index+1:]
except ValueError: …
Ugh, all wrong in the worst way. The code probably works, but it is
completely unnecessary. Claude's claim that “you need to actually
implement support for the -- separator” is flat wrong. I pointed
this out and Claude got more confused. Oh well, nobody is perfect!
Lessons learned
A long time ago, when syntax-coloring editors were still new, I tried
one and didn't like it, then tried again a few years later and
discovered that I liked it better than I had before, and not for the
reasons that anyone had predicted or that I would have been able to
predict.
(I wrote an article about the surprising reasons to use the syntax coloring.)
This time also. As usual, an actual experiment produced unexpected
results, because the world is complicated and interesting. Some of
the results were unsurprising, but some were not anything I would have
thought of beforehand.
Claude's code is good enough, but it is not a magic oracle
Getting Claude to write most of the code was a lot faster and easier
than writing it myself. This is good! But I was dangerously tempted
to just take Claude's code at face value instead of checking it
carefully. I quickly got used to flying along at great speed, and it
was tough to force myself to slow down and be methodical, looking over
everything as carefully as I would if Claude were a real junior
programmer. It would be easy for me to lapse into bad habits,
especially if I were tired or ill. I will have to be wary.
Fortunately there is already a part of my brain trained to deal with
bright kids who lack experience, and I think perhaps that part of my brain
will be able to deal effectively with Claude.
I did not notice any mistakes on Claude's part — at least this time.
At one point my testing turned up what appeared to be a bug, but it
was not. The testing was still time well-spent.
Claude remembers the manual better than I do
Having Claude remember stuff for me, instead of rummaging the
manual, is great. Having Claude stub out an argument parser,
instead of copying one from somewhere else, was pure win.
Partway along I was writing a test script and I wanted to use that
Bash flag that tells Bash to quit early if any of the subcommands
fails. I can never remember what that flag is called. Normally I
would have hunted for it in one of my own shell scripts, or groveled
over the 378 options in the bash manual. This time I just asked in
plain English “What's the bash option that tells the script to abort
if a command fails?” Claude told me, and we went back to what we were
doing.
Claude can talk about code with me, at least small pieces
Claude easily does simple refactors. At least at this scale, it got
them right. I was not expecting this to work as well as it did.
When I told Claude to stop commenting every line, it did. I
wonder, if I had told it to use if not expr only for Boolean
expressions, would it have complied? Perhaps, at least for a
while.
When Claude wrote code I wasn't sure about, I asked it what it was
doing and at least once it explained correctly. Claude had written
parser.add_argument(
"-p", "--parallel",
nargs="?",
const=5,
type=int,
default=1,
help="Run up to N commands in parallel (default: 5)"
)
Wait, I said, I know what the const=5 is doing, that's so that if
you have --parallel with no number it defaults to 5. But what is
the --default doing here? I just asked Claude and it told me:
that's used if there is no --parallel flag at all.
This was much easier than it would have been for me to pick over
the argparse manual to figure out how to do this in the first
place.
More thoughts
On a different project, Claude might have done much worse. It might
have given wrong explanations, or written wrong code. I think that's
okay though. When I work with human programmers, they give wrong
explanations and write wrong code all the time. I'm used to it.
I don't know how well it will work for larger systems. Possibly pretty
well if I can keep the project sufficiently modular that it doesn't get
confused about cross-module interactions. But if the criticism is
“that LLM stuff doesn't work unless you keep the code extremely
modular” that's not much of a criticism. We all need more
encouragement to keep the code modular.
Programmers often write closely-coupled modules knowing that it is bad
and it will cause maintenance headaches down the line, knowing that the
problems will most likely be someone else's to deal with. But what if
writing closely-coupled modules had an immediate cost today, the cost
being that the LLM would be less helpful and more likely to mess up
today's code? Maybe programmers would be more careful about letting
that happen!
Will my programming skill atrophy?
Folks at Recurse Center were discussing this question.
I don't think it will. It will only atrophy if I let it. And I have a
pretty good track record of not letting it. The essence of
engineering is to pay attention to what I am doing and why, to try to
produce a solid product that satisifes complex constraints, to try
to spot problems and correct them. I am not going to stop doing
this. Perhaps the problems will be different ones than they were
before. That is all right.
Starting decades ago I have repeatedly told people
You cannot just paste code with no understanding of
what is going on and expect it to work.
That was true then without Claude and it is true now with Claude. Why
would I change my mind about this? How could Claude change it?
Will I lose anything from having Claude write that complex
parser.add_argument call for me? Perhaps if I had figured it out
on my own, on future occasions I would have remembered the const=5 and default=1
specifications and how they interacted. Perhaps.
But I suspect that I have figured it out on my own in the past, more
than once, and it didn't stick. I am happy with how it went this time.
After I got Claude's explanation, I checked its claimed behavior pretty
carefully with a stub program, as if I had been reviewing a
colleague's code that I wasn't sure about.
The biggest win Claude gave me was that I didn't know about this
ProcessPoolExecutor thing before, and now I do. That is going to
make me a better programmer. Now I know something about useful that
I didn't know before, and I have a pointer to documentation I know I
should study.
My skill at writing ad-hoc process pool managers might atrophy, but if
it does, that is good. I have already written too many ad-hoc
process pool managers. It was a bad habit, I should have stopped long
ago, and this will help me stop.
Conclusion
This works.
Perfectly? No, it's technology, technology never works perfectly.
Have you ever used a computer?
Will it introduce new problems? Probably, it's new technology, and
new technology always introduces new problems.
But is it better than what we had before? Definitely.
I still see some programmers turning up their noses at this technology
as if they were sure it was a silly fad that would burn itself out
once people came to their senses and saw what a terrible idea it was.
I think that is not going to happen, and those nose-turning-up people,
like the people who pointed out all the drawbacks and unknown-unknowns
of automobiles as compared to horse-drawn wagons, are going to look
increasingly foolish.
Suppose a centrifuge has slots, arranged in a circle around the
center, and we have test tubes we wish to place into the slots.
If the tubes are not arranged symmetrically around the center, the
centrifuge will explode.
(By "arranged symmetrically around the center, I mean that if the
center is at , then the sum of the positions of the tubes
must also be at .)
Let's consider the example of . Clearly we can arrange ,
, , or tubes symmetrically:
Equally clearly
we can't arrange only . Also it's easy to see we can do tubes if
and only if we can also do tubes, which rules out .
From now on I will write to mean the problem of balancing
tubes in a centrifuge with slots. So and are possible, and and are
not. And is solvable if and only if is.
It's perhaps a little surprising that is possible.
If you just ask this to someone out of nowhere they might
have a happy inspiration: “Oh, I'll just combine the solutions for
and , easy.” But that doesn't work because two groups
of the form and always overlap.
For example, if your group of is the
slots then you can't also have your group of be
, because slot already has a tube in it.
The
other balanced groups of are blocked in the same way. You
cannot solve the puzzle with ; you have to do as
below left.
The best way to approach this is to do , as below right.
This is easy,
since the triangle only blocks three of the six symmetric pairs.
Then you replace the holes with tubes and the tubes with holes to
turn into .
Given and , how can we decide whether the centrifuge can be
safely packed?
Clearly you can solve when is a multiple of , but the example
of (or ) shows this isn't a necessary condition.
A generalization of this is that is always solvable
if since you can easily
balance tubes at positions , then do another tubes one position over, and
so on. For example, to do you just put first four tubes
in slots and the next four one position over, in slots
.
An interesting counterexample is that the strategy for ,
where we did , cannot be extended to . One
would want to do , but there is no way to arrange the tubes
so that the group of doesn't conflict with the group of ,
which blocks one slot from every pair.
But we can see that this must be true without even considering the
geometry. is the reverse of , which
impossible: the only nontrivial divisors of are and
, so must be a sum of s and s, and is not.
You can't fit tubes when , but again the reason is
a bit tricky. When I looked at directly, I did a case analysis
to make sure that the -group and the -group would always
conflict. But again there was an easier was to see this: and
clearly won't work, as is not a sum of s and s.
I wonder if there's an example where both and are not obvious?
For , every works except and the always-impossible .
What's the answer in general? I don't know.
Addenda
20250502
Now I am amusing myself thinking about the perversity of a centrifuge
with a prime number of slots, say . If you use it at all, you must
fill every slot. I hope you like explosions!
While I did not explode any centrifuges in university chemistry, I did
once explode an expensive Liebig condenser.
Omar Antolín points out an important consideration I missed:
it may be necessary
to subtract polygons. Consider . This is obviously
possible since . But there is a more interesting
solution. We can add the pentagon to the
digons and to obtain the solution
$${0,5,6,10,12,18, 20, 24, 25}.$$
Then from this we can subtract the triangle to obtain $${5, 6, 12, 18, 24, 25},$$ a solution to
which is not a sum of regular polygons:
Thanks to Dave Long for pointing out a small but significant error,
which I have corrected.
The GHC developers are very pleased to announce the availability
of the final release for GHC 9.10.2. Binary distributions, source
distributions, and documentation are available at downloads.haskell.org and
via GHCup.
GHC 9.10.2 is a bug-fix release fixing over 50 issues of a variety of
severities and scopes, including:
Significantly improved performance when dynamically loading Haskell symbols (#23415).
Fixing a bug where the simplifier sometimes destroyed join points during float out, which could impact performance (#24768).
Reduced memory fragmentation in the non-moving GC’s segment allocator, improving resident set size by up to 26% for some applications (#24150).
Added new flags to control speculative evaluation (-fspec-eval and -fspec-eval-dictfun) to work around performance regressions (#25606).
Fixed several platform-specific issues, including segfaults with FFI on PowerPC (#23034) and improved code
generation for AArch64 with multiway branches now using jump tables (#19912)
And many more!
A full accounting of these fixes can be found in the release notes. As
always, GHC’s release status, including planned future releases, can be found on
the GHC Wiki status.
We would like to thank Well-Typed, Tweag I/O, Juspay, QBayLogic, Channable,
Serokell, SimSpace, the Haskell Foundation, and other anonymous contributors
whose on-going financial and in-kind support has facilitated GHC maintenance
and release management over the years. Finally, this release would not have
been possible without the hundreds of open-source contributors whose work
comprise this release.
As always, do give this release a try and open a ticket if you see
anything amiss.
At work I’ve been researching
how to improve the ergonomics of prompt engineering and I wanted to share
and open source some of what I’ve done. This initial post is about how
I’ve been experimenting with using bidirectional type inference
to streamline prompt chaining.
“Prompt chaining” is a prompt engineering technique that splits a
larger task/prompt into multiple smaller tasks/prompts which are chained
together using code. For example, instead of prompting a model to
generate a poem in one prompt like this:
Write a poem based off this idea:
${idea}
… by following this process:
First think through the form, stanza count, lines per stanza, and
rhyme scheme
Then choose a poetic style (tone, voice, and literary devices) based
on the poem’s form
Then write a complete poem based on that plan
… you can split it into smaller prompts, like this:
structure prompt:
Plan the structure of a new poem based on this idea
${idea}
Describe its form, stanza count, lines per stanza, and rhyme
scheme
style prompt:
Given this poem structure:
Form: ${structure.form}
Stanzas: ${structure.stanzaCount}
Lines per stanza: ${structure.linesPerStanza}
Rhyme scheme: ${structure.rhymeScheme}
Choose a poetic style: tone, voice, and literary devices to
emphasize
poem prompt:
Write a complete poem based on this idea:
${idea}
Structure:
Form: ${structure.form}
Stanzas: ${structure.stanzaCount}
Lines per stanza: ${structure.linesPerStanza}
Rhyme scheme: ${structure.rhymeScheme}
Style:
Tone: ${style.stone}
Voice: ${style.voice}
Literary Devices: ${style.literaryDevices}
Why might you want to do this?
to improve the quality of the results
Models perform better when working on more constrained subproblems.
Splitting a larger prompt into smaller prompts helps the model stay
focused at each step.
to introspect intermediate results
This comes in handy when you want to log, validate, or correct
intermediate results.
to perform actions in between prompts
You might want to take the output of one prompt, use that to call
some tool, then use the output of that tool to decide what the next
prompt should be, which you can’t do with a single prompt.
In other words, prompt chaining unlocks greater accuracy,
control, and flexibility for prompt engineering.
The problem
The main issue with prompt chaining is that it is a huge pain in the
ass; if you start do anything a little bit complicated you need to start
using structured outputs (i.e. JSON), which adds a whole lot of
boilerplate to the process:
you have to define the schema for each intermediate step of the
process
You typically do this by defining your data model in your host
programming language (e.g. a Pydantic model in Python) or directly
defining your JSON schema
You have to instruct the model to produce JSON and explain the
shape of the expected output
(Depending on the framework) you have to decode the JSON into
your data model
For small prompt chaining pipelines this isn’t too hard, but it
starts to get annoying to define all these schemas when you scale this
up to more sophisticated prompt chaining pipelines.
So as a thought experiment I wanted to create a research prototype
that handled all of that for you so that you didn’t need to specify any
schemas at all. In other words I wanted to build a programming language
that harnessed bidirectional type inference to perform
schema inference for prompts with structured JSON outputs.
Example
I’ll cut to the case by showing the above prompt chain written as a
program in this language:
let concatSep = https://raw.githubusercontent.com/Gabriella439/grace/refs/heads/main/prelude/text/concatSep.ffgletlines= concatSep "\n"let generatePoem idea =let structure = prompt { model:"gpt-4o" , text:lines [ "Plan the structure of a new poem based on this idea:" , "" , idea , "" , "Describe its form, stanza count, lines per stanza, and rhyme scheme." ] }let renderedStructure =lines [ "- Form: "+ structure.form , "- Stanzas: "+Real/show (structure.stanzaCount :Integer) , "- Lines per stanza: "+Real/show (structure.linesPerStanza :Integer) , "- Rhyme scheme: "+ structure.rhymeScheme ]let style = prompt { model:"gpt-4o" , text:lines [ "Given this poem structure:" , renderedStructure , "" , "Choose a poetic style: tone, voice, and literary devices to emphasize." ] }let renderedStyle =lines [ "- Tone: "+ style.tone , "- Voice: "+ style.voice , "- Literary Devices: "+ concatSep ", " style.literaryDevices ]let poem :Text= prompt { model:"gpt-4o" , text:lines [ "Write a complete poem based on this idea:" , "" , idea , "" , "Structure:" , renderedStructure , "" , "Style:" , renderedStyle ] }in { structure: structure, style: style, poem: poem }in generatePoem "The stillness of early morning"
If you run the above example, you might get an output like this:
{"structure":{"form":"Free verse with a sprinkle of rhyme","linesPerStanza":4,"rhymeScheme":"ABCB","stanzaCount":5},"style":{"literaryDevices":["imagery","alliteration","personification","enjambment"],"tone":"nostalgic","voice":"reflective"},"poem":"### The Stillness of Early Morning\n\n**I** \nIn the gentle rise of dawn's embrace, \nA whisper of winds through sleepy skies, \nThe world lies suspended in an echo \nOf yesterday's dreams, reluctantly saying goodbye.\n\n**II** \nSoft orange light graces fields of dew, \nCradling tenacious tendrils of the past. \nMorning breathes upon the waiting world \nWith promises that this serenity will last.\n\n**III** \nWaves of silence surge across the meadow, \nNature's symphony gently cradles rest. \nA chorus of leaves sings its soft refrain, \nComforted by morning's tender caress.\n\n**IV** \nThe silence dips to caress weary trees, \nWhose branches waltz to whispers soft and slow. \nEach tendril holds close the morning's intention, \nBefore the day's demands, the chaos, start to show.\n\n**V** \nEarly rays dance upon a tranquil sea, \nA canvas painted through a dreamy haze. \nTime sits by, savoring the gentle hush, \nAs the world awakens, stirs, and finally plays."}
… and if you’re curious what the formatted poem looks like:
The Stillness of Early
Morning
I
In the gentle rise of dawn’s embrace,
A whisper of winds through sleepy skies,
The world lies suspended in an echo
Of yesterday’s dreams, reluctantly saying goodbye.
II
Soft orange light graces fields of dew,
Cradling tenacious tendrils of the past.
Morning breathes upon the waiting world
With promises that this serenity will last.
III
Waves of silence surge across the meadow,
Nature’s symphony gently cradles rest.
A chorus of leaves sings its soft refrain,
Comforted by morning’s tender caress.
IV
The silence dips to caress weary trees,
Whose branches waltz to whispers soft and slow.
Each tendril holds close the morning’s intention,
Before the day’s demands, the chaos, start to show.
V
Early rays dance upon a tranquil sea,
A canvas painted through a dreamy haze.
Time sits by, savoring the gentle hush,
As the world awakens, stirs, and finally plays.
Type inference
The sample Grace program hardly specifies any types (mainly the final
expected type for the poem: Text). The reason
this works is because Grace supports bidirectional type
inference, which means that Grace can work backwards from how
intermediate results are used to infer their schemas.
I’ll illustrate this with a contrived Grace example:
let numbers = prompt{ text:"Give me two numbers" }in { x: numbers.x , y: numbers.y , sum: numbers.x + numbers.y :Integer }
… which might produce an output like this:
$ grace interpret ./numbers.ffg
{"x":7,"y":14,"sum":21}
When Grace analyzes this program the type checker works backwards
from this expression:
numbers.x + numbers.y :Integer
… and reasons about it like this:
the addition produces an Integer, therefore
numbers.x and numbers.y must also be
Integers
therefore numbers is a record with two fields,
x and y, both of which are
Integers
… or using Grace syntax, the inferred type of numbers
is: { x: Integer, y: Integer }
therefore the output of the prompt command must have
the same type
… and then Grace generates a JSON schema for the prompt which looks
like this:
Of course, you can specify types if you want (and they’re
more lightweight than schemas in traditional prompt chaining
frameworks). For example:
$ grace repl>>> prompt{ text:"Give me a first and last name" } : { first: Text, last: Text }{"first":"Emily", "last": "Johnson" }>>> prompt{ text:"Give me a list of names" } : List Text["Alice","Bob","Charlie","Diana","Ethan","Fiona","George","Hannah","Isaac","Jack"]
However in our original example we don’t need to specify intermediate
types because when the type-checker sees this code:
let structure = prompt { model:"gpt-4o" , text:lines [ "Plan the structure of a new poem based on this idea:" , "" , idea , "" , "Describe its form, stanza count, lines per stanza, and rhyme scheme." ] }let renderedStructure =lines [ "- Form: "+ structure.form , "- Stanzas: "+Real/show (structure.stanzaCount :Integer) , "- Lines per stanza: "+Real/show (structure.linesPerStanza :Integer) , "- Rhyme scheme: "+ structure.rhymeScheme ]
… the compiler can reason backwards from how the
structure value is used to infer that the JSON schema for
the prompt needs to be:
This doesn’t actually run any tools (I haven’t added any
callable tools to my work-in-progress branch yet), but just renders the
tool use as a string for now:
$ grace interpret ./tools.ffg
["curl https://api.example.com/data","ls -l -a"]
However, the idea is that you can model a tool as a sum type with one
constructor per callable tool, and in the above example the type checker
infers that the sum type representing one tool call is:
… but since we List/map the call function
over the output of the prompt the type checker infers that
the prompt needs to generate a List of tool
calls:
prompt{ text:"Call some tools" } :List<HttpRequest: …, ShellCommand: … >
… and then Grace does some magic under the hood to convert that type
to the equivalent JSON schema.
What’s particularly neat about this example is that the prompt is so
incredibly bare (“Call some tools”) because all the information the
model needs is present in the schema.
Schema-driven prompting
We can explore this idea of using the schema to drive the prompt
instead of prose using an example like this:
prompt{ text:"Generate some characters for a story", model:"gpt-4o" }:List { "The character's name":Text , "The most memorable thing about the character":Text , "The character's personal arc":Text }
[{"The character's name":"Aveline Thatcher","The character's personal arc":"Aveline starts as a skeptical journalist who doubts the stories of mythical creatures. Over time, she becomes a firm believer, risking her career to uncover the truth and protect these creatures.","The most memorable thing about the character":"The intricate tattoo of a phoenix on her forearm that seems to glow when she discovers hidden truths."},{"The character's name":"Kelan Frost","The character's personal arc":"A former rogue alchemist who turns hero after he inadvertently creates a dangerous substance. Driven by guilt, Kelan seeks redemption by finding an antidote and saving his village.","The most memorable thing about the character":"His iridescent blue eyes that seem to see into one's soul, a side effect of his alchemical experiments."},{"The character's name":"Luciana Blair","The character's personal arc":"Luciana is a reclusive artist who initially fears the world outside her home. After a mysterious vision rejuvenates her, she sets out on a journey of self-discovery, ultimately finding both her voice and courage.","The most memorable thing about the character":"Her ability to paint scenes before they happen, which she attributes to the visions she sees in her dreams."},{"The character's name":"Ezra Hartman","The character's personal arc":"Once a charismatic but self-centered lawyer, Ezra is confronted with a moral crisis that forces him to reevaluate his values. He chooses a path of integrity, becoming an advocate for justice.","The most memorable thing about the character":"His perfectly tailored suits that slowly become more casual, symbolizing his transformation and shifting priorities."},{"The character's name":"Seraphine Mora","The character's personal arc":"Seraphine is a young music prodigy who loses her hearing after an accident. Battling despair, she learns to embrace a new way of 'hearing' music through vibrations and her other senses.","The most memorable thing about the character":"The ethereal way she 'dances' with the music, using her entire body to express each note's emotion."}]
Grace is a superset of JSON and since JSON supports arbitrary field
names so does Grace! Field names in Grace support arbitrary
capitalization, punctuation, and whitespace as long as you quote them,
and we can use the field names to “smuggle” the description of each
field into the schema.
Conclusion
Hopefully this gives you some idea of why I’ve begun to think of
prompt chaining as a programming languages problem. Type inference is
just the beginning and I think it is possible to use a domain-specific
programming language not just to simplify the code but to ultimately
unlock greater reasoning power.
I’m going to continue to use Grace as a research vehicle for prompt
chaining but my LLM-enabled branch
of Grace (like Grace itself) is not really intended to be used in
production and I created it mainly as a proof-of-concept for where I’d
like prompt chaining frameworks to go. If I do end up eventually
productionizing this research I will create a proper fork with its own
name and the whole works.
In January, we presented our work on Explicit Level Imports at the
Trends in Functional Programming symposium.
We’re pleased to announce that the paper was awarded the the John McCarthy Prize for best paper overall!
The paper introduces the ExplicitLevelImports extension to GHC, which gives
programmers fine-grained control over which modules and dependencies are
required by Template Haskell. For instance, in the following example, the
splice import tells the compiler that Control.Lens.TH is needed only at compile
time and not at runtime:
import splice Control.Lens.TH (makeLenses)
import App (S)
data T = MkT { foo :: S }
$(makeLenses ''T )
By taking advantage of this extra information, the compiler can perform less work
in certain situations. In one benchmark, we modified pandoc to use ExplicitLevelImports and
compilation time was halved when using -fno-code. For full details, read the paper
or check out
GHC Proposal #682.
The implementation has landed in
GHC MR !14241,
so ExplicitLevelImports will be available in the next major release of GHC (9.14).
Explicit Level Imports
Matthew Pickering, Rodrigo Mesquita, Adam Gundry
TFP 2025 (PDF) (Awarded John McCarthy Prize for best paper overall)
Abstract. Cross-stage persistence rules are commonly admitted in multi-
stage programming languages. These rules codify the assumption that all
module and package dependencies are available at all stages. However,
in practice, only a small number of dependencies may be needed at each
particular stage.
This paper introduces Explicit Level Imports, a mechanism which gives
programmers precise control about which dependencies are required at
each stage. Imports are annotated with a modifier which brings identifiers
into scope at a specific level. This precision means it is straightforward
for the compiler to work out what is exactly needed at each stage, and
only provide that. The result is faster compilation times and the potential
for improved cross-compilation support.
We have implemented these ideas in GHC Haskell, consider a wide variety
of practical considerations in the design, and finally demonstrate that the
feature solves a real-world issue in a pragmatic way.
Google have stopped supporting the Chart API so all of the mathematics notation below is missing. There is a PDF version of this article at GitHub.
There are many introductions to the Expectation-Maximisation algorithm.
Unfortunately every one I could find uses arbitrary seeming tricks that seem to be plucked out of a hat by magic.
They can all be justified in retrospect, but I find it more useful to learn from reusable techniques that you can apply to further problems.
Examples of tricks I've seen used are:
Using Jensen's inequality. It's easy to find inequalities that apply in any situation. But there are often many ways to apply them. Why apply it to this way of writing this expression and not that one which is equal?
Substituting in the middle of an expression. Again, you can use just about anywhere. Why choose this at this time? Similarly I found derivations that insert a into an expression.
Majorisation-Minimisation. This is a great technique, but involves choosing a function that majorises another. There are so many ways to do this, it's hard to imagine any general purpose method that tells you how to narrow down the choice.
My goal is to fill in the details of one key step in the derivation of the EM algorithm in a way that makes it inevitable rather than arbitrary.
There's nothing original here, I'm merely expanding on a stackexchange answer.
Generalities about EM
The EM algorithm seeks to construct a maximum likelihood estimator (MLE) with a twist: there are some variables in the system that we can't observe.
First assume no hidden variables.
We assume there is a vector of parameters that defines some model.
We make some observations .
We have a probability density that depends on .
The likelihood of given the observations is .
The maximum likelhood estimator for is the choice of that maximises for the we have observed.
Now suppose there are also some variables that we didn't get to observe.
We assume a density .
We now have
where we sum over all possible values of .
The MLE approach says we now need to maximise
One of the things that is a challenge here is that the components of might be mixed up among the terms in the sum.
If, instead, each term only referred to its own unique block of , then the maximisation would be easier as we could maximise each term independently of the others.
Here's how we might move in that direction.
Consider instead the log-likelihood
Now imagine that by magic we could commute the logarithm with the sum.
We'd need to maximise
One reason this would be to our advantage is that often takes the form where is a simple function to optimise.
In addition, may break up as a sum of terms, each with its own block of 's.
Moving the logarithm inside the sum would give us something we could easily maximise term by term.
What's more, the for each is often a standard probability distribution whose likelihood we already know how to maximise.
But, of course, we can't just move that logarithm in.
Maximisation by proxy
Sometimes a function is too hard to optimise directly.
But if we have a guess for an optimum, we can replace our function with a proxy function that approximates it in the neighbourhood of our guess and optimise that instead.
That will give us a new guess and we can continue from there.
This is the basis of gradient descent.
Suppose is a differentiable function in a neighbourhood of .
Then around we have
We can try optimising with respect to within a neighbourhood of .
If we pick a small circular neighbourhood then the optimal value will be in the direction of steepest descent.
(Note that picking a circular neighbourhood is itself a somewhat arbitrary step,
but that's another story.)
For gradient descent we're choosing because it matches both the value and derivatives of at .
We could go further and optimise a proxy that shares second derivatives too, and that leads to methods based on Newton-Raphson iteration.
We want our logarithm of a sum to be a sum of logarithms.
But instead we'll settle for a proxy function that is a sum of logarithms.
We'll make the derivatives of the proxy match those of the original function
precisely so we're not making an arbitrary choice.
Write
The are constants we'll determine.
We want to match the derivatives on either side of the
at :
On the other hand we have
To achieve equality we want to make these expressions match.
We choose
Our desired proxy function is:
So the procedure is to take an estimated and obtain a new estimate
by optimising this proxy function with respect to .
This is the standard EM algorithm.
It turns out that this proxy has some other useful properties.
For example, because of the concavity of the logarithm,
the proxy is always smaller than the original likelihood.
This means that when we optimise it we never optimise ``too far''
and that progress optimising the proxy is always progress optimising the
original likelihood.
But I don't need to say anything about this as it's all part of the standard literature.
Afterword
As a side effect we have a general purpose optimisation algorithm that has nothing to do with statistics. If your goal is to compute
you can iterate, at each step computing
where is the previous iteration.
If the take a convenient form then this may turn out to be much easier.
Note
This was originally written as a PDF using LaTeX. It'll be available here for a while. Some fidelity was lost when converting it to HTML.
Google have stopped supporting the Chart API so all of the mathematics notation below is missing. There is a PDF version of this article at GitHub.
Preface
Functional programming encourages us to program without mutable state.
Instead we compose functions that can be viewed as state transformers.
It's a change of perspective that can have a big impact on how we reason about our code.
But it's also a change of perspective that can be useful in mathematics and I'd like to give an example: a really beautiful technique that alows you to sample from the infinite limit of a probability distribution without needing an infinite number of operations.
(Unless you're infinitely unlucky!)
Markov Chains
A Markov chain is a sequence of random states where each state is drawn from a random distribution that possibly depends on the previous state, but not on any earlier state.
So it is a sequence such that for all .
A basic example might be a model of the weather in which each day is either sunny or rainy but where it's more likely to be rainy (or sunny) if the previous day was rainy (or sunny).
(And to be technically correct: having information about two days or earlier doesn't help us if we know yesterday's weather.)
Like imperative code, this description is stateful.
The state at step depends on the state at step .
Probability is often easier to reason about when we work with independent identically drawn random variables and our aren't of this type.
But we can eliminate the state from our description using the same method used by functional programmers.
Let's choose a Markov chain to play with.
I'll pick one with 3 states called , and and with transition probabilities given by
where
Here's a diagram illustrating our states:
Implementation
First some imports:
> {-# LANGUAGE LambdaCase #-}
> {-# LANGUAGE TypeApplications #-}
> data ABC = A | B | C deriving (Eq, Show, Ord, Enum, Bounded)
We are now in a position to simulate our Markov chain.
First we need some random numbers drawn uniformly from [0, 1]:
> uniform :: (RandomGen gen, MonadState gen m) => m Double
> uniform = state random
And now the code to take a single step in the Markov chain:
> step :: (RandomGen gen, MonadState gen m) => ABC -> m ABC
> step A = do
> a <- uniform
> if a < 0.5
> then return A
> else return B
> step B = do
> a <- uniform
> if a < 1/3.0
> then return A
> else if a < 2/3.0
> then return B
> else return C
> step C = do
> a <- uniform
> if a < 0.5
> then return B
> else return C
Notice how the step function generates a new state at random in a way that depends on the previous state.
The m ABC in the type signature makes it clear that we are generating random states at each step.
We can simulate the effect of taking steps with a function like this:
> steps :: (RandomGen gen, MonadState gen m) => Int -> ABC -> m ABC
> steps 0 i = return i
> steps n i = do
> i <- steps (n-1) i
> step i
We can run for 100 steps, starting with , with a line like so:
*Main> evalState (steps 3 A) gen
B
The starting state of our random number generator is given by gen.
Consider the distribution of states after taking steps.
For Markov chains of this type, we know that as goes to infinity the distribution of the th state approaches a limiting "stationary" distribution.
There are frequently times when we want to sample from this final distribution.
For a Markov chain as simple as this example, you can solve exactly to find the limiting distribution.
But for real world problems this can be intractable.
Instead, a popular solution is to pick a large and hope it's large enough.
As gets larger the distribution gets closer to the limiting distribution.
And that's the problem I want to solve here - sampling from the limit.
It turns out that by thinking about random functions instead of random states we can actually sample from the limiting distribution exactly.
Some random functions
Here is a new version of our random step function:
> step' :: (RandomGen gen, MonadState gen m) => m (ABC -> ABC)
> step' = do
> a <- uniform
> return $ \case
> A -> if a < 0.5 then A else B
> B -> if a < 1/3.0
> then A
> else if a < 2/3.0 then B else C
> C -> if a < 0.5 then B else C
In many ways it's similar to the previous one.
But there's one very big difference: the type signature m (ABC -> ABC) tells us that it's returning a random function, not a random state.
We can simulate the result of taking 10 steps, say, by drawing 10 random functions, composing them, and applying the result to our initial state:
> steps' :: (RandomGen gen, MonadState gen m) => Int -> m (ABC -> ABC)
> steps' n = do
> fs <- replicateA n step'
> return $ foldr (flip (.)) id fs
Notice the use of flip.
We want to compose functions , each time composing on the left by the new .
This means that for a fixed seed gen, each time you increase by 1 you get the next step in a single simulation:
(BTW I used replicateA instead of replicateM to indicate that these are independent random draws.
It may be well known that you can use Applicative instead of Monad to indicate independence but I haven't seen it written down.)
*Main> [f A | n <- [0..10], let f = evalState (steps' n) gen]
[A,A,A,B,C,B,A,B,A,B,C]
When I first implemented this I accidentally forgot the flip.
So maybe you're wondering what effect removing the flip has?
The effect is about as close to a miracle as I've seen in mathematics.
It allows us to sample from the limiting distribution in a finite number of steps!
Here's the code:
> steps_from_past :: (RandomGen gen, MonadState gen m) => Int -> m (ABC -> ABC)
> steps_from_past n = do
> fs <- replicateA n step'
> return $ foldr (.) id fs
We end up building .
This is still a composition of independent identically distributed functions and so it's still drawing from exactly the same distribution as steps'.
Nonetheless, there is a difference: for a particular choice of seed, steps_from_past n no longer gives us a sequence of states from a Markov chain.
Running with argument draws a random composition of functions.
But if you increase by 1 you don't add a new step at the end.
Instead you effectively restart the Markov chain with a new first step generated by a new random seed.
Try it and see:
*Main> [f A | n <- [0..10], let f = evalState (steps_from_past n) gen]
[A, A, A, A, A, A, A, A, A, A]
Maybe that's surprising.
It seems to get stuck in one state.
In fact, we can try applying the resulting function to all three states.
*Main> [fmap f [A, B, C] | n <- [0..10], let f = evalState (steps_from_past n) gen]
[[A,B,C],[A,A,B],[A,A,A],[A,A,A],[A,A,A],[A,A,A],[A,A,A],[A,A,A],[A,A,A],[A,A,A],[A,A,A]]
In other words, for large enough we get the constant function.
Think of it this way:
If f isn't injective then it's possible that two states get collapsed to the same state.
If you keep picking random f's it's inevitable that you will eventually collapse down to the point where all arguments get mapped to the same state.
Once this happens, we'll get the same result no matter how large we take .
If we can detect this then we've found the limit of as goes to infinity.
But because we know composing forwards and composing backwards lead to draws from the same distribution, the limiting backward composition must actually be a draw from the same distribution as the limiting forward composition.
That flip can't change what probability distribution we're drawing from - just the dependence on the seed.
So the value the constant function takes is actually a draw from the limiting stationary distribution.
We can code this up:
> all_equal :: (Eq a) => [a] -> Bool
> all_equal [] = True
> all_equal [_] = True
> all_equal (a : as) = all (== a) as
> test_constant :: (Bounded a, Enum a, Eq a) => (a -> a) -> Bool
> test_constant f =
> all_equal $ map f $ enumFromTo minBound maxBound
This technique is called coupling from the past.
It's "coupling" because we've arranged that different starting points coalesce.
And it's "from the past" because we're essentially asking answering the question of what the outcome of a simulation would be if we started infinitely far in the past.
> couple_from_past :: (RandomGen gen, MonadState gen m, Enum a, Bounded a, Eq a) =>
> m (a -> a) -> (a -> a) -> m (a -> a)
> couple_from_past step f = do
> if test_constant f
> then return f
> else do
> f' <- step
> couple_from_past step (f . f')
We can now sample from the limiting distribution a million times, say:
*Main> let samples = map ($ A) $ evalState (replicateA 1000000 (couple_from_past step' id)) gen
We can now count how often A appears:
*Main> fromIntegral (length $ filter (== A) samples)/1000000
0.285748
That's a pretty good approximation to , the exact answer that can be found by finding the eigenvector of the transition matrix corresponding to an eigenvalue of 1.
> gen = mkStdGen 669
Notes
The technique of coupling from the past first appeared in a paper by Propp and Wilson.
The paper Iterated Random Functions by Persi Diaconis gave me a lot of insight into it.
Note that the code above is absolutely not how you'd implement this for real.
I wrote the code that way so that I could switch algorithm with the simple removal of a flip.
In fact, with some clever tricks you can make this method work with state spaces so large that you couldn't possibly hope to enumerate all starting states to detect if convergence has occurred.
Or even with uncountably large state spaces.
But I'll let you read the Propp-Wilson paper to find out how.
In those articles I showed how you could build up the Clifford algebras like so:
type Cliff1 = Complex R
type Cliff1' = Split R
type Cliff2 = Quaternion R
type Cliff2' = Matrix R
type Cliff3 = Quaternion Cliff1'
type Cliff3' = Matrix Cliff1
type Cliff4 = Quaternion Cliff2'
type Cliff4' = Matrix Cliff2
type Cliff5 = Quaternion Cliff3'
...
I used CliffN as the Clifford algebra for a negative definite inner product and
CliffN' for the positive definite case.
It's not a completely uniform sequence in the sense that CliffN is built from CliffN' for dimension two lower and you use a mix of Matrix and Quaternion.
The core principle making this work is that for type constructors implemented like Matrix, Quaternion etc. we have the property that
eg. Matrix (Quaternion Float) is effectively the same thing as Matrix FloatQuaternion Float.
But John Baez pointed out to me that you can build up the CliffN algebras much more simply enabling us to use these definitions:
> type Cliff1 = Complex Float
> type Cliff2 = Complex Cliff1
> type Cliff3 = Complex Cliff2
> type Cliff4 = Complex Cliff3
> type Cliff5 = Complex Cliff4
...
Or even better:
> type family Cliff (n :: Nat) :: * where
> Cliff 0 = Float
> Cliff n = Complex (Cliff (n - 1))
But there's one little catch.
We have to work, not with the tensor product, but the super tensor product.
We define Complex the same way as before:
> data Complex a = C a a deriving (Eq, Show)
Previously we used a definition of multiplication like this:
instance Num a => Num (Complex a) where
C a b * C c d = C (a * c - b * d) (a * d + b * c)
We can think of C a b in Complex R as representing the element .
The definition of multiplication in a tensor product of algebras is defined by .
So we have .
This means that line of code we wrote above defining * for Complex isn't simply a definition of multiplication of complex numbers, it says how to multiply in an algebra tensored with the complex numbers.
Let's go Super!
A superalgebra is an algebra graded by where is the ring of integers modulo 2.
What that means is that we have some algebra that can be broken down as a direct sum (the subscripts live in ) with the property that multiplication respects the grading, ie. if is in and is in then is in .
The elements of are called "even" (or bosonic) and those in "odd" (or fermionic). Often even elements commute with everything and odd elements anticommute with each other but this isn't always the case. (The superalgebra is said to be supercommutative when this happens. This is a common pattern: a thing X becomes a superX if it has odd and even parts and swapping two odd things introduces a sign flip.)
The super tensor product is much like the tensor product but it respects the grading.
This means that if is in and is in then is in .
From now on I'm using to mean super tensor product.
Multiplication in the super tensor product of two superalgebras and is now defined by the following modified rule:
if is in and is in then .
Note that the sign flip arises when we shuffle an odd left past an odd .
The neat fact that John pointed out to me is that
.
We have to modify our definition of * to take into account that sign flip.
I initially wrote a whole lot of code to define a superalgebra as a pair of algebras with four multiplication operations and it got a bit messy.
But I noticed that the only specifically superalgebraic operation I ever performed on an element of a superalgebra was negating the odd part of an element.
So I could define SuperAlgebra like so:
class SuperAlgebra a where
conjugation :: a -> a
where conjugation is the negation of the odd part.
(I'm not sure if this operation corresponds to what is usually called conjugation in this branch of mathematics.)
But there's a little efficiency optimization I want to write.
If I used the above definition, then later I'd often find myself computing a whole lot of negates in a row.
This means applying negate to many elements of large algebraic objects even
though any pair of them cancel each other's effect.
So I add a little flag to my conjugation function that is used to say we want an extra negate and we can
accumulate flips of a flag rather than flips of lots of elements.
> class SuperAlgebra a where
> conjugation :: Bool -> a -> a
Here's our first instance:
> instance SuperAlgebra Float where
> conjugation False x = x
> conjugation True x = negate x
This is saying that the conjugation is the identity on Float but if we
want to perform an extra flip we can set the flag to True.
Maybe I should call it conjugationWithOptionalExtraNegation.
And now comes the first bit of non-trivial superalgebra:
> instance (Num a, SuperAlgebra a) => SuperAlgebra (Complex a) where
> conjugation e (C a b) = C (conjugation e a) (conjugation (not e) b)
We consider to be even and to be odd. When we apply the conjugation to then we can just apply it directly to .
But that flips the "parity" of (because tensor product respects the grading) so we need to swap when we use the conjugation.
And that should explain why conjugation is defined the way it is.
Now we can use the modified rule for defined above:
> instance (Num a, SuperAlgebra a) => Num (Complex a) where
> fromInteger n = C (fromInteger n) 0
> C a b + C a' b' = C (a + a') (b + b')
> C a b * C c d = C (a * c - conjugation False b * d)
> (conjugation False a * d + b * c)
> negate (C a b) = C (negate a) (negate b)
> abs = undefined
> signum = undefined
For example, conjugation False is applied to the first on the RHS because implicitly represents an term and when expanding out the product we shuffle the (odd) in left of . It doesn't get applied to the second because and remain in the same order.
That's it!
Tests
I'll test it with some examples from Cliff3:
> class HasBasis a where
> e :: Integer -> a
> instance HasBasis Float where
> e = undefined
> instance (Num a, HasBasis a) => HasBasis (Complex a) where
> e 0 = C 0 1
> e n = C (e (n - 1)) 0
> make a b c d e f g h =
> C (C (C a b) (C c d))
> (C (C e f) (C g h))
The implementation of multiplication looks remarkably like it's the Cayley-Dickson construction.
It can't be (because iterating it three times gives you a non-associative algebra but the Clifford algebras are associative).
Nonetheless, I think comparison with Cayley-Dickson may be useful.
Efficiency
As mentioned above, before I realised I just needed the conjugation operation I wrote the above code with an explicit split of a superalgebra into two pieces intertwined by four multiplications.
I think the previous approach may have a big advantage - it may be possible to use variations on the well known "speed-up" of complex multiplication that uses three real multiplications instead of four.
This should lead to a fast implementation of Clifford algebras.
Also be warned: you can kill GHC if you turn on optimization and try to multiply elements of high-dimensional Clifford algebras.
I think it tries to inline absolutely everything and you end up with a block of code that grows exponentially with .
Note also that this code translates directly into many languages.
From my perspective, one of the biggest open problems in implementing
programming languages is how to add a type system to the language
without significantly complicating the implementation.
For example, in my tutorial Fall-from-Grace
implementation the type checker logic accounts for over half of the
code. In the following lines of code report I’ve highlighted the modules
responsible for type-checking with a ‡:
That’s 2684 lines of code (≈51%) just for type-checking (and believe
me: I tried very hard to simplify the type-checking code).
This is the reason why programming language implementers will be
pretty keen to just not implement a type-checker for their language, and
that’s how we end up with a proliferation of untyped programming
languages (e.g. Godot or Nix), or ones that end up with a type system
bolted on long after the fact (e.g. TypeScript or Python). You can see
why someone would be pretty tempted to skip implementing a type system
for their language (especially given that it’s an optional language
feature) if it’s going to balloon the size of their codebase.
So I’m extremely keen on implementing a “lean” type checker that has
a high power-to-weight ratio. I also believe that a compact type checker
is an important foundational step for functional programming to “go
viral” and displace imperative programming. This post outlines one
approach to this problem that I’ve been experimenting with1.
Unification
The thing that bloats the size of most type-checking implementations
is the need to track unification variables. These variables are
placeholders for storing as-yet-unknown information about something’s
type.
For example, when a functional programming language infers the type
of something like this Grace expression:
(λx → x) true
… the way it typically works is that it will infer the type of the
function (λx → x) which will be:
λx → x : α → α
… where α is a unification variable (an unsolved type).
So you can read the above type annotation as saying “the type of
λx → x is a function from some unknown input type
(α) to the same output type (α).
Then the type checker will infer the type of the function’s input
argument (true) which will be:
true :Bool
… and finally the type checker will combine those two pieces of
information and reason about the final type like this:
the input to the function (true) is a
Bool
therefore the function’s input type (α) must also be
Bool
therefore the function’s output type (α) must also be
Bool
therefore the entire expression’s type is Bool
… which gives the following conclusion of type inference:
(λx → x) true :Bool
However, managing unification variables like α is a lot
trickier than it sounds. There are multiple unification
algorithms/frameworks in the wild but the problem with all of them is
that you have to essentially implement a bespoke logic programming
language (with all of the complexity that entails). Like, geez, I’m
already implementing a programming language and I don’t want to have to
implement a logic programming language on top of that just to power my
type-checker.
So there are a couple of ways I’ve been brainstorming how to address
this problem and one idea I had was: what if we could get rid of
unification variables altogether?
Deleting unification
Alright, so this is the part of the post that requires some
familiarity/experience with implementing a type-checker. If you’re
somebody new to programming language theory then you can still keep
reading but this is where I have to assume some prior knowledge
otherwise this post will get way too long.
The basic idea is that you start from the “Complete and Easy”
bidirectional type checking algorithm which is a type checking
algorithm that does use unification variables2 but
is simpler than most type checking algorithms. The type checking rules
look like this (you can just gloss over them):
Now, delete all the rules involving unification variables. Yes, all
of them. That means that all of the type-checking judgments from Figures
9 and 10 are gone and also quite a few rules from Figure 11 disappear,
too.
Surprisingly, you can still type check a lot of code with what’s
left, but you lose two important type inference features if you do
this:
you can no longer infer the types of lambda arguments
you can no longer automatically instantiate polymorphic
code
… and I’ll dig into those two issues in more detail.
Inferring lambda argument
types
You lose the ability to infer the type of a function like this one
when you drop support for unification variables:
λx → x ==False
Normally, a type checker that supports unification can infer that the
above function has type Bool → Bool, but (in general) a
type checker can no longer infer that when you drop unification
variables from the implementation.
This loss is not too bad (in fact, it’s a pretty common
trade-off proposed in the bidirectional type checking literature)
because you can make up for it in a few ways (all of which are easy and
efficient to implement in a type checker):
You can allow the input type to be inferred if the lambda is
given an explicit type annotation, like this:
λx → x ==False:Bool → Bool
More generally, you can allow the input type to be inferred if the
lambda is checked against an expected type (and a type annotation is one
case, but not the only case, where a lambda is checked against an
expected type).
We’re going to lean on this pretty heavily because it’s pretty
reasonable to ask users to provide type annotations for function
definitions and also because there are many situations where we can
infer the expected type of a lambda expression from its immediate
context.
You can allow the user to explicitly supply the type of the
argument
… like this:
λ(x :Bool) → x ==False
This is how Dhall works,
although it’s not as ergonomic.
You can allow the input type to be inferred if the lambda is
applied to an argument
This is not that interesting, but I’m mentioning it for completeness.
The reason it’s not interesting is because you won’t often see
expressions of the form (λx → e) y in the wild, because
they can more idiomatically be rewritten as
let x = y in e.
Instantiating polymorphic
code
The bigger issue with dropping support for unification variables is:
all user-defined polymorphic functions now require explicit type
abstraction and explicit type application, which is a
major regression in the type system’s user experience.
For example, in a language with unification variables you can write
the polymorphic identity function as:
Most programmers do NOT want to program in a
language where they have to explicitly manipulate type variables in this
way. In particular, they really hate explicit type application. For
example, nobody wants to write:
map { x :Bool, … large record … } Bool (λr → r.x) rs
So we need to figure out some way to work around this limitation.
The trick
However, there is a solution that I believe gives a high
power-to-weight ratio, which I will refer to as “keyword” type
checking:
add a bunch of built-in functions
Specifically, add enough built-in functions to cover most use cases
where users would need a polymorphic function.
add special type-checking rules for those built-in functions when
they’re fully saturated with all of their arguments
These special-cased type-checking rules would not require unification
variables.
still require explicit type abstraction when these built-in
functions are not fully saturated
Alternatively, you can require that built-in polymorphic functions
are fully saturated with their arguments and make it a parsing error if
they’re not.
still require explicit type abstraction and explicit type
application for all user-defined (i.e. non-builtin) polymorphic
functions
optionally, turn these built-in functions into keywords or
language constructs
I’ll give a concrete example: the map function for
lists. In many functional programming languages this map
function is not a built-in function; rather it’s defined within the host
language as a function of the following type:
map: ∀(a b :Type) → (a → b) → List a → List b
What I’m proposing is that the map function would now
become a built-in function within the language and you would now apply a
special type-checking rule when the map function is fully
saturated:
Γ ⊢ xs ⇒ List a Γ ⊢ f ⇐ a → b
───────────────────────────────
Γ ⊢ map f xs ⇐ List b
In other words, we’re essentially treating the map
built-in function like a “keyword” in our language (when it’s fully
saturated). Just like a keyword, it’s a built-in language feature that
has special type-checking rules. Hell, you could even make it an actual
keyword or language construct (e.g. a list comprehension) instead of a
function call.
I would even argue that you should make each of these special-cased
builtin-functions a keyword or a language construct instead of a
function call (which is why I call this “keyword type checking” in the
first place). When viewed through this lens the restrictions that these
polymorphic built-in functions (A) are saturated with their arguments
and (B) have a special type checking judgment are no different than the
restrictions for ordinary keywords or language constructs (which also
must be saturated with their arguments and also require special type
checking judgments).
To make an analogy, in many functional programming languages the
if/then/else construct has this
same “keyword” status. You typically don’t implement it as a user-space
function of this type:
ifThenElse : ∀(a :Type) → Bool → a → a → a
Rather, you define if as a language construct and you
also add a special type-checking rule for if:
Γ ⊢ b ⇐ Bool Γ ⊢ x ⇒ a Γ ⊢ y ⇐ a
────────────────────────────────────
Γ ⊢ if b then x else y ⇒ a
… and what I’m proposing is essentially greatly exploding the number
of “keywords” in the implementation of the language by turning a whole
bunch of commonly-used polymorphic functions into built-in functions (or
keywords, or language constructs) that are given special type-checking
treatment.
For example, suppose the user were to create a polymorphic function
like this one:
let twice = λ(a :Type) → λ(x : a) → [ x, x ]in twice (ListBool) (twice Bool true)
That’s not very ergonomic to define and use, but we also can’t
reasonably expect our programming language to provide a
twice built-in function. However, our language could
provide a generally useful replicate builtin function (like
Haskell’s
replicate function):
replicate: ∀(a :Type) → Natural → a → List a
… with the following type-checking judgment:
Γ ⊢ n ⇐ Natural Γ ⊢ x ⇒ a
───────────────────────────
Γ ⊢ replicate n x ⇒ List a
… and then you would tell the user to use replicate
directly instead of defining their own twice function:
replicate2 (replicate2 true)
… and if the user were to ask you “How do I define a
twice synonym for replicate 2” you would just
tell them “Don’t do that. Use replicate 2 directly.”
Conclusion
This approach has the major upside that it’s much easier to implement
a large number of keywords than it is to implement a unification
algorithm, but there are other benefits to doing this, too!
It discourages complexity and fragmentation in user-space
code
Built-in polymorphic functions have an ergonomic advantage over
user-defined polymorphic functions because under this framework type
inference works better for built-in functions. This creates an ergonomic
incentive to stick to the “standard library” of built-in polymorphic
functions, which in turn promotes an opinionated coding style across all
code written in that language.
You might notice that this approach is somewhat similar in spirit to
how Go handles polymorphism which is to say: it doesn’t handle
user-defined polymorphic code well. For example, Go provides a few
built-in language features that support polymorphism (e.g. the
map data structure and for loops) but if users ask for any
sort of user-defined polymorphism then the maintainers tell them they’re
wrong for wanting that. The main difference here is that (unlike Go) we
do actually support user-defined polymorphism; it’s not forbidden, but
it is less ergonomic than sticking to the built-in utilities that
support polymorphism..
It improves error messages
When you special-case the type-checking logic you can also
special-case the error messages, too! With general-purpose unification
the error message can often be a bit divorced from the user’s intent,
but with “keyword type checking” the error message is not only more
local to the problem but it can also suggest highly-specific tips or
fixes appropriate for that built-in function (or keyword or language
construct).
It can in some cases more closely match the expectations of
imperative programmers
What I mean is: most programmers coming from an imperative and typed
background are used to languages where (most of the time) polymorphism
is “supported” via built-in language constructs and keywords and
user-defined polymorphism might be supported but considered “fancy”.
Leaning on polymorphism via keywords and language constructs would
actually make them more comfortable using polymorphism instead of trying
to teach them how to produce and consume user-defined polymorphic
functions.
For example, in a lot of imperative languages the idiomatic solution
for how to do anything with a list is “use a for loop” where you can
think of a for loop as a built-in keyword that supports polymorphic
code. The functional programming equivalent of “just use a for loop”
would be something like “just use a list comprehension” (where a list
comprehension is a “keyword” that supports polymorphic code that we can
give special type checking treatment).
That said, this approach is still more brittle than unification and
will require more type annotations in general. The goal here isn’t to
completely recover the full power of unification but rather to get
something that’s not too bad but significantly easier to
implement.
I think this “keyword type checking” can potentially occupy a “low
tech” point in the type checking design space for functional programming
languages that need to have efficient and compact implementations
(e.g. for ease of embedding). Also, this can potentially provide a
stop-gap solution for novice language implementers that want
some sort of a type system but they’re not willing to commit to
implementing a unification-based type system.
There’s also variation on this idea which Verity Scheel has been
exploring, which is to provide userland support for defining new
functions with special type-checking rules and there’s a post from her
outlining how to do that:
The other approach is to create
essentially an “ABNF for type checkers” that would let you write
type-checking judgments in a standard format that could generate the
corresponding type-checking code in multiple languages. That’s still a
work-in-progress, though.↩︎
I believe some people might take
issue with calling these unification variables because they consider
bidirectional type checking as a distinct framework from unification.
Moreover, in the original bidirectional type checking paper they’re
called “unsolved” variables rather than unification variables. However,
I feel that for the purpose of this post it’s still morally correct to
refer to these unsolved variables as unification variables since their
usage and complexity tradeoffs are essentially identical to unification
variables in traditional unification algorithms.↩︎
Since I was a child, I’ve been playing the French
Horn. I still play, and I
take it quite seriously. I’m lucky enough to play with some good
ensembles, and I perform many concerts each year.
When learning difficult music, I often practise with a
metronome. A metronome is a
device that clicks or beeps regularly. You can set how often it
clicks; for example you might set it to click 80 times a minute. The
tempo (or speed) of a piece of music is often specified by the
composer telling you how many beats per minute they want. This is
guidance and not sacrosanct: you don’t normally have to play at
exactly this tempo, and frequently music requires some implicit
variation of tempo for it to be successful. But it certainly is an
important piece of information from the composer, telling you at what
basic speed the piece should be going.
The problem with metronomes is that they can’t change their speed by
themselves. For complex music, the speed could be changing a lot, or,
maybe the number of beats per bar changes. This is annoying because it
means for some pieces of music you have to keep stopping playing,
readjust the metronome, and then continue on. There are also gradual
changes in tempo in music: a part of a piece might gently get faster
or slower. No metronome can cope with this: essentially, metronomes
know nothing about the piece of music you’re playing.
It does nearly all the musical things you would want it to. As this is
mainly a technical blog though, here I’ll focus on those aspects.
It could have been an app; I’ve built apps before. But the faff of
getting onto the play store, or the app store is just not worth
it. The development tools are heavyweight and annoying. Sending out
new versions requires approval processes, and you have to convince
people to install something before they can use it. So I wanted this
to be browser based. Also, modern web browsers are pretty amazing –
great features and well designed APIs. Yep, all the old APIs are
terrible and awful to work with, but everything that’s at all recent
is pretty great.
Drawing music in a browser is quite a challenge though. The way I’m
doing it is I’m building an SVG, client-side. This was the first thing
I started on: trying to figure out how to draw music in a browser, how
to be able to detect clicks, and make it all interactive. The client
side code is all generated from
TypeScript using the plain tsc to do
the translation to JavaScript. I can’t stand complex tool-chains, and
modern browsers are absolutely fine with loading modules (and you can
do some really nice things with import
maps
as we’ll see). I’m not even minimising the JavaScript: I’ve written
the server myself; the modules are sent over the wire gzipped and I
have correct cache-control settings using immutable and
“cache-busting”,
so minimising the source just makes debugging life harder for no real
gain.
A score is essentially a list of blocks. I’m using a
CRDT
(the fugue list
CRDT) to
allow local-first editing (and even offline editing). Dirty blocks get
sent over a websocket and stored on the server, using LMDB which is
all very normal for me.
The server has a neat part of its design: when you compile the server,
all the static assets are embedded into the binary, thus making it a
single self-contained executable. Now those assets (HTML, CSS, images,
JavaScript etc) are just normally named files, but they can also be Go
templates. When the server starts up, it works through these static
assets, building HTTP routes for them. The HTTP routes contain in
their paths the hashcode of the file – this is necessary for the
cache
busting. If
the asset is a template, the server knows how to run the template, and
critically, I provide a url function in the template engine so that
templates can get the URL of some other asset including its
hashcode. So this means that if some HTML file needs to link to some
CSS file, the HTML file as built into the server can be a template. At
start up, this template gets run, it can invoke this url function,
and it can find out the final URL of the CSS file. And of course this
URL now influences the hashcode of the HTML file itself. This also
plays very nicely with the
integrity
attribute you can put on all sorts of things these days.
So it all works out rather nicely: if you consider the tree of file A
importing files B and C, and file B imports file D, then if I make
some change to file D, then it means its hashcode changes, and so its
URL changes. This propagates up to file B, and from there to file A
(but no change to file C). So it’s safe to serve all these static
assets with immutable cache-control headers and rely on this lovely
hashcode chaining. All of this work is done once, each time the server
starts-up. And it’ll all explode and stop working if there’s ever a
cycle in the graph of file imports.
Now in practice, it seems that references between HTML, CSS, images,
or JavaScript don’t seem to create cycles – at least I’ve not had a
problem so far. But between JavaScript modules, it’s much more common,
as you’d likely expect. But here, import
maps
come to the rescue: in my TypeScript/JavaScript, I just import modules
normally. I have a function in the template engine which knows how to
generate an import-map of all my JavaScript modules, which gets
injected into the top HTML page. This import-map provides both the
rewriting of paths (to add hashcodes onto the paths), and also
provides the integrity
section. This
solves the problem of circular imports because it means the JavaScript
itself never needs to contain the hashcode of any module it
imports. Yet, if I change some JavaScript module, then its hashcode
changes, which means the import-map changes, and so again, the browser
is forced into correctly fetching the updated resource.
A couple of weekends ago, I was up visiting my parents and I wanted to
demonstrate this thing to them (they’re also musicians). They have
computers running Windows. I tried loading up a demo score, and it
just didn’t work. Their browsers were up to date. Nothing of note in
the server logs, so I opened up the browser console and found errors
from deserialisation of data coming over the websocket: it was
claiming the data was corrupted. I’d never seen this in my own
development and use.
Checking a few other things, and I spotted that the source HTML for
the page had had some additional <script> elements added to it:
something was injecting some JavaScript. And then the penny dropped:
this is MITM
behaviour by some shitty anti-virus software – in this case,
AVG. Some quick web searching, and yep, those products are also known
for dicking around with websocket traffic: if you’re sending binary
messages and you’re compressing the stream, it’s apparently quite
common that the anti-virus software intercepts the traffic, and then
screws up the framing leading your own code to face corrupted
data. Completely ridiculous.
In my case, disabling compression on the websocket was enough to
prevent the corruption, and I then established that even for big
scores, the initial load would be maybe 25kB of data over the
websocket, so not compressing it isn’t terrible.
What made me laugh though was this: the browser console was telling me
both about the corrupted data, and also about the fact the browser was
refusing to run some script, due to it violating
CSP
settings. It took me a moment to realise that the script that wasn’t
being run, was the script that the anti-virus software was injecting
into my pages! Now, import-maps can’t be external files, they have to
be inline in the HTML. But there’s no way I’m serving HTML pages with
a CSP header with script-src 'unsafe-inline'. Instead, I’m issuing a
script-src CSP header with 'self' and also the hashcode of
import-map itself.
What this says to the browser is that it can trust the import-map
(because hashing it will give a hashcode that matches the CSP header),
the import-map itself has its integrity for every JavaScript module it
needs to load, and the CSP headers also tell the browser that it’s OK
to load JavaScript modules from the same domain (this is the
'self'). But, this does not give permission for the browser to run
arbitrary bits of crap JavaScript that some awful anti-virus thing has
injected! So, by making use of CSP and import-maps, you can defeat
attackers from tampering with your website and code!
In my functional programming course to Master Students of
Telecom Nancy,
I like to use parsing as an example of monadic programming, relying on the
megaparsec library.
My only concern with megaparsec is that its official tutorial
is long: at the time I’m writing, it’s 15000 words long.
Unlike the official megaparsec tutorial, this blog post is intended to be smaller,
and is aimed at an audience with only a basic understanding of Haskell and monadic programming.
My running example is a parser for a domain-specific language that I designed for the class. This language uses primitive drawing commands to represent ASCII art roguelike maps. It looks like this:
Here, HLine x y len and VLine x y len draw horizontal and vertical walls respectively.
The Start x y command marks the player’s starting point and Cell x y ~ places special terrain.
Roguelike maps typically consist of rectangular rooms and connecting corridors, where
walls are shown as #, water as ~, and walkable spaces as dots (.)
For example, the snippet above draws a map with two connected rooms.
The room on the left contains the player’s start location (>), while some water appears in the lower right corner of the room on the right:
Walkable floor cells are omitted from the domain-specific language,
as they can be inferred by computing the set of cells reachable from the starting point.
In implementations of roguelikes, maps like this one
are translated into an array of arrays of symbols, with some symbols being
walkable (e.g. dot cells and water cells) and some symbols being blockers (walls).
The top-level array is then used to compute possible moves and collisions.
The Parsec monad
To use megaparsec, we define our main monad type using the Parsec e s a type.
It has three arguments:
The type of errors returned by the parser,
the type of stream accepted as input by the parser, and
the type of data returned upon successful parsing of an input stream.
For a simple parser, we define:
The error type to be Text, for simplicity. In a production parser, you would use a structured error type,
that distinguishes the different error cases; so that you can handle them differently.
The input stream to be Text, because this is the most idiomatic choice in the Haskell ecosystem:
import Data.Text(Text)import Text.MegaparsectypeError=TexttypeInput=Text-- | @Parser a@ is a parser that accepts @Text@ as input and returns an @a@ upon-- successful parsing.typeParsera=ParsecErrorInputa
Our first parser
Parsers are built from
primitive combinators (e.g. lookAhead, notFollowedBy, end of file eof)
and combinators derived from them (e.g. oneOf, anySingle, satisfy).
These combinators are designed to consume a few symbols, not complex structures (more on this later).
Combinators return parsers in any MonadParsec monad,
which means that they have a signature where the head is MonadParsec e s m => ...
and the return type is of the form m a1.
In our context, it suffices to know that m a is instantiated to Parser a, so we can
use these combinators for our parsers.
Let’s parse the different kinds of symbols we usually find in ASCII art roguelike maps,
using the anySingle function, which parses a single token. In our case, since the input type
is Text, the type of tokens is Char (see the ShareInput case of
Stream’s documentation,
as well as the instances of Stream):
-- | A symbol in the map of an ASCII roguelikedataSymbol=-- | A wall, depicted by a # characterWall|-- | A water cell, depicted by a ~ characterWaterderiving(Eq,Show)-- | A parser for the symbol of a single cell. Used in 'parseElement' below.parseSymbol::ParserSymbolparseSymbol=doc<-anySinglecasecof'#'->returnWall'~'->returnWater_->fail$"Unknown symbol: "<>[c]-- See below for how to avoid this case altogether (in parseLineElement)
Parser combinators
By virtue of MonadParsecs
being monads, parsers can be built using functions that are common in monadic Haskell code
(including functions from Functor, Applicative, etc.). Let’s demonstrate this
to build a parser for more advanced roguelike map constructs:
dataElement=-- | Horizontal wall, starting at @(x,y)@ with @length@ cells (ending at @(x+length-1,y)@)HorizontalLineIntIntInt|-- | Vertical wall, starting at @(x,y)@ with @length@ cells (ending at @(x,y+length-1)@)VerticalLineIntIntInt|-- | A cell at @(x,y)@ with a symbolCellIntIntSymbol|-- | The starting point of the playerStartIntIntderiving(Eq,Show)
The parser for the HorizontalLine and VerticalLine cases can be written as follows:
import Control.Monad(void)import Control.Monad.Extra(when)import Text.Megaparsec.Charimport Text.Megaparsec.Char.LexerparseLineElement::ParserElementparseLineElement=doconstructor<-choice[string"HLine">>returnHorizontalLine,string"VLine">>returnVerticalLine]space1-- One or more spacex<-decimalspace1y<-decimalspace1len<-decimalwhen(len<1)$fail$"Length must be greater than 0, but got "<>showlenreturn$constructorxylen
The first two lines either parse the string HLine or the string VLine and use the
choice
function to encode the two possibilities. Also, because each line in a do block encodes a step
in the computation, writing monadic parsers is natural: each line consumes some of the input,
until enough is consumed to return the desired value. Another example of using a regular monadic function
is to use when
to stop parsing when an incorrect value is consumed.
Running parsers
Since our parser takes Text as input, it can be tested in a pure context.
Megaparsec provides the runParser
function for this. To be able to print errors of our parser, our error type must be an instance of ShowErrorComponent;
and then we can define a convenient runMyParser function that returns either an error or the parsed value:
import Data.Text(pack,unpack)-- | Instance required for 'runMyParser'instanceShowErrorComponentErrorwhereshowErrorComponent=unpack-- | A variant of megaparsec's 'runParser', instantiated to our context.-- Successfully parses an @a@ or returns an error message.runMyParser::Parsera->Input->EitherTextarunMyParserparserinput=caserunParserparser""inputofLefterr->Left$pack$errorBundlePrettyerrRightx->Rightx
Parsing expressions, lists, etc.
Megaparsec not only provides building blocks for parsing tokens and combining parsers. It also provides
parsers for common constructs found in programming languages and domain-specific languages, such as expressions
and lists. Megaparsec does this by relying on the
parser-combinators
package.
I don’t want to go into the details of parsing expressions here (e.g. parsing 1 + 2 - 3…), but let me emphasize that it is
a bad idea to write your own expression parser. Instead, think about what kind of operators you need and encode them, using the
Operator
type.
List parsing, on the other hand, is done with various
sep… functions.
In our case of roguelike maps, we allow different elements to be separated by a semicolon, or by one or more newlines.
This is encoded as follows:
parseElements::Parser[Element]parseElements=parseElement`sepBy1`separatorwhereseparator=dohspace-- Optional horizontal (non-newline) spacechoice[void$char';',void$someeol]-- Either a single ';' or many newlineshspaceparseElement::ParseElementparseElement=choice[parseLineElement,parseStart,parseCell]whereparseStart=dovoid$string"Start"space1(x,y)<-parseCoordreturn$StartxyparseCell=dovoid$string"Cell"space1(x,y)<-parseCoordspace1symbol<-parseSymbolreturn$CellxysymbolparseCoord=dox<-decimalspace1y<-decimalreturn(x,y)
Conclusion
We’ve presented how to parse simple constructs using megaparsec and how to run our parsers.
This blog post is less than 1500 words long: mission accomplished presenting megaparsec in a shorter way than the official tutorial 🥳
New languages are coming out all the time, some experimental, some
industrial, others are purpose built for a specific domain. No single
language has the people-power or scope to try every cool new feature, so
a critical step in designing a new language is to observe how
experimental features have borne themselves out in practice.
As the saying goes, good [language designers] copy, great [language
designers] steal.
If you've heard anything about the Unison Language it's not a
surprise to you that it innovates in many areas. Unison very much tries
to reinvent Human-Compiler interactions for the 21st century, and in
that pursuit has spawned fully integrated ecosystem between the
compiler, codebase-manager, language server, version control and package
manager.
While some of these features are still too new to have proven their
worth (but we have our fingers crossed); there are aspects that I think
new languages should certainly consider as part of their designs.
A Fully
Interactive and Incremental Compiler
With the modern era of language servers and programming assistants,
developers greatly benefit from instant feedback on their work. With
traditional batch compilers it's all too tempting to go for a coffee, or
a walk, or a YouTube binge every time you kick off a big build. The
context-switching induced by switching tasks while compiling wastes
developer time by paging things in and out of their working memory, not
to mention: it just feels bad. After the build finishes, the
developer is left with a giant wall of text, sentenced to dig through a
large list of compiler errors trying to find some root-cause error in
the file they're working on.
Unison has a fully interactive compilation experience. The
language-server is typechecking your scratch-file on every keystroke
providing error feedback right in your editor, and offering helpful
information via hover-hints which use your codebase and typechecking
info to help you orient yourself. It can even partially typecheck the
file to suggest which types or operators you may want to fill into a
given slot.
Once you're happy with a chunk of code, you can check it in to the
codebase and it won't be compiled again unless you want to change it, or
an update is automatically propagated into it from a downstream
change.
While most languages won't adopt Unison's scratch-file and codebase
model; having an interactive compiler with good support for caching of
already-compiled-assets is a huge boon to productivity in any
language.
On the topic of the language server, Unison's language server is
built directly into the compiler. This ensures we avoid the awkward
disagreements between the LSP and compiler that sometimes happen in
other languages. It can also help to avoid duplicate work, many
languages are running the compiler independently and in their LSP at the
same time without sharing any of the work between them, causing
redundant work and a waste of precious resources.
Codebase API
It's the compiler's job to understand your code intimately. It knows
exactly how every definition is linked together, even if you don't! In
many languages it can be frustrating to know that this information
exists deep within the compiler, but not having any access to it
yourself!
Unison stores all your code as structured data within your codebase
and exposes the ability for you to ask it useful questions about your
code, exposing that precious understanding to you as a developer.
Unison allows searching by type, finding the dependencies of a
definition, or inverting that relationship to finding all definitions
which depend on a definition.
Via the UCM CLI you can use utilities like text.find to
search only string constants, or find to search only
definition names.
Some codebase data is provided via an API which is exposed from the
interactive UCM compiler, allowing developers to write tooling to
customize their workflow. For example, check out this VS
Code plugin someone wrote to view codebase definitions in the
sidebar. In other languages you'd typically need to write a scrappy
Regex or re-compile the code in a subprocess in order to achieve
something similar.
It doesn't have to be an API, it could be a parquet file or a SQLite
database or any number of things, the important part is that a language
exposes its one-true-source of information about the codebase in some
structured format for third-party tools to build upon.
Smart docs
It doesn't matter how great your language's package ecosystem is if
nobody can figure out how to use it! Documentation is critical for
helping end users understand and use functionality in your language, but
it has a fatal flaw: documentation isn't compiled and falls out of date
with the code.
In Unison, docs are a data-type within the language itself. This
means that docs can be generated dynamically by running Unison
code! We've leveraged this ability to enable embedding typechecked
runnable code examples into your docs. These examples are compiled
alongside the rest of your program, so they're guaranteed to be
kept up to date, and the outputs from your example code is run
and updated whenever the source definitions change.
You can also write code which generates documentation based
on your real application code. For example, you could write code which
crawls your web-server's implementation and collects all the routes and
parameters the server defines and displays them nicely as
documentation.
Unison goes one step further here by providing special support for
the documentation format on Unison Share, ensuring any definitions
mentioned in docs and code examples are hyper-linked to make for a
seamless package-browsing experience.
As an example of how far this can go, check out this
awesome project by community contributor Alvaro which generates
mermaid graphs in the docs representing the behaviour of simulations.
The graphs are generated from the same underlying library code so they
won't go out of date.
Get stealing
This subset of topics doesn't touch on Unison's ability system,
continuation capturing, or code serialization so I'll probably need at
least a part 2!
Hopefully you learned something 🤞! Did you know I'm currently writing a book? It's all about Lenses and Optics! It takes you all the way from beginner to optics-wizard and it's currently in early access! Consider supporting it, and more posts like this one by pledging on my Patreon page! It takes quite a bit of work to put
these things together, if I managed to teach your something or even just entertain you for a minute or two
maybe send a few bucks my way for a coffee? Cheers! �
In this episode Mike Sperber and Niki Vazou talk with Sandy Maguire, lead compiler engineer at Manifold Valley. They talk about the benefits of using Haskell of course, about all the books Sandy has written, on effects and the problem with monads, on combinator libraries and programming with laws.
The GHC developers are very pleased to announce the availability
of the release candidate for GHC 9.10.2. Binary distributions, source
distributions, and documentation are available at downloads.haskell.org and
via GHCup.
GHC 9.10.2 is a bug-fix release fixing over 50 issues of a variety of
severities and scopes. A full accounting of these fixes can be found in the
release notes. As always, GHC’s release status, including planned future
releases, can be found on the GHC Wiki status.
This release candidate will have a two-week testing period. If all goes well
the final release will be available the week of 1 May 2025.
We would like to thank Well-Typed, Tweag I/O, Juspay, QBayLogic, Channable,
Serokell, SimSpace, the Haskell Foundation, and other anonymous contributors
whose on-going financial and in-kind support has facilitated GHC maintenance
and release management over the years. Finally, this release would not have
been possible without the hundreds of open-source contributors whose work
comprise this release.
As always, do give this release a try and open a ticket if you see
anything amiss.
A few months ago, I announced that the GHC wasm
backend added support for Template Haskell and ghci. Initially, the
ghci feature only supported running code in nodejs and accessing the
nodejs context, and I’ve been asked a few times when ghci was going to work in
browsers in order to allow live-coding the frontend. Sure, why not? I promised
it in the last blog post’s wishlist. After all, GHCJS used to support
GHCJSi for browsers almost 10 years ago!
I was confident this could be done with moderate effort. Almost all the
pieces are already in place: the external interpreter logic in GHC is
there, and the wasm dynamic linker already works in nodejs. So just
make it runnable in browsers as well, add a bit of logic for
communicating with GHC and we’re done right? Well, it still took a few
months for me to land it…but finally here it is!
To keep this post within reasonable length, I will only introduce the
user-facing aspects of the wasm ghci browser mode and won’t cover the
underlying implementation. The rest of the post is an example ghci
session followed by a series of bite sized subsections, each covering
one important tip about using this feature.
How to use it
The ghc-wasm-meta repo provides user-facing
installation methods for the GHC wasm backend. Here we’ll go with the
simplest nix-based approach:
$ nix shell 'gitlab:haskell-wasm/ghc-wasm-meta?host=gitlab.haskell.org'
$ wasm32-wasi-ghc --interactive -fghci-browser
GHCi, version 9.12.2.20250327: https://www.haskell.org/ghc/ :? forhelp
Open http://127.0.0.1:38827/main.html or import http://127.0.0.1:38827/main.js to boot ghci
The -fghci-browser flag enables the browser mode. There are a couple
of other related flags which you can read about in the user
manual, but for now, let’s open that page to
proceed. You’ll see a blank page, but you can press F12 to open the
devtools panel and check the network monitor tab to see that it’s
sending a lot of requests and downloading a bunch of wasm modules.
Within a few seconds, the initial loading process should be complete,
and the ghci prompt should appear in the terminal and accept user
commands.
Let’s start with the simplest:
ghci>putStrLn"hello firefox"ghci>
The message is printed in the browser’s devtools console. That’s not
impressive, so let’s try something that only works in a browser:
The above code implements logic to export a Haskell IO () function
to a JavaScript synchronous callback that can be attached as a
button’s client event listener. Synchronous callbacks always attempt
to run Haskell computations to completion, which works fine as long as
the exported Haskell function’s main thread does not block indefinitely,
like waiting for an async JSFFI import to resolve or be rejected. You
can read more about JSFFI in the user manual, but let’s
carry on with this example:
Now, the button is attached to a simple counter in Haskell that prints
an incrementing integer to the console each time the button is
clicked. And that should be sufficient for a minimal demo! Now, there
are still a couple of important tips to be mentioned before we wrap up
this post:
Hot reloading
Just like native ghci, you can perform hot reloading:
ghci> :r
Ok, no modules to be reloaded.
ghci> btn
<interactive>:15:1: error: [GHC-88464]
Variable not in scope: btn
Reloading nukes all bindings in the current scope. But it doesn’t
magically undo all the side effects we’ve performed so far: if you
click on the button now, you’ll notice the counter is still working
and the exported Haskell function is still retained by the JavaScript
side! And this behavior is also consistent with native ghci:
hot-reloading does not actually wipe the Haskell heap, and there
exist tricks like foreign-store to persist values
across ghci reloads.
For the wasm ghci, things like foreign-store should work, though you
can allocate a stable pointer and print it, then reconstruct the
stable pointer and dereference it after a future reload. Since wasm
ghci runs in a JavaScript runtime after all, you can also cook your
global variable by assigning to globalThis. Or locate the element
and fetch its event handler, it should be the same Haskell callback
exported earlier which can be freed by freeJSVal.
So, when you do live-coding that involve some non-trivial back and
forth calling between JavaScript and Haskell, don’t forget that hot
reloads don’t kill old code and you need to implement your own logic
to disable earlier callbacks to prevent inconsistent behavior.
Loading object code
The wasm ghci supports loading GHC bytecode and object code. All the
code you type into the interactive session is compiled to bytecode.
The code that you put in a .hs source file and load via command line
or :l commands can be compiled as object code if you pass
-fobject-code to ghci.
I fixed the ghci debugger for all 32-bit cross targets
since the last blog post. Just like native ghci, debugger features
like breakpoints now work for bytecode. If you don’t use the ghci
debugger, it’s recommended that you use -fobject-code to load
Haskell modules, since object code is faster and more robust at
run-time.
Interrupting via ^C
My GHC patch that landed the ghci browser
mode also fixed a previous bug in wasm ghci: ^C was not handled at all
and would kill the ghci session. Now, the behavior should be
consistent with native ghci. With or without -fghci-browser, if
you’re running a long computation and you press ^C, an async exception
should interrupt the computation and unblock the ghci prompt.
Read the :doc, Luke
Among the many changes I landed in GHC since last blog post, one of
them is adding proper haddock documentation to all user-facing things
exported by GHC.Wasm.Prim. Apart from the GHC user manual, the
haddock documentation is also worth reading for users. I haven’t set
up a static site to serve the haddock pages yet, but they are already
accessible in ghci via the :doc command. Just try import GHC.Wasm.Prim and check :doc JSVal or :doc freeJSVal, then you
can read them in plain text.
As the Haskell wasm user community grows, so will the frustration with
lack of proper documentation. I’m slowly improving that. What you see
in :doc will continue to be polished, same for the user manual.
Importing an npm library in ghci
You can use JavaScript’s dynamic import() function as an async JSFFI
import. If you want to import an npm library in a ghci session, the
simplest approach is using a service like esm.run which
serves pre-bundled npm libraries as ES modules over a CDN.
If you have a local npm project and want to use the code there, you
need to do your own bundling and start your own development server
that serves a page to make that code somehow accessible (e.g. via
globalThis bindings). But how does that interact with the wasm ghci?
Read on.
Using ghci to debug other websites
The browser mode works by starting a local HTTP server that serves
some requests to be made from the browser side. For convenience, that
HTTP server accepts CORS requests from any origin, which means
it’s possible to inject the main.js startup script into browser tabs
of other websites and use the wasm ghci session to debug those
websites! Once you fire up a ghci session, just open the devtools
console of another website and drop a
import("http://127.0.0.1:38827/main.js") call, if that website
doesn’t actively block third-party scripts, then you can have more fun
than running it in the default blank page.
All JavaScript code for the GHC wasm backend consists of proper ES modules
that don’t pollute the globalThis namespace. This principle has
been enforced since day one, which allows multiple Haskell wasm
modules or even wasm ghci sessions to co-exist in the same page! It
works fine as long as you respect their boundaries and don’t attempt
to do things like freeing a JSVal allocated elsewhere, but even if
you only have one wasm module or ghci session, the “no global variable”
principle should also minimize the interference with the original page.
In my opinion, being able to interact with other websites is the most
exciting aspect of the browser mode. Sure, for Haskell developers that
want to experiment with frontend development, using ghci should
already be much easier than setting up a playground project and
manually handling linker flags, wrapper scripts, etc. But there’s even
greater potential: who said the website itself needs to be developed
in Haskell? Haskell can be used to test websites written in foreign
tech stacks, and testing backed by an advanced type system is
undoubtedly one of our core strengths! You can use libraries like
quickcheck-state-machine or
quickcheck-dynamic to perform state machine
property testing interactively, which has much greater potential of
finding bugs than just a few hard coded interactions in JavaScript.
No host file system in wasm
The default nodejs mode of wasm ghci has full access to the host file
system, so you can use Haskell APIs like readFile to operate on any
host file path. This is no longer the case for browser mode: the only
handles available are stdout/stderr, which output to the devtools
console in a line-buffered manner, and there’s no file to read/write
in wasm otherwise. The same restriction also applies to Template
Haskell splices evaluated in a browser mode ghci session, so splices
like $(embedFile ...) will fail.
This is a deliberate design choice. The dev environment backed by ghci
browser mode should be as close as possible to the production
environment used by statically linked wasm modules, and the production
environment won’t have access to the host file system either. It would be
possible to add extra plumbing to expose the host file system to ghci
browser mode, but that is quite a bit of extra work and also makes the
dev environment less realistic, so I’d like to keep the current design
for a while.
If you need to read a local asset, you can serve the asset via another
local HTTP server and fetch it in ghci. If you have modules that use
splices like embedFile, those modules should be pre-compiled to
object code and loaded later in ghci.
Don’t press F5
It’s very important that the browser page is never refreshed. The
lifetime of the browser tab is supposed to be tied to the ghci
session. Just exit ghci and close the tab when you’re done, but
refreshing the page would completely break ghci! A lot of shared state
between the browser side and host side is required to make it work,
and refreshing would break the browser side of the state.
Likewise, currently the browser mode can’t recover from network
glitches. It shouldn’t be a concern when you run GHC and the browser
on the same machine, but in case you use SSH port forwarding or
tailscale to establish the GHC/browser connection over an unstable
network, once the WebSocket is broken then the game is over.
This is not ideal for sure, but supporting auto-recovery upon network
issues or even page reloads is incredibly challenging, so let’s live
with what is supported for now.
Doesn’t work on Safari yet
Currently the browser mode works fine for Firefox/Chrome, including
desktop/mobile versions and all the forks with different logos and
names. Sadly, Safari users are quite likely to see spurious crashes
with a call_indirect to a null table entry error in the console.
Rest assured, normal statically-linked Haskell wasm modules still work
fine in Safari.
This is not my fault, but WebKit’s! I’ve filed a WebKit
bug and if we’re lucky, this may be looked into on their
side and get fixed eventually. If not, or if many people complain
loudly, I can implement a workaround that seems to mitigate the WebKit
bug to make the browser mode work in Safari too. That’ll be extra
maintenance burden, so for now, if you’re on macOS, your best bet is
installing Firefox/Chrome and using that for ghci.
Huge libraries don’t work yet
How large is “huge”? Well, you can check the source code of
V8, SpiderMonkey and
JavaScriptCore. In brief: there are limits agreed upon
among major browser engines that restrict a wasm module’s
import/export numbers, etc, and we do run into those limits
occasionally when the Haskell library is huge. For instance, the
monolithic ghc library exceeds the limit, and so does the profiling way of
ghc-internal. So cost-center profiling doesn’t work for the ghci
browser mode yet, though it does work for statically linked wasm
modules and ghci nodejs mode.
Unfortunately, this issue is definitely not a low hanging fruit even
for me. I maintain a nodejs fork that patches the V8 limits so that
the Template Haskell runner should still work for huge libraries, but
I can’t do the same for browsers. A fundamental fix to sidestep the
browser limits would be a huge amount of work. So I’ll be prioritizing
other work first. If you need to load a huge library in the browser,
you may need to split it into cabal sublibraries.
Wishlist, as usual
My past blog posts usually ends with a “what comes next” section. This
one is no exception. The browser mode is in its early days, so it’s
natural to find bugs and other rough edges, and there will be
continuous improvement in the coming months. Another thing worth
looking into is profiling: modern browsers have powerful profilers,
and it would be nice to integrate our own profiling and event log
mechanism with browser devtools to improve developer experience.
The next big thing I’ll be working on is threaded RTS
support. Currently all Haskell wasm modules are
single-threaded and runs in the browser main thread, but there may
exist workloads that can benefit from multiple CPU cores. Once this is
delivered, Haskell will also become the first functional language with
multi-core support in wasm!
You’re welcome to join the Haskell wasm Matrix room
to chat about the GHC wasm backend and get my quick updates on this
project.
Today, 2025-04-16, at 1830 UTC (11:30 am PDT, 2:30 pm EDT, 7:30 pm GMT, 20:30 CET, …)
we are streaming the 42th episode of the Haskell Unfolder live on YouTube.
Functional programming is programming with mathematical functions, mapping
inputs to outputs. By contrast, logic programming—perhaps best known from
the language Prolog—is programming with mathematical relations between
values, without making a distinction between inputs and outputs. In this
two-year anniversary episode of the Haskell Unfolder we take a look at
typedKanren, an embedding of the logic programming language miniKanren in
Haskell. We will see how we can use it to write a type checker for a simple
functional language in a few lines of code.
About the Haskell Unfolder
The Haskell Unfolder is a YouTube series about all things Haskell hosted by
Edsko de Vries and Andres Löh, with episodes appearing approximately every two
weeks. All episodes are live-streamed, and we try to respond to audience
questions. All episodes are also available as recordings afterwards.
Purely functional list concatenation, xs ++ ys in Haskell syntax, is well known to be linear time
in the length of the first input and constant time in the length of the second, i.e. xs ++ ys is
O(length xs). This leads to quadratic complexity if we have a bunch of left associated uses of
concatenation.
The ancient trick to resolve this is to, instead of producing lists, produce list-to-list functions
a la [a] -> [a] or ShowS = String -> String = [Char] -> [Char]. “Concatenation” of “lists”
represented this way is just function composition which is a constant time operation. We can lift a
list xs to this representation via the section (xs ++). This will still lead to O(length xs)
amount of work to apply this function, but a composition of such functions applied to a list will
always result in a fully right associated expression even if the function compositions aren’t
right associated.
In the last several years, it has become popular to refer to this technique as “difference lists”.
Often no justification is given for this name. When it is given, it is usually a reference to the
idea of difference lists in logic programming. Unfortunately, other than both techniques giving rise
to efficient concatenation, they have almost no similarities.
Functional Lists
To start, I want to do a deeper analysis of the “functional lists” approach, because I think what it
is doing is a bit misunderstood and, consequently, oversold1. Let’s see how we would model this approach in an OO
language without higher-order functions, such as early Java. I’ll use strings for simplicity, but it
would be exactly the same for generic lists.
This is just a straight, manual implementation of closures for (.) and (++) (specialized to
strings). Other lambdas not of the above two forms would lead to other implementations of
PrependTo. Let’s say, however, these are the only two forms that actually occur, which is mostly
true in Haskell practice, then another view on this OO code (to escape back to FP) is that it is an
OOP encoding of the algebraic data type:
dataPrependTo=ComposePrependToPrependTo|PrependStringprependTo ::PrependTo->String->StringprependTo (Compose left right) end = prependTo left (prependTo right end)prependTo (Prepend s) end = s ++ end
We could have also arrived at this by defunctionalizing a typical example of the technique. Modulo
some very minor details (that could be resolved by using the Church-encoded version of this), this
does accurately reflect what’s going on in the technique. Compose is clearly constant time. Less
obviously, applying these functional lists requires traversing this tree of closures – made
into an explicit tree here. In fact, this reveals that this representation could require arbitrarily
large amounts of work for a given size of output. This is due to the fact that prepending an empty
string doesn’t increase the output size but still increases the size of the tree. In practice,
it’s a safe assumption that, on average, at least one character will be prepended per leaf of the
tree which makes the overhead proportional to the size of the output.
This tree representation is arguably better than the “functional list” representation. It’s less
flexible for producers, but that’s arguably a good thing because we didn’t really want arbitraryString -> String functions. It’s more flexible for consumers. For example, getting the head of
the list is a relatively efficient operation compared to applying a “functional list” and taking
the head of the result even in an eager language. (Laziness makes both approaches comparably
efficient.) Getting the last element is just the same for the tree version, but, even with laziness,
is much worse for the functional version. More to the point, this concrete representation allows
the concatenation function to avoid adding empty nodes to the tree whereas (.) can’t pattern
match on whether a function is the identity function or not.
This view makes it very clear what the functional version is doing.
Difference Lists in Prolog
List append is the archetypal example of a Prolog program due to the novelty of its “invertibility”.
For our purposes, viewing this as a function of the first two arguments, this is exactly the usual
functional implementation of list concatenation with exactly the same problems. We could, of course,
encode the defunctionalized version of the functional approach into (pure) Prolog. This would
produce:
(I’ll be ignoring the issues that arise due to Prolog’s untyped nature.)
However, this being a logic programming language means we have additional tools available to use
that functional languages lack. Namely, unification variables. For an imperative (destructive)
implementation of list concatenation, the way we’d support efficient append of linked lists is we’d
keep pointers to the start and end of the list. To append two lists, we’d simply use the end
pointer of the first to update the end of the first list to point at the start of the second. We’d
then return a pair consisting of the start pointer of the first and the end pointer of the second.
This is exactly how Prolog difference lists work, except instead of pointers, we use unification
variables which are more principled. Concretely, we represent a list as a pair of lists, but the
second list will be represented by an unbound unification variable and the first list contains
that same unification variable as a suffix. This pair is often represented using the infix
operator (“functor” in Prolog terminology), -, e.g. Xs - Ys. We could use diff(Xs, Ys) or
some other name. - isn’t a built-in operator, it’s just a binary constructor essentially.
At the level of logic, there are no unification variables. The constraints above mean that Xs - Ys
is a list Xs which contains Ys as a suffix.
The name “difference list” is arguably motivated by the definition of concatenation in this
representation.
concat(Xs-Ys,Ys-Zs,Xs-Zs).
This looks a lot like |Xs - Ys + Ys - Zs = Xs - Zs|. If the suffix component of the first argument
is unbound, like it’s supposed to be, then this is a constant-time operation of binding that
component to Ys. If it is bound, then we need to unify which, in the worst-case, is O(length Ys)
where the length is up to either nil or an unbound variable tail2.
We also have the unit of concat, i.e. the empty
list via3:
empty(Xs-Xs).
See the footnote, but this does in some way identify Xs - Ys with the “difference” of Xs and
Ys.
We get back to a “normal” list via:
to_list(Xs- [],Xs).% or more generally,prepend_to(Xs-Ys,Ys,Xs).
to_list is a constant-time operation, no matter what. Note, to_list binds the suffix component
of the difference list. This means that the first input no longer meets our condition to be a
difference list. In other words, to_list (and prepend_to) consumes the difference list.
More precisely, it constrains the possible suffixes the list could be.
Indeed, any operation that binds the suffix component of a difference list consumes it. For example,
concat consumes its first argument.
Of course, it still makes logical sense to work with the difference list when its suffix component
is bound, it’s just that its operational interpretation is different. More to the point, given a
difference list, you cannot prepend it (via prepend_to or concat) to two different lists to get
two different results.
Converting from a list does require traversing the list since we need to replace the nil node, i.e.
[], with a fresh unbound variable. Luckily, this is exactly what append does.
from_list(Xs,Ys-Zs) :- append(Xs,Zs,Ys).
from_list also suggests this “difference list” idea. If all of Xs, Ys, and Zs are ground
terms, then from_list(Xs, Ys - Zs) holds when append(Xs, Zs, Ys) holds. Exactly when if our
invariants are maintained, i.e. that Zs is a suffix of Ys. Writing these relations more
functionally and writing append as addition, we’d have:
If we did want to “duplicate” a difference list, we’d essentially need to convert it to a (normal)
list with to_list, and then we could use from_list multiple times on that result. This would,
of course, still consume the original difference list. We’d also be paying O(length Xs) for every
duplicate, including to replace the one we just consumed4.
That said, we can prepend to a list to a difference list without consuming it. We can perform
other actions with the risk of (partially) consuming the list, e.g. indexing into the list. Indexing
into the list would force the list to be at least a certain length, but still allow prepending to
any list that will result in a final list at least that long.
Comparison
I’ll start the comparison with a massive discrepancy that we will ignore going forward. Nothing
enforces that a value of type ShowS actually just appends something to its input. We could use
abstract data type techniques or the defunctionalized version to avoid this. To be fair, difference
lists also need an abstraction barrier to ensure their invariants, though their failure modes are
different. A difference list can’t change what it is based on what it is prepended to.
Functional Representation
Difference Lists
constant-time concatenation
constant-time concatenation
constant-time conversion from a list (though you pay for it later)
O(n) conversion from a list
persistent
non-persistent, requires linear use
represented by a tree of closures
represented by a pair of a list and a unification variable
O(n) (or worse!) conversion to a list
constant-time conversion to a list
defunctionalized version can be implemented in pretty much any language
requires at least single-assignment variables
unclear connection to being the difference of two lists (which two lists?)
mathematical, if non-obvious, connection to being the difference of two (given) lists
As an illustration of the difference between persistent and non-persistent uses, the function:
double f = f . f
is a perfectly sensible function on ShowS values that behaves exactly as you’d expect. On the
other hand:
double(In,Out) :- concat(In,In,Out).
is nonsense that will fail the occurs check (if it is enabled, otherwise it will create a cyclic
list) except for when In is the empty difference list.
Conclusion
I hope I’ve illustrated that the functional representation is not just not difference lists, but
is, in fact, wildly different from difference lists.
This functional representation is enshrined into Haskell via the ShowS type and related functions,
but I’d argue the concrete tree representation is actually clearer and better. The functional
representation is more of a cute trick that allows us to reuse existing functions. Really, ShowS
should have been an abstract type.
Difference lists are an interesting example of how imperative ideas can be incorporated into a
declarative language. That said, difference lists come with some of the downsides of an imperative
approach, namely the lack of persistence.
As far as I’m aware, there isn’t an unambiguous and widely accepted name for this functional
representation. Calling it “functional lists” or something like that is, in my opinion, very
ambiguous and potentially misleading. I think the lack of a good name for this is why “difference
lists” started becoming popular. As I’ve argued, using “difference list” in this context is even
more misleading and confusing.
If people really want a name, one option might be “delta list”. I don’t think this term is used.
It keeps the intuitive idea that the functional representation represents some “change” to a list,
a collection of deltas that will all be applied at once, but it doesn’t make any false reference to
difference lists. I’m not super into this name; I just want something that isn’t “difference list”
or otherwise misleading.
To be clear, it’s still much, much,
better than using plain concatenation.↩︎
Such a length relation couldn’t
be written in pure Prolog but can in actual Prolog.↩︎
For those algebraically minded, this almost makes concat and empty into another
monoid except concat is partial, but such a partial monoid is just a category! In other words,
we have a category whose objects are lists and whose homsets are, at most, singletons containing
Xs - Ys for Hom(Xs, Ys). If we maintain our invariant that we have Xs - Ys only when Ys is a
suffix of Xs, this thin category is exactly the category corresponding to the reflexive,
transitive “has suffix” relation. We could generalize this to any monoid via a “factors through”
relation, i.e. |\mathrm{Hom}(m, n)| is inhabited if and only if |\exists p. m = pn| which you can
easily prove is a reflexive, transitive relation given the monoid axioms. However, for a general
monoid, we can have a (potentially) non-thin category by saying |p \in \mathrm{Hom}(m,n)| if and
only if |m = pn|. The category will be thin if and only if the monoid is cancellative. This is
exactly the slice category of the monoid viewed as a one-object category.↩︎
Again, in actual Prolog, we could
make a duplicate without consuming the original, though it would still take O(length Xs) time using
the notion of length mentioned before.↩︎
Rust has always felt like a strange beast, culturally speaking. The community is
made of a mix of people with very different perspectives, including anything
from hardcore low-level kernel hackers to category-theorist and functional
programming gurus. This is also what makes this community so fertile: whether
you’re coming from C, Haskell or TypeScript, you’re likely to learn a lot from
other perspectives.
I’d like to add my modest contribution by introducing a pattern coming from the
functional programming world, recursion schemes1. Recursion
schemes are a design pattern for representing and traversing recursive data structures
(typically trees) which help factor the common part of recursive traversals,
making transformations nicer to write, to read and to compose.
Even in the functional programming world, recursion schemes are not so
well-known. Like monads, they are usually presented in Haskell with frightening
words like zygohistomorphic prepromorphisms. It’s a pity because
recursion schemes can be both simple, useful and practical. I’d even argue that
in Rust, the most interesting part is perhaps the representation technique, more
than the traversal, despite the latter being the original and the usual
motivation for using recursion schemes.
In this post, we’ll work through a concrete example to introduce recursion
schemes and what they can do. We’ll point to a more real life example of how we
use them in the implementation of the Nickel configuration language,
and we’ll discuss the pros and cons of using recursion schemes in the particular
context of Rust.
(In)flexible representations
Let’s say you’re writing a JSON parser library. You’ll need to expose a type
representing JSON values. For the sake of argument, let’s assume that you
support an extension of the JSON language with pairs, so you can write {"foo": ("hello","world")}. Here’s a natural representation:
This data structure is recursive: JSON values can contain other JSON values. We
thus have to use Box (or any other indirection) around recursive occurrences
of JsonValue. Otherwise, this enum would have an infinite size (excepted for
Array and Object since Vec and HashMap add their own indirection, but
it’s somehow luck).
Now, user requestor asks that your parser adds location information to the
output, because they validate some user-provided configuration and would like to
point to specific items on error. This is a reasonable
request which is sadly very hard
to satisfy in the serde ecosystem. Anyway, our parser isn’t interfacing
with serde, so we can add span information:
You can go different ways about this. We could have added a second argument to
each constructor of the enum, such as in String(String, Span), to avoid the
additional Spanned layer, but that would be a lot of repetition. We could also
have moved Box to data: Box<T>. Still, the general idea is that we now have
two layers:
a struct layer gathering the JSON data and the span together;
the original enum layer, the core of JSON, which is almost unchanged.
So far, so good. But user conservator is now complaining that you’ve spoiled
their performance. They’re using JSON as a machine exchange format and don’t
care about position information. Could you restore the old representation and a
way to produce it, ignoring spans?
Unfortunately, we had to change JsonValue. Copy-pasting the original
JsonValue enum under a different name is possible, but it’s unsatisfying, as
we now have multiple copies to maintain. It also doesn’t scale. Beside adding
position information, you might want to have a value representation that uses
Rc instead of Box, because you’re going to need to keep reference to
arbitrary nodes during some complex transformation.
The functorial representation
The recursion schemes pattern has two components: a representation technique and
a transformation technique. I believe the representation part is particularly
interesting for Rust, so let’s start with that.
We’ll try to make our JSON representation more generic to accommodate for the
different variations that we mentioned in the previous section. The fundamental
idea is to replace the recursive occurrences of JsonValue within itself,
Box<JsonValue> (or JsonValue for Array and Object), by a generic
parameter T. Doing so, we’re defining just one layer of a JSON tree where
recursive children can be anything, not necessarily JSON values (we use the F
suffix for that generic version because it’s technically a functor, but that
doesn’t really matter).
This is precisely a single node of a JSON tree, that is either a leaf or a
marker of a node with children but without actually including them.
If we set T = Box<JsonValueF<T>>, we get back the original JsonValue.
But wait, you can’t define the generic parameter T to be something which
depends on T itself! In fact we can, but we need to introduce an extra
indirection:
The price to pay is an additional struct layer, so you need to match on
value.data, and wrap new values as JsonValue { data: JsonValueF::Number(0) }. Note that this layer doesn’t have any cost at run-time.
Another difference is that we now box the values in Array and Object,
which isn’t needed. For now I’ll just ignore that, but you could take a second
generic parameter U to represent the occurrences of T that don’t need an
indirection if this really matters to you.
If we extend our intermediate layer a bit, we can get SpannedValue!
This idea of putting a self-referential type within JsonValueF is referred to
as tying the knot. The power of this approach is that you can keep the core
JsonValueF type unchanged. This applies to any tree-like recursive structure.
Some methods can be implemented only once on JsonValueF for any T, say
is_string or is_number. With additional trait constraints on T, we can
write more involved functions, still operating on the generic functor
representation.
Let’s now see how to traverse our JSON values.
Traversals
The strong point of recursion schemes is to provide an interface for traversing
recursive structures that let you focus on what the function actually does,
which is otherwise mixed with how the recursion is done. The idea is to use
generic combinators which factor out the plumbing of recursive traversals.
Let’s count the number of String nodes in a JSON value, the naive way.
We’ll see how to write this function in the style of recursion schemes. First, we need to
define one core combinator: map.
map takes a JsonValueF<T>, a function f from T to U and returns a
JsonValue<U>. That is, map takes a JSON layer where all the direct children
(the recursive occurrences in our full type) are of some type T and applies
f to transform them to something of type U. This is the secret sauce for
defining traversals.
map isn’t specific to JsonValueF. It can be defined mechanically for any
functor representation (e.g. through a macro) of a data structure.
Note that there’s no recursion in sight: there can’t be, because T and U are
entirely generic and could very well be (), but we saw that JsonValueF<()>
is a single node. map only operates at the current layer.
The trick is that f can use map itself. Let’s see how to use it for
count_strings:
If you look closely, there’s no more recursion in the body of the pattern
matching. It’s factored out in the map call. Let’s break down this example:
map, given a function from T to U, promises you that it can transform
the direct children of type T in JsonValueF<T> to U, providing
JsonValueF<U>. We use it immediately with a recursive call to
count_strings, which can indeed transform the direct children from a
Box<JsonValue> to a u32. If the children have children itself,
count_strings will do that recursively as its first action, down to the
leaves.
Once we’ve reduced potential children of deeper layers to u32s, we get a
JsonValueF<u32>. We sum its content at the current layer.
There is a catch though: our count_strings function takes an owned argument,
which consumes the original JSON value. I’ll come back to that later.
While I find the second version of count_strings a little cleaner, the
difference between the two isn’t really astonishing.
As a more compelling example, let’s define a generic bottom-up traversal
function on JsonValue. This traversal is able to map — that is to rewrite —
nodes (more exactly entire subtrees). map_bottom_up takes a generic
transformation f and applies this function to every subtree starting from the
leaves. You could use such a function to apply program transformations
or optimizations on an abstract syntax tree.
implJsonValue{pubfnmap_bottom_up(self:JsonValue, f:implFnMut(JsonValue)->JsonValue)->JsonValue{let data =self.data.map(|v|Box::new(v.map_bottom_up(f)));f(JsonValue{ data })}}
This example is quite remarkable: it’s almost a one-liner and there is no
pattern matching at all! Once again, the structural recursion is entirely
factored out in the map function. We implemented map_bottom_up on
JsonValue directly, but with some trait constraints on T, we can write a
more generic version JsonValueF that works on both the Boxed and Rced
version (the arena one is more tricky as it requires an explicit allocator).
This example is only scratching the surface.
Mapping is just one example: another common traversals are folds (known as
catamorphisms in the recursion schemes jargon), which generalize the well-known
Iterator::fold from sequences to trees. In fact, count_strings would make
more sense as a fold, but we’ll leave that for another time.
Are recursion schemes useful in Rust?
Haskell has a number of features that make recursion schemes particularly nice
to use and to compose, not the least of which is garbage collection. You don’t
have to think about ownership; it’s references all the way down. Recursive data
structures are easy to express.
On the other side, there is Rust, which culturally doesn’t like recursive
functions that much, for good and bad reasons2. Though sometimes
recursion is hard to avoid, especially on tree-like data structures.
An important issue is that our count_strings consumes its argument, which is
unacceptable in practice. It is possible to write a version of map that takes
a value by reference, and thus similarly for count_strings, but it’s not
entirely straightforward nor free. You can find a by-reference version and more
explanations in our associated repository. At any rate, you can always
write specific traversals manually without resorting to the recursion schemes
way if needed. It’s not an all or nothing approach.
In fact, even if you don’t use map at all, the functor representation alone is
quite useful.
How we use recursion schemes in Nickel
In the implementation of the Nickel configuration language, we use the functor
representation for the abstract syntax tree of a static type.
Here are the stages we went through:
In the parser and most of the Nickel pipeline, we used to have a simple
Box-based, owned representation, akin to JsonValue.
However, during type inference, the Nickel typechecker needs to handle new
type constructions, in particular unification
variables. Those are as-of-yet unknown types, similar
to unknowns in an algebraic equation. Extending the base representation is
readily done as for SpannedJsonValue:
pubenumUnifType{Concrete(Box<TypeF<UnifType>>),/// A unification variable.UnifVar(VarId),//.. rigid type variables, etc.}
More recently, we’ve split the historical, all-powerful unique representation
of expressions (including Nickel types) into two intermediate ones. The new
initial representation is arena-allocated, which makes it natural to use bare
references as the recursive indirection instead of allocating in the heap
through e.g. Box. This is easy with recursion schemes: that is precisely
the ArenaJsonValue example. For a smooth transition, we need to temporarily
keep the old Box-ed Type representation in parts of the codebase, but
having different representations co-exist is a basic feature of recursion
schemes.
We use map-based traversal typically to substitute type variables (that is,
a Nickel generic type, as our T in Rust) for a concrete type and similar
rewriting operations. We have variants of the core map function that can also
thread mutable state, raise errors, or both. Traversal by reference are
implemented manually, with a plain recursive function.
On the downside, type and core function definitions can be a bit verbose and
tricky to get right. For example, Nickel’s TypeF has sub-components that themselves
contain types leading to 4 generic parameters. There are multiple possibilities
for Box placement in particular, only some of them are correct and they are
subtly different. Though once you’ve defined a new variant, this complexity is
mostly hidden from the consumers of your API. It can still manifest as terrible
Rust type errors sometimes if, God forbid, you’ve put a Box at the wrong
place.
Conclusion
We’ve introduced recursion schemes, a design pattern for representing and
traversing recursive data structures. While the traversal part isn’t as good a
fit as in purer functional languages like Haskell, it can still be useful in
Rust. The representation part is particularly relevant, making it easy to define
variations on a recursive data structure with different ownership models or
metadata. We’ve shown how we use recursion schemes in Nickel, and while there
are performance and complexity trade-offs to consider, they can bring value for
moderately complex tree types that need to be extended and transformed in
various ways.
Rust allocates on the stack by default, which makes it easier to overflow
(though the stack can be configured to be larger at compile time). However,
I have the impression that there’s a misleading idea that recursive
functions perform poorly. For tree transformations at least, the iterative
version is usually harder to write and can require explicitly representing
the context on the heap through an auxiliary data structure such as a
zipper, which is likely to perform worse. The stack can overflow, and
(recursive) functions call aren’t entirely free either, but in terms of
allocation, deallocation and locality, the stack is also hard to beat!↩
Do you use an LLM for coding? Do you maintain a personal benchmark based on problems you have posed the LLM? The purpose of this blog post is to convince you should do this: that you can do so with marginal effort on top of your day-to-day vibe coding and that you will get both short and long term benefits from making your own personal benchmark exist.
I started thinking about benchmarks for coding in part with my frustration with the discourse around LLMs in the public squares I frequent (Reddit and Twitter). People often want to know "what's the best model" or "what's the best coding IDE"? One might imagine that the way to answer this question would be to test the models on a variety of problems from real world uses of the LLM for coding, and then compare how well various systems do on this. Indeed, whenever a new SOTA model releases, the lab will usually tell you about the model's performance against a few well known coding benchmarks. Problem solved?
Of course not! In fact, for the most part, no one really talks about benchmarks when comparing models. Why? I argue the most popular benchmarks measure tasks that are largely different from what a user wants out of an LLM. For example, take the recent Gemini 2.5 Pro release. In their headline table, they test against LiveCodeBench, Aider Polyglot and SWE-bench Verified. Both LiveCodeBench and Aider Polyglot derive their problems from contest programming and pedagogical exercises (respectively), while SWE-bench assesses bug fixes to preexisting codebases. While useful, this is only a small slice things people want to do with LLMs.
Wouldn't it be great if you had your own, personal benchmark, based on problems you actually care about? If you are tweaking your .cursorrules, you could run your benchmark to see if a change you made helped or not. When a new model comes out, you could spend a few bucks to run your eval and make a decision if you should switch your daily driver. And then on social media, if you wanted to stan the new model, instead of asking the model to drop a ball inside a rotating hexagon or vagueposting about how the new model is incredible, you could just post your benchmark results.
It's a collection of nearly 100 tests I've extracted from my actual conversation history with various LLMs.
There are two defining features of this benchmark that make it interesting. Most importantly, I've implemented a simple dataflow domain specific language to make it easy for me (or anyone else!) to add new tests that realistically evaluate model capabilities. This DSL allows for specifying both how the question should be asked and also how the answer should be evaluated. Most questions are evaluated by actually running the code the model writes but the framework supports a bunch of other evaluation methods as well. And then, directly as a result of this, I've written nearly 100 tests for different situations I've actually encountered when working with LLMs as assistants.
I have been working on my own benchmark based off of Carlini's benchmark, and I can confirm that this works well for the traditional style of coding eval, where you have a one-shot task that generates and executes the code against some test cases. My basic strategy is to vibe code as usual, but whenever I give an LLM a task that it isn't able to one shot, I consider adding it to the benchmark. In more detail:
I only add a task if a SOTA LLM failed it. This ensures the benchmark consists of all appropriate difficulty problems: easy enough that I thought an LLM should be able to do it, but hard enough that a SOTA model failed on it. I don't need problems that are too hard (this is already well covered by well known benchmarks like SWE-Bench or SWE-Lancer), and I don't mind if my problems saturate because, hey, that means the models are that much better for my use cases!
After I have added the task to the benchmark, I can use the benchmark runner to tell if changing the model, tweaking the prompt, or even just running the prompt again at nonzero temperature can make it pass. Indeed, it's helpful to find some configuration that makes the eval pass, as this is good for debugging issues in the evaluation function itself... also it means you have working code for whatever task you were working on. Conversely, you can make the task harder by leaving things out from the prompt.
Writing the test is the labor intensive part, but you can always vibe code a test. Importantly, you have a failing implementation (your initial generation) and some way you (manually?) determined that the implementation was wrong, so just turn this into your evaluation function! (And for all you yak shaving aficionados, if the model fails to vibe code your test, well, you have another task for your benchmark!)
For example, the other day I needed to take an asciinema recording and convert it into a sequence of frames rendered as plain text. However, the only project for doing these conversations was agg, which converts recordings into animated gifs. In
agg_to_text, I ask an LLM to take agg's source code and create a new program which dumps the frames as plain text rather than gif images. The reason why this task is difficult, is because there is some discretion in deciding when to emit a frame, and with my original prompt the LLM didn't precisely replicate the original behavior in agg. While working on the benchmark, I realized that instructing the model specifically about how frame batching worked was enough to get it to preserve the original behavior. But I don't think I should need to do this: thus this task. (P.S. If this test saturates, well, I can always make it harder by removing the agg source code from the prompt.)
The ability to benchmark one shot tasks is here today, but I would like to speculate a bit about what lies beyond them. In particular, most of my LLM coding activity involves asking the LLM to make changes to a pre-existing project, which makes it less amenable to "single prompt creates self contained program". (Also, I usually only ask one-shot questions that the LLM can answer, so most of them would never go in my benchmark.)
In short, how can I extract tasks from my day-to-day work? There seems to be two big extra levers we have:
Codebase tasks. This is the heavy-weight approach: you record the Git commit of your codebase at the time you prompted for some new feature to be added, and then when you want to run an eval on a new model you just check out the codebase at that commit and let the end-to-end system go. You'll typically want to execute the modified code, which means you'll also need a way to reliably setup the runtime environment for the code; things like lockfiles can help a lot here.
Transcript tasks. You don't actually need the entire codebase to be available to ask an LLM for a completion; you only need the conversation transcript up to the point of the critical generation. If the transcript is mostly your agent system reading in files for context, you can end up with a relatively system generic prompt that can tell you something about other systems. Of course, if you want to actually run the change, you still need the full codebase, which is why this approach is much more amenable if you're going to do some static analysis on the output. For example, if a model keeps adding try: ... except: ... blocks that are suppressing errors, you can take some transcripts where you've caught the model red-handed doing this and make an eval that checks if the model is still doing this. I suspect testing on transcripts works best for testing if changing prompts or rules improves performance, since the transcript itself will put the model into some particular latent space and if it were a different model they might have made different choices leading to a different latent space. Transcripts from thinking models are especially susceptible to this!
I have started adapting Carlini's framework to work better for these cases, although I would love to be told someone has already solved this problem for me. In particular, I am very excited about using transcript tasks to evaluate whether or not things I add to my prompts / triggered rules are helping or not. Current SOTA model instruction following isn't great and I regularly catch models doing behaviors that I explicitly told them not to in the system prompt. I have started some initial analysis over all of my chat logs to find cases where the model misbehaved, although I haven't quite worked out how I want to build an eval out of it.
One word of warning: to make transcript tasks, you need an AI coding system that doesn't obscure how it assembles its underlying prompts (which rules out most of the popular closed source AI code editors.)
I started building evals for a selfish reason: I wanted to be able to tell if modifications to my prompts were doing anything. But I also think there is a broader opportunity that arises if we also publish these benchmarks to the world.
For one, building a real world benchmark on use cases we care about is a way to communicate to the people training AI models whether or not they are doing well or not. Historical evals have focused on LeetCoding, and consequently we have models that would ace any big tech interview and yet on real world tasks will drive you off a cliff at the first opportunity. And this is not just free labor for the top labs: if you believe in open source models, one of the biggest barriers to good small models is having really high quality data. We, the OSS vibe coding community, can directly help here.
I think there is a tremendous opportunity for the open source community to really push the state of the art in coding evaluations. There's only so many benchmarks that I, personally, can create, but if everyone is making benchmarks I could eventually imagine a universe of benchmarks where you could curate the problems that are relevant to your work and quickly and cheaply judge models in this way: a Wikipedia of Coding Benchmarks.
To summarize: every time an LLM fails to solve a problem you ask it for, this is a potential new benchmark. As long as there is a way to automate testing if the LLM has solved the problem, you can turn this into a benchmark. Do this for yourself, and you can quickly have a personal benchmark with which to evaluate new models. Do this at scale, and you can help push the frontier in coding models.
Haskell is the world’s best programming language1, but let’s
face the harsh reality that a lot of times in life you’ll have to write in other
programming languages. But alas you have been fully Haskell-brained and
lost all ability to program unless it is type-directed, you don’t even know how
to start writing a program without imagining its shape as a type first.
Well, fear not. The foundational theory behind Algebraic Data Types and
Generalized Algebraic Data Types (ADTs and GADTs) are so fundamental that
they’ll fit (somewhat) seamlessly into whatever language you’re forced to write.
After all, if they can fit profunctor
optics in Microsoft’s Java code, the sky’s the limit!
This is an “April Fools” joke in the tradition of my previous
one in some of these ways that we are going to twist these other languages
might seem unconventional or possibly ill-advised… but also the title is
definitely a lie: these languages definitely should have them! :D
Normal ADTs
As a reminder, algebraic Data Types (ADTs) are products and sums; that’s why
they’re algebraic, after all!
Product Types
Products are just immutable structs, which pretty much every language
supports — as long as you’re able to make sure they are never mutated.
This is much simpler in languages where you can associate functions with
data, like OOP and classes. For example, this is the common “value object”
pattern in java (roughly related to the java bean2):
In this case, not only are these ADTs (algebraic data types), they’re also
ADTs (abstract data types): you are meant to work with them
based on a pre-defined abstract interface based on type algebra, instead of
their internal representations.
Sum Types
If your language doesn’t support sum types, usually the way to go is with the
visitor pattern: the underlying implementation is hidden, and the only
way to process a sum type value is by providing handlers for every branch — a
pattern match as a function, essentially. Your sum values then basically
determine which handler is called.
For example, we can implement it for a network address type that can either
be IPv4 or IPv6. Here we are using C++ just for generics and lambdas with
closures, for simplicity, but we’ll discuss how this might look in C later.
Note that in this way, the compiler enforces that we handle every branch.
And, if we ever add a new branch, everything that ever consumes
IPAddress with an IPAddressVisitor will have to add a
new handler.
In a language without generics or powerful enough polymorphism, it’s
difficult to enforce the “pure” visitor pattern because you can’t ensure that
all branches return the same type.
One common pattern is to have an “effectful” visitor pattern, where the point
isn’t to return something, but to execute something on the payload of
the present branch. This is pretty effective for languages like C, javascript,
python, etc. where types aren’t really a rigid thing.
For example, this might be how you treat an “implicit nullable”:
This is basically for_ from Haskell: You can do something like
conditionally launch some action if the value is present.
visitMaybe( () =>console.log("Nothing to request"), (reqPayload) =>makeRequest("google.com", reqPayload), maybeRequest);
On a simpler note, if your language as subtyping built in (maybe with classes
and subclasses) or some other form of dynamic dispatch, you can implement it in
terms of that, which is nice in python, java, C++, etc.
interface ExprVisitor<R>{ R visitLit(int value); R visitNegate(Expr unary); R visitAdd(Expr left, Expr right); R visitMul(Expr left, Expr right);}abstractclass Expr {publicabstract<R> R accept(ExprVisitor<R> visitor);}
Alternatively, you’re in a language where lambdas are easy, instead of
tupling up the visitor, you could just have accept itself take a
number of arguments corresponding to each constructor:
//Alternative definition without an explicit Visitorclassabstract classExpr { public abstract <R>R accept(Function<int,R> visitLit,Function<Expr,R> visitNegate,BiFunction<Expr,Expr,R> visitAdd,BiFunction<Expr,Expr,R> visitMul );}
(Note that C++ doesn’t allow template virtual methods — not because it’s not
possible within the language semantics and syntax, but rather because the
maintainers are too lazy to add it — so doing this faithfully requires a bit
more creativity)
Now, if your language has dynamic dispatch or subclass polymorphism, you can
actually do a different encoding, instead of the tagged union. This will work in
languages that don’t allow or fully support naked union types, too. In this
method, each constructor becomes a class, but it’s important to only
allow access using accept to properly enforce the sum type
pattern.
class Lit extends Expr {privatefinalint value;publicLit(int value){this.value= value;}@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitLit(value);}}class Negate extends Expr {privatefinal Expr unary;publicNegate(Expr unary){this.unary= unary;}@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitNegate(unary);}}class Add extends Expr {privatefinal Expr left;privatefinal Expr right;publicAdd(Expr left, Expr right){this.left= left;this.right= right;}@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitAdd(left, right);}}class Mul extends Expr {privatefinal Expr left;privatefinal Expr right;publicMul(Expr left, Expr right){this.left= left;this.right= right;}@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitMul(left, right);}}
(But, just wanted to note that if you actually are working in java,
you can actually do something with sealed classes, which allows exhaustiveness
checking for its native switch/case statements.)
Alternatively you could make all of the subclasses anonymous and expose them
as factory methods, if your language allows it:
abstractclass Expr {publicabstract<R> R accept(ExprVisitor<R> visitor);publicstatic Expr lit(int value){returnnewExpr(){@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitLit(value);}};}publicstatic Expr negate(Expr unary){returnnewExpr(){@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitNegate(unary);}};}publicstatic Expr add(Expr left, Expr right){returnnewExpr(){@Overridepublic<R> R accept(ExprVisitor<R> visitor){return visitor.visitAdd(left, right);}};}// ... etc}
Passing around function references like this is actually pretty close to the
scott encoding of our data type — and for non-recursive types, it’s essentially
the church encoding.
Recursive Types
Speaking of recursive types…what if your language doesn’t allow recursive
data types? What if it doesn’t allow recursion at all, or what if recursively
generated values are just annoying to deal with? Just imagine writing that
Expr type in a language with explicit memory management, for
example. Or, what if you wanted a way to express your recursive types in a more
elegant and runtime-safe manner?
One thing you can instead do is have your visitor be in its “catamorphism”,
or church encoding. Instead of having the “visitor” take the recursive
sub-values, instead have it return the result of recursively applying
itself.
Let’s do this in dhall, one of the most famous non-recursive
languages. Dhall does have native sum types, so we won’t worry about
manually writing a visitor pattern. But it does not have recursive data
types.
Let’s define a type like:
dataExpr=LitNatural|AddExprExpr|MulExprExpr
But we can’t define data types in dhall that refer to themselves. So instead,
we can define them in their “church encoding”: give what you would do with an
Expr to consume it, where the consumption function is given as if
it were recursively applied.
Note that ExprF r is essentially
ExprVisitor<R>, except instead of add being
Expr -> Expr -> r, it’s r -> r -> r: the
input values aren’t the expression, but rather the results of recursively
folding on the expression. In fact, our original non-recursive
ExprVisitor<R> (to be more precise, the
R accept(ExprVisitor<R>)) is often called the “scott
encoding”, as opposed to the recursive “church encoding” fold.
For value creation, you take the visitor and recursively apply:
And finally, using the data type involves providing the
handler to fold up from the bottom to top. Note that
add : \(left : Natural) -> \(right : Natural) -> left + right
already assumes that the handler has been applied to the sub-expressions, so you
get Naturals on both sides instead of Expr.
This pattern is useful even in languages with good datatype recursion, like
Haskell — it’s actually the recursion-schemes
refactoring of a recursive data type, and it can be useful to have it live
alongside your normal recursive types. I’ve written this blog
post talking about how useful this pattern is to have alongside your normal
recursive types.
This pattern is pretty portable to other languages too, as long as you can
scrounge together something like Rank-N types:
interface ExprFold<R>{ R foldLit(int value); R foldNegate(R unary); R foldAdd(R left, R right); R foldMul(R left, R right);}interface Expr {publicabstract<R> R accept(ExprFold<R> fold);publicstatic Expr lit(int value){returnnewExpr(){@Overridepublic<R> R accept(ExprFold<R> fold){return fold.foldLit(value);}};}publicstatic Expr negate(Expr unary){returnnewExpr(){@Overridepublic<R> R accept(ExprFold<R> fold){return fold.foldNegate(unary.accept(fold));}};}// etc.}
By “Rank-N types” here, I mean that your objects can generate polymorphic
functions: given an Expr, you could generate an
<R> R accept(ExprFold <R> fold) for any R,
and not something pre-determined or pre-chosen by your choice of representation
of Expr.
Generalized Algebraic Data Types
You’ve implemented ADTs in your language of choice, or you are currently in a
language with native ADTs. Life is good, right? Until that sneaky voice starts
whispering in your hear: “we need more type safety.” You resist that urge, maybe
even get a lot done without it, but eventually you are compelled to give in and
embrace the warm yet harsh embrace of ultimate type safety. Now what?
Singletons and Witnesses
In Haskell, singletons are essentially enums used to associate a value with a
reifiable type. “Reifiable” here means that you can take the runtime value of a
singleton and use it to bring evidence to the type-level. I ran into a
real-world usage of this while writing https://coronavirus.jle.im/, a web-based data visualizer of
COVID-19 data (source here) in
purescript. I needed a singleton to represent scales for scatter plots
and linking them to the data that can be plotted. And, not only did it need to
be type-safe in purescript (which has ADTs but not GADTs), it had to be
type-safe in the javascript ffi as well.
Here’s how it might look in Haskell:
-- | Numeric typesdataNType ::Type->TypewhereNInt ::NTypeIntNDouble ::NTypeDoubleNPercent ::NTypePercent-- | Define a scaledataScale ::Type->TypewhereScaleDate ::ScaleDateScaleLinear ::Bool->NType a ->Scale a -- ^ whether to include zero in the axis or notScaleLog ::NType a ->Scale a
You’d then run it like this:
plot ::Scale a ->Scale b -> [(a, b)] ->Canvas
So, we have the type of the input tuples being determined by the
values you pass to plot:
But let’s say we only had ADTs. And then we’re passing them down to a
javascript FFI which only has structs and functions. We could drop the
type-safety and instead error on runtime, but…no. Type unsafety is not
acceptable.
The fundamental ability we want to gain is that if we pattern match on
ScaleDate, then we knowa has to be
Date. If we match on NInt, we know that ahas to be Int.
For the sake of this example, we’re going to be implementing a simpler
function in purescript and in javascript: a function that takes a scale type and
a list of points prints the bounds. In Haskell, this looks like:
(Pretend the Percent type is just a newtype-wrapped
Float or something)
There are at least two main approaches to do this. We’ll be discussing
runtime equality witnesses and Higher-Kinded Eliminators.
Runtime Witnesses and Coyoneda
Embedding
The Yoneda Lemma
is one of the most powerful tools that Category Theory has yielded as a branch
of math, but its sibling coyoneda
is one of the most useful Haskell abstractions.
This doesn’t give you GADTs, but it’s a very lightweight way to “downgrade”
your GADTs into normal GADTs which is appropriate if you don’t need the full
power.
The trick is this: if you have MyGADT a, and you know you are
going to be using it to produceas, you can do a covariant
coyoneda transform.
For example, if you have this type representing potential data sources:
dataSource ::Type->TypewhereByteSource ::Handle->SourceWordStringSource ::FilePath->SourceStringreadByte ::Handle->IOWordreadString ::FilePath->IOStringreadSource ::Source a ->IO areadSource = \caseByteSource h -> readByte hStringSource fp -> readString fp
You could instead turn Source into a non-GADT by making it a
normal parameterized ADT and adding a X -> a field, which is a
type of CPS transformation:
dataSource a =ByteSourceHandle (Word-> a)|StringSourceFilePath (String-> a)byteSource ::Handle->SourceWordbyteSource h =ByteSource h idstringSource ::FilePath->SourceStringstringSource fp =StringSource fp idreadSource ::Source a ->IO areadSource = \caseByteSource h out -> out <$> readByte hStringSource fp out -> out <$> readString fp
A nice benefit of this method is that Source can now have a
Functor instance, which the original GADT could not.
And, if MyGADT a is going to be consumingas, you can do the contravariant
coyoneda transform:
dataSink a =ByteSinkHandle (a ->Word)|StringSinkFilePath (a ->String)
And, if you are going to be both consuming and producing as, you
can do the invariant coyoneda transform
dataInterface a =ByteInterfaceHandle (Word-> a) (a ->Word)|StringInterfaceFilePath (String-> a) (Word-> a)
However, in practice, true equality involves being able to lift
under injective type constructors, and carrying every single
continuation is unwieldy. We can package them up together with a runtime
equality witness.
This is something we can put “inside” NInt such that, when we
pattern match on a NType a, the type system can be assured that
a is an Int.
You need some sort of data of type IsEq a b with functions:
refl :: IsEq a a
to :: IsEq a b -> a -> b
sym :: IsEq a b -> IsEq b a
trans :: IsEq a b -> IsEq b c -> IsEq a c
inj :: IsEq (f a) (f b) -> IsEq a b
If you have to and sym you also get
from :: IsEq a b -> b -> a.
From all of this, we can recover our original
IsEq a Word -> Word -> a and
IsEq a Word -> a -> Word functions, saving us from having to
put two functions.
Your language of choice might already have this IsEq. But one of
the more interesting ways to me is Leibniz equality (discussed a lot in this
Ryan Scott post), which works in languages with higher-kinded polymorphism.
Leibniz quality in languages with higher-kinded polymorphism means that
a and b are equal if
forall p. p a -> p b: any property of a is also
true of b.
In Haskell, we write this like:
newtypeLeibniz a b =Leibniz (forall p. p a -> p b)refl ::Leibniz a arefl =Leibnizid
The only possible way to construct a ‘Leibniz’ is with both type parameters
being the same: You can only ever create a value of type
Leibniz a a, never a value of Leibniz a b where
b is not a.
You can prove that this is actually equality by writing functions
Leibniz a b -> Leibniz b a and
Leibniz a b -> Leibniz b c -> Leibniz a c (this
Ryan Scott post goes over it well), but in practice we realize this equality
by safely coercing a and b back and forth:
newtypeIdentity a =Identity { runIdentity :: a }to ::Leibniz a b -> a -> bto (Leibniz f) = runIdentity . f .IdentitynewtypeOp a b =Op { getOp :: b -> a }from ::Leibniz a b -> b -> afrom (Leibniz f) = getOp (f (Opid))
So, if your language supports higher-kinded Rank-2 types, you have a
solution!
There are other solutions in other languages, but they will usually all be
language-dependent.
Let’s write everything in purescript. The key difference is we use
map (to isNumber) :: Array a -> Array Number, etc., to get our
Array as something we know it has the type of.
importText.PrintfnewtypeLeibniz a b =Leibniz (forall p. p a -> p b)to ::Leibniz a b -> a -> bfrom ::Leibniz a b -> b -> adataNType a =NInt (Leibniz a Int)|NNumber (Leibniz a Number)|NPercent (Leibniz a Percent)typeAxisBounds a = { minValue :: a , minLabel ::String , maxValue :: a , maxLabel ::String }displayNumericAxis ::NType a ->Array a ->AxisBounds adisplayNumericAxis = \caseNInt isInt -> \xs ->let xMin =minimum$map (to isInt) xs xMax =maximum$map (to isInt) xsshowInt=showin { minValue: xMin , minLabel:showInt xMin , maxValue: xMax , maxLabel:showInt xMax }NNumber isNumber -> \xs ->let xMin =minimum$map (to isNumber) xs xMax =maximum$map (to isNumber) xs showFloat = printf (Proxy ::Proxy"%.4f") -- it works a little differentlyin { minValue: xMin , minLabel: showFloat xMin , maxValue: xMax , maxLabel: showFloat xMax }NPercent isPercent -> \xs ->let xMin =minimum$map (to isPercent) xs xMax =maximum$map (to isPercent) xs showPercent = printf (Proxy ::Proxy"%.1f%%") <<< (_ *100.0)in { minValue: xMin , minLabel: showPercent xMin , maxValue: xMax , maxLabel: showPercent xMax }
To work with our [a] as if it were [Int], we have
to map the coercion function over it that our Leibniz a Int gave
us. Admittedly, this naive way adds a runtime cost of copying the array. But we
could be more creative with finding the minimum and maximum in this way in
constant space and no extra allocations.
And, if we wanted to outsource this to the javascript FFI, remember that
javascript doesn’t quite have sum types, so we can create a quick visitor:
typeNVisitor a r = { nvInt ::Leibniz a Int-> r , nvNumber ::Leibniz a Number-> r , nvPercent ::Leibniz a Percent-> r }typeNAccept a =forall r.NVisitor a r -> rtoAccept ::NType a ->NAccept atoAccept =case _ ofNInt isInt -> \nv -> nv.nvInt isIntNNumber isNumber -> \nv -> nv.nvNumber isNumberNPercent isPercent -> \nv -> nv.nvPercent isPercentforeign import _formatNumeric :: forall a. Fn2 (NAccept a) a StringformatNumeric ::NType a -> a ->StringformatNumeric nt = runFn2 _formatNumeric (toAccept nt)
Admittedly in the javascript we are throwing away the “GADT type safety”
because we throw away the equality. But we take what we can — we at least retain
the visitor pattern for sum-type type safety and exhaustiveness checking. I
haven’t done this in typescript yet so there might be a way to formalize Leibniz
equality to do this in typescript and keep the whole chain type-safe from top to
bottom.
Higher-Kinded Eliminators
This is essentially the higher-kinded version of the visitor pattern, except
in dependent type theory these visitors are more often called “eliminators” or
destructors, which is definitely a cooler name.
In the normal visitor you’d have:
dataUser=TheAdmin|MemberIntdataUserHandler r =UH { uhTheAdmin :: r , uhMember ::Int-> r }
But note that if you have the right set of continuations, you have something
that is essentially equal to User without having to actually use
User:
typeUser'=forall r.UserHandler r -> rfromUser ::User->User'fromUser = \caseTheAdmin-> \UH{..} -> uhTheAdminMember userId -> \UH{..} -> uhMember userIdtoUser ::User'->FootoUser f = f $UH { fhTheAdmin =TheAdmin, fhMember =Member }
This means that User is actually equivalent to
forall r. UserHandler r -> r: they’re the same type, so if your
language doesn’t have sum types, you could encode it as
forall r. UserHandler r -> r instead. Visitors, baby.
But, then, what actually does the r type variable represent
here, semantically? Well, in a UserHandler r, r is the
“target” that we interpret into. But there’s a deeper relationship between
r and User: A UserHandler r essentially
“embeds” a User into an r. And, a
UserHandler r -> r is the application of that embedding to an
actual User.
If we pick r ~ (), then UserHandler () embeds
User into (). If we pick r ~ String, then
UserHandler () embeds User into String
(like, “showing” it). And if we pick r ~ User, a
UserHandler User embeds a User into…itself?
So here, r is essentially the projection that we view the user
through. And by making sure we are forall r. UserHandler r -> r
for allr, we ensure that we do not lose any information:
the embedding is completely 1-to-1. It lets you “create” the User
faithfully in a “polymorphic” way.
In fact, to hammer this home, some people like to use the name of the type as
the type variable: UserHandler user:
-- | The same thing as before but with things renamed to prove a pointdataMakeUser user =MakeUser { uhTheAdmin :: user , uhMember ::Int-> user }typeUser'=forall user.MakeUser user -> user
The forall user. lets us faithfully “create” a User
within the system we have, without actually having a User data
type. Essentially we can imagine the r in the forall r
as “standing in” for User, even if that type doesn’t actually
exist.
Now, here’s the breakthrough: If we can use forall (r :: Type)
to substitute for User :: Type, how about we use a
forall (p :: Type -> Type) to substitute for a
Scale :: Type -> Type?
dataScale ::Type->TypewhereScaleDate ::ScaleDateScaleLinear ::Bool->LType a ->Scale aScaleLog ::NType a ->Scale adataScaleHandler p a =SH { shDate :: p Date , shLinear ::Bool->NType a -> p a , shLog ::NType a -> p a }typeScale' a =forall p.ScaleHandler p a -> p afromScale ::Scale a ->Scale' afromScale = \caseScaleDate-> \SH{..} -> shDateScaleLinear hasZero lt -> \SH{..} -> shLinear hasZero ltScaleLog nt -> \SH{..} -> shLog nttoScale ::Scale' a ->Scale atoScale f = f $SH { shDate =ScaleDate, shLinear =ScaleLinear, shLog =ScaleLog }
So in our new system, forall p. ScaleHandler p a -> p a is
identical to Scale: we can use p a to substitute in
Scale in our language even if our language itself cannot support
GADTs.
So let’s write formatNType in purescript. We no longer have an
actual Scale sum type, but its higher-kinded church encoding:
typeNType a =forall p. { int :: p Int , number :: p Number , percent :: p Percent } -> p atypeScale a =forall p. { date :: p Date , linear ::Bool->NType a -> p a , log ::NType a -> p a } -> p antInt ::NTypeIntntInt nth = nth.intntNumber ::NTypeNumberntNumber nth = nth.numberntPercent ::NTypePercentntPercent nth = nth.percentformatNType ::NType a -> a ->StringformatNType nt = fwhereOp f = nt { int:Opshow , number:Op$ printf (Proxy"%.4f") , percent:Op$ printf (Proxy"%.1f%%") <<< (_ *100.0) }
Here we are using
newtypeOp b a =Op (a -> b)
as our “target”: turning an NType a into an
Op String a. And an Op String a is an
a -> String, which is what we wanted! The int field
is Op String Int, the number field is
Op String Number, etc.
In many languages, using this technique effectively requires having a newtype
wrapper on-hand, so it might be unwieldy in non-trivial situations. For example,
if we wanted to write our previous axis function which is
NType a -> [a] -> String, we’d have to have a newtype wrapper
for [a] -> String that has a as its argument:
newtypeOpList b a =Op ([a] -> b)
or you could re-use Compose:
newtypeCompose f g a =Compose (f (g a))
and your p projection type would be Compose Op [].
So, you don’t necessarily have to write a bespoke newtype wrapper, but you do
have to devote some brain cycles to think it through (unless you’re in a
language that doesn’t need newtype wrappers to have this work, like we’ll
discuss later).
By the way, this method generalizes well to multiple arguments: if you have a
type like MyGADT a b c, you just need to project into a
forall (p :: k1 -> k2 -> k3 -> Type).
I believe I have read somewhere that the two methods discussed here (runtime
equality witness vs. higher-kinded eliminator) are not actually fully identical
in their power, and there are GADTs where one would work and not the other … but
I can’t remember where I read this and I’m also not big-brained enough to figure
out what those situations are. But if you, reader, have any idea, please let me
know!
Existential Types
Let’s take a quick break to talk about something that’s not
technically related to GADTs but is often used alongside them.
What if we wanted to store a value with its NType and hide the
type variable? In Haskell we’d write this like:
dataNType ::Type->TypewhereNInt ::NTypeIntNDouble ::NTypeDoubleNPercent ::NTypePercentdataSomeNType=forall a.SomeNType (NType a) aformatNType ::NType a -> a ->StringformatNType nt x =...formatSomeNType ::SomeNType->StringformatSomeNType (SomeNType nt x) = formatNType nt xmyFavoriteNumbers :: [SomeNType]myFavoriteNumbers = [SomeNTypeNInt3, SomeNTypeNDoublepi]
But what if our language doesn’t have existentials? Remember, this is
basically a value SomeNType that isn’t a Generic, but
contains both a NType a and an a of the
same variable.
One strategy we have available is to CPS-transform our existentials into
their CPS form (continuation-passing style form). Basically, we write exactly
what we want to do with our contents if we pattern matched on them.
It’s essentially a Rank-N visitor pattern with only a single constructor:
typeSomeNType=forall r. (forall a.NType a -> a -> r) -> rsomeNType ::NType a -> a ->SomeNTypesomeNType nt x = \f -> f nt xformatSomeNumeric ::SomeNType->StringformatSomeNumeric snt = snt \nt x -> formatNumeric nt x
You can imagine, syntactically, that snt acts as its “own”
pattern match, except instead of matching on
SomeNType nt x -> .., you “match” on
\nt x -> ..
This general pattern works for languages with traditional generics like Java
too:
interface SomeNTypeVisitor<R>{<A> R visit(NType<A> nt, A val);}interface SomeNType {publicabstract<R> R accept(SomeNTypeVisitor<R> visitor);// One option: the factory methodpublicstatic<A> SomeNType someNType(NType<A> nt, A val){returnnewSomeNType(){@Overridepublic<R> R accept(SomeNTypeVisitor<R> visitor){return visitor.visit(nt, val);}};}}// Second option: the subtype hiding a type variable, which you have to always// make sure to upcast into `SomeNType` after creatingclass SomeNTypeImpl<A>extends SomeNType {private NType<A> nt;private A val;publicSomeNTypeImpl(NType<A> nt, A val){this.nt= nt;this.val= val;}@Overridepublic<R> R accept(SomeNTypeVisitor<R> visitor){return visitor.visit(nt, val);}}
Does…anyone write java like this? I tried committing this once while at
Google and I got automatically flagged to be put on a PIP.
Recursive GADTs
The climax of this discussion: what if your language does not support GADTs
or recursive data types?
We’re going to be using dhall as an example again, but note that the
lessons applied here are potentially useful even when you do have
recursive types: we’re going to be talking about a higher-kinded church
encoding, which can be a useful form of your data types that live alongside your
normal recursive ones.
Let’s imagine Expr as a GADT, where Expr a
represents an Expr that evaluates to an a:
dataExpr ::Type->TypewhereNatLit ::Natural->ExprNaturalBoolLit ::Bool->ExprBoolAdd ::ExprNatural->ExprNatural->ExprNaturalLTE ::ExprNatural->ExprNatural->ExprBoolTernary ::ExprBool->Expr a ->Expr a ->Expr aeval ::Expr a -> aeval = \caseNatLit n -> nBoolLit b -> bAdd x y -> eval x + eval yLTE a b -> eval a <= eval bTernary b x y ->if eval b then eval x else eval y
Adding this type variable ensures that our Expr is type-safe:
it’s impossible to Add an Expr Bool, and the two
branches of a Ternary must have the same result type, etc. And, we
can write eval :: Expr a -> a and know exactly what type will be
returned.
Now, let’s combine the two concepts: First, the church encoding, where our
handlers take the “final result” of our fold r instead of the
recursive value Expr. Second, the higher-kinded eliminator pattern
where we embed Expr :: Type -> Type into
forall (p :: Type -> Type).
Again, now instead of add taking Expr, it takes
p Natural: the “Natural result of the fold”.
p not only stands in for what we embed Expr into, it
stands in for the result of the recursive fold. That’s why in eval,
the first arguments of add are the Natural results of
the sub-evaluation.
These values can be created in the same way as before, merging the two
techniques, sending the handlers downstream:
If all of this is difficult to parse, try reviewing both the recursive ADT
section and the higher-kinded eliminator section and making sure you understand
both well before tackling this, which combines them together!
Admittedly in Haskell (and purescript) this is a lot simpler because we don’t
have to explicitly pass in type variables:
dataExprF p =ExprF { natLit ::Natural-> p Natural , boolLit ::Bool-> p Bool , add :: p Natural-> p Natural-> p Natural , ternary ::forall a. p Bool-> p a -> p a -> p a }typeExpr a =forall p.ExprF p a -> p aeval ::Expr a -> aeval e = runIdentity $ e { natLit =Identity , boolLit =Identity , add = \(Identity x) -> \(Identity y) ->Identity (x + y) , ternary = \(Identity b) -> \(Identity x) -> \(Identity y) ->if b then x else y }ternary ::ExprBool->Expr a ->Expr a ->Expr aternary b x y handlers = handlers.ternary (b handlers) (x handlers) (y handlers)
But one nice thing about the dhall version that’s incidental to dhall is that
it doesn’t require any extra newtype wrappers like the Haskell one does. That’s
because type inference tends to choke on things like this, but dhall doesn’t
really have any type inference: all of the types are passed explicitly. It’s one
of the facts about dhall that make it nice for things like this.
Congratulations
In any case, if you’ve made it this far, congratulations! You are a master of
ADTs and GADTs. Admittedly every language is different, and some of these
solutions have to be tweaked for the language in question. And, if your program
gets very complicated, there is a good chance that things will become
ergonomically unfeasible.
But I hope, at least, that this inspires your imagination to try to bring
your haskell principles, techniques, standards, practices, and brainrot into the
language of your choice (or language you are forced to work with).
And, if you ever find interesting ways to bring these things into a language
not discussed here (or a new interesting technique or pattern), I would
absolutely love to hear about it!
Until next time, happy “Haskelling”!
Special Thanks
I am very humbled to be supported by an amazing community, who make it
possible for me to devote time to researching and writing these posts. Very
special thanks to my supporter at the “Amazing” level on patreon, Josh Vera! :)
I bet you thought there was going be some sort of caveat in this
footnote, didn’t you?↩︎
I didn’t think I’d ever write “java bean” non-ironically on my
blog, but there’s a first time for everything.↩︎
Be aware that this implementation is not necessarily
appropriately lazy or short-circuiting in Ternary: it might
evaluate both sides returning the chosen branch.↩︎
To visit a tree or graph in breadth-first order, there are two main
implementation approaches: queue-based or level-based.
Our goal here is to develop a level-based approach where the levels of
the breadth-first walk are constructed compositionally and dynamically.
Compositionality means that for every node, its descendants—the other nodes
reachable from it—are defined by composing the descendants of its children.
Dynamism means that the children of a node are generated only when that node
is visited; we will see that this requirement corresponds to asking for a
monadic unfold.
A prior solution, using the Phases applicative functor,
is compositional but not dynamic in that sense. The essence of Phases
is a zipping operation in free applicative functors.
What if we did zipping in free monads instead?
A breadth-first walk explores the tree level by level; every level contains the
nodes at the same distance from the root. The list of levels of a tree can be defined
recursively—it is a fold. For a tree Node x l r, the first level contains
just the root node x, and the subsequent levels are obtained by appending the
levels of the subtrees l and r pairwise.
(We can’t just use zipWith because it throws away the end of a list when the
other list is empty.)
Finally, we concatenate the levels together to obtain the list of nodes in
breadth-first order.
toListBF::Treea-> [a]toListBF=concat.levels
Thanks to laziness, the list will indeed be produced by walking the tree in
breadth-first order.
So far so good.
The above function lets us fold a tree in breadth-first order.
The next level of difficulty is to traverse a tree, producing a tree
with the same shape as the original tree, only with modified labels.
This has the exact same type as traverse, which you might obtain with
deriving (Foldable, Traversable). The stock-derived Traversable—enabled
by the DeriveTraversable extension—is a depth-first traversal, but the laws
of traverse don’t specify the order in which nodes should be visited,
so you could make it a breadth-first traversal if you wanted.
“Breadth-first numbering” is a special case of “breadth-first traversal”
where the arrow (a -> m b) is specialized to a counter.
Okasaki presents a “numbering” solution based on queues and another solution
based on levels.
Both are easily adaptable to the more general “traversal” problem as we will
soon see.
There is a wonderful Discourse thread from 2024 on the topic of
breadth-first traversals.
The first post gives an elegant breadth-first numbering algorithm
which also appears in the appendix of Okasaki’s paper,
but sadly it does not generalize from “numbering” to
“traversal” beyond the special case m = State s.
Last but not least, another level-based solution to the breadth-first traversal
problem can be found in the
tree-traversals library by Noah Easterly.
It is built around an applicative transformer named Phases,
which is a list of actions—imagine the type “[m _]”—where each
element m _ represents one level of the tree.
The Phases applicative enables a compositional definition of a
breadth-first traversal, similarly to the levels function above:
the set of nodes reachable from the root is defined by combining the sets of
nodes reachable from its children. This concern of compositionality
is one of the main motivations behind this post.
Non-standard terminology
The broad family of algorithms being discussed is typically called
“breadth-first search” (BFS) or “breadth-first traversal”,
but in general these algorithms are not “searching” for anything,
and in Haskell, “traversal” is reserved for “things like traverse”.
Instead, this post will use “walks” as a term encompassing folds, traversals,
unfolds, or any concept that can be qualified with “breadth-first”.
Problem statement: Breadth-first unfolds
Both the fold toListBF and the traversal traverseBF had in common that they
receive a tree as an input. This explicit tree makes the notion of levels
“static”. With unfolds, we will have to deal with levels that exist only
“dynamically” as the result of unfolding the tree progressively.
To introduce the unfolding of a tree, it is convenient to introduce its “base
functor”. We modify the tree type by replacing the recursive tree fields with
an extra type parameter:
An unfold generates a tree from a seed and a
function which expands the seed into a leaf or a node containing more seeds.
A pure unfold—or anamorphism—can be defined readily:
The order in which nodes are evaluated depends on
how the resulting tree is consumed. Hence unfold
is neither inherently “depth-first” nor “breadth-first”.
The situation changes if we make the unfold monadic.
unfoldM::Monadm=> (s->m (TreeFas)) ->s->m (Treea)
An implementation of unfoldM must decide upon an ordering between actions.
To see why adding an M to unfold imposes an ordering,
contemplate the fact that these expressions have the same meaning:
Node a (unfold f l) (unfold f r)
= ( let tl = unfold f l in
let tr = unfold f r in
Node a tl tr )
= ( let tr = unfold f r in
let tl = unfold f l in
Node a tl tr )
whereas these monadic expressions do not have the same meaning in general:
( unfoldM f l >>= \tl ->
unfoldM f r >>= \tr ->
pure (Node a tl tr) )
/=
( unfoldM f r >>= \tr ->
unfoldM f l >>= \tl ->
pure (Node a tl tr) )
Without further requirements, there is an “obvious” definition of unfoldM,
which is a depth-first unfold:
We unfold the left subtree l fully before unfolding the right one r.
The problem is to define a breadth-firstunfoldM.
If you want to think about this problem on your own, you can stop reading here.
The rest of this post presents solutions.
Queue-based unfold
The two breadth-first numbering algorithms in Okasaki’s paper can
actually be generalized to breadth-first unfolds.
Here is the first one that uses queues (using the function (<+) for “push” and
pattern-matching on (:>) for “pop”):
If you’re frowning upon the use of error—as you should be—you can replace
error with dummy values here (Empty, Leaf), but
(1) that won’t be possible with tree structures that must be non-empty
(e.g., if Leaf contained a value) and (2) this is dead code, which
is harmless but no more elegant than making it obvious with error.
The correctness of this solution is also not quite obvious.
There are subtle ways to get this implementation wrong:
should the recursive call be b2 <+ b1 <+ q or b1 <+ b2 <+ q?
Should the pattern be p :> t1 :> t2 or p :> t2 :> t1?
For another version of this challenge, try implementing the unfold for another
tree type, such as finger trees or rose trees, without getting lost in the
order of pushes and pops (by the way, this is Data.Tree.unfoldTreeM_BF in
containers). The invariant is not complex but there is room for mistakes.
I believe that the compositional approach that will be presented later is more
robust on that front, although it is admittedly a subjective quality for which
is difficult to make a strong case.
Some uses of unfolds
Traversals from unfolds
One sense in which unfoldM is a more difficult problem than traverse is
that we can use unfoldM to implement traverse.
We do have to make light of the technicality that there is a Monad constraint
instead of Applicative, which makes unfoldM not suited to implement the
Traversable class.
A depth-first unfold gives a depth-first traversal:
We can use a tree unfold to explore a graph.
This usage distinguishes unfolds from folds and traversals,
which only let you explore trees.
Given a type of vertices V, a directed graph is represented by a function
V -> F V, where F is a functor which describes the arity of each node.
The obvious choice for F is lists, but we will stick to TreeF here
so we can just reuse this post’s unfoldM implementations.
The TreeF functor restricts us graphs where each node has zero or two
outgoing edges; it is a weird restriction, but we will make do for the sake of
example.
An ASCII drawing of a graph
+-------+
v |
+->1--->2--->3 |
| | | ^ |
| v v | |
| 4--->5--->6--+
| | | ^
| +----|----+
| |
+-------+
The graph drawn above turns into the following function, where every vertex
is mapped either to NodeF with the same vertex as the first argument followed
by its two adjacent vertices, or to LeafF if it has no outgoing edges or does
not belong to the graph.
If we simply feed that function to unfold, we will get the infinite tree
of all possible paths from a chosen starting vertex.
To obtain a finite tree, we want to keep track of vertices that we have
already visited, using a stateful memory. The following function wraps graph,
returning LeafF also if a vertex has already been visited.
Applying unfoldM_BF to that function produces a “breadth-first tree”
of the graph, an encoding of the trajectory of a breadth-first walk through the
graph. “Breadth-first trees” are a concept from graph theory with well-studied
properties.
-- Visit `graph` in breadth-first orderbfGraph_Q::Int->TreeIntbfGraph_Q= (`evalState`Set.empty) .unfoldM_BF_QvisitGraph
This post is a compilable Literate Haskell file. You can run all of the tests
and benchmarks in here. The source repository provides the necessary
configuration to build it with cabal.
$ cabal build breadth-first-unfolds
Test cases can then be selected with the -p option and a pattern
(see the tasty documentation for details).
Run all tests and benchmarks by passing no option.
$ cabal exec breadth-first-unfolds -- -p "/Q-graph/||/S-graph/"
All
Q-graph: OK
S-graph: OK
“Global” level-based unfold
The other solution from Okasaki’s paper can also be adapted into a monadic unfold.
The starting point is to unfold a list of seeds [s] instead of a single seed:
we can traverse the list with the expansion function s -> m (TreeF a s) to
obtain another list of seeds, the next level of the breadth-first unfold,
and keep going.
Iterating this process naively yields a variant of monadic unfold without a
result. This no-result variant can be generalized from TreeF to
any foldable structure:
Modifying this solution to create the output tree requires a little more thought.
We must keep hold of the intermediate list of ts :: [TreeF a s] to
reconstruct trees after the recursive call returns.
This solution is less brittle than the queue-based solution because
we always traverse lists left-to-right.
To avoid the uses of error in reconstruct,
you can probably create a specialized data structure in place of [TreeF a s],
but that is finicky in its own way.
In search of compositionality
Both of the solutions above (the queue-based and the “monolithic” level-based unfolds)
stem from a global view of breadth-first walks: we are iterating on a list or a
queue which holds all the seeds from one or two levels at a time.
That structure represents a “front line” between visited and unvisited
vertices, and every iteration advances the front line a little: with a queue we
advance it one vertex at a time, with a list we advance the whole front line
in an inner loop—one call to traverse—before recursing.
The opposite local view of breadth-first order is exemplified by the earlier
levels function: it only produces a list of lists of the vertices
reachable from the current root. It does so recursively, by composing
together the vertices reachable from its children. Our goal here is to find a
similarly local, compositional implementation of breadth-first unfolds.
Rather than defining unfoldM directly, which sequences the computations on
all levels into a single computation, we will introduce an intermediate
function weave that keeps levels separate—just as toListBF is defined
using levels.
The result of weave will be in an as yet unknown applicative functor F m
depending on m.
And because levels are kept separate, weave only needs
a constraint Applicative m to compose computations on the same level.
The goal is to implement this signature, where the result type F is also an
unknown:
With only what we know so far, a bit of type-directed programming leads to the
following incomplete definition. We have constructed something of type
m (F m (Tree a)), while we expect F m (Tree a):
To fill the hole _, we postulate the following primitive, weft,
as part of the unknown definition of F:
weft::Applicativem=>m (Fma) ->Fma
Intuitively, F m represents “multi-level computations”.
The weft function constructs a multi-level (F m)-computation from
one level of m-computation which returns the subsequent levels
as an (F m)-computation.
We fill the hole with weft, completing the definition of weave:
The function weave defines a multi-level computation which represents
a breadth-first walk from a seed s:
the first level of the walk is f s, expanding the initial seed;
the auxiliary function weaveF constructs the remaining levels from
the initial seed’s expansion:
if the seed expands to LeafF, there are no more seeds,
and we terminate with an empty computation (pure);
if the seed expands to NodeF, we obtain two sub-seeds l and r,
they generate their own weaves recursively (weave f l and weave f r),
and we compose them (liftA2).
One way to think about weft is as a generalization of the following primitives:
we can “embed” m-computations into F m,
and we can “delay” multi-level (F m)-computations, shifting the
m-computation on each level to the next level.
The key law relating these two operations is that embedded computations
and delayed computations commute with each other:
embed u *> delay v = delay v <* embed u
The embed and delay operations are provided by the Phases applicative
functor that I mentioned earlier, which enables breadth-first traversals,
but not breadth-first unfolds. Thus, weft is a strictly more expressive
primitive than embed and delay.
Eventually, we will run a multi-level computation as a single m-computation
so that we can use weave to define unfoldM. The runner function will be
called mesh:
mesh::Monadm=>Fma->ma
It is characterized by this law which says that mesh executes the first
level of the computation u :: m (F m a), then executes the remaining levels
recursively:
mesh (weft u) = u >>= mesh
Putting everything together, weave and mesh combine into a breadth-first unfold:
It remains to find an applicative functor F equipped with weft and mesh.
The weave applicative
A basic approach to design a type is to make some of the operations it
should support into constructors. The weave applicative WeaveS has
constructors for pure and weft:
dataWeaveSma=EndSa|WeftS (m (WeaveSma))
(The suffix “S” stands for Spoilers. Read on!)
We instantiate the unknown functor F with WeaveS.
typeF=WeaveS
Astute readers will have recognized WeaveS as the free monad.
Just as Phases has the same type definition as the free applicative functor but
a different Applicative instance, we will give WeaveS an Applicative
instance that does not coincide with the Applicative and Monad instances of
the free monad.
Starting with the easy functions,
weft is WeftS, and the equation for mesh above is basically its definition.
We just need to add an equation for EndS.
Recall that WeaveS represents multi-level computations.
Computations are composed level-wise with the following liftS2.
The interesting case is the one where both arguments are WeftS: we compose
the first level with liftA2, and the subsequent ones with liftS2
recursively.
liftS2 will be the liftA2 in WeaveS’s Applicative instance.
The Functor and Applicative instances show that WeaveS is an
applicative transformer: for every applicative functor m,
WeaveS m is also an applicative functor.
The outer weft constructor was moved into the recursive calls.
The result type has an extra m, which makes it more apparent that
we always start with a call to f. It’s the same vibe as replacing the type
[a] with NonEmpty a when we know that a list will always have at least one
element; weaveS always produces at least one level of computation.
We also replace (<$>) with its flipped version (<&>) for aesthetic reasons:
we can apply it to a lambda without parentheses, and that change makes the
logic flow naturally from left to right: we first expand the seed s using
f, and continue depending on whether the expansion produced LeafF or NodeF.
To define unfoldM, instead of applying mesh directly, we chain it with
(>>=).
That solution is Obviously Correct™, but it has a terrible flaw:
it does not run in linear time!
We can demonstrate this by generating a “thin” tree whose height
is equal to its size.
The height h is the seed of the unfolding, and we generate a NodeF as long
as it is non-zero, asking for a decreased height h - 1 on the right,
and a zero height on the left.
$ cabal exec breadth-first-unfolds -- -p "S-thin"
All
S-thin
1x: OK
27.6 μs ± 2.6 μs, 267 KB allocated, 317 B copied, 6.0 MB peak memory
10x: OK
2.90 ms ± 181 μs, 23 MB allocated, 178 KB copied, 7.0 MB peak memory, 105.35x
Multiplying the height by 10x makes the function run 100x slower.
Dramatically quadratic.
Complexity analysis
We can compare this implementation with level from earlier, which is linear-time.
In particular, looking at zipLevels with liftS2—which play similar
roles—there is a crucial difference when one of the arguments is empty
([] or EndS):
zipLevels simply returns the other argument, whereas liftS2 calls (<$>),
continuing the recursion down the other argument.
So zipLevels stops working after reaching the end of either argument, whereas
liftS2 walks to the end of both arguments. There is at least one
call to liftS2 on every level which will walk to the bottom of the tree,
so we get a quadratic lower bound Ω(height2).
Out of sight, out of mind
The problematic combinators are fmap and liftS2, which weaveS uses to
construct the unfolded tree. If we don’t care about that tree—wanting only
the effect of a monadic unfold—then we can get rid of the complexity
associated with those combinators.
With no result to return, we remove the a type parameter from the definition
of WeaveS, yielding the oblivious (“O”) variant:
dataWeaveOm=EndO|WeftO (m (WeaveOm))
We rewrite mesh into meshO, reducing a WeaveO m computation
into m () instead of m a.
To implement a breadth-first walk, we modify weaveS above by replacing
liftA2 (Node a) with (<>). Note that the type parameter a is no longer in
the result. It was only used in the tree that we decided to forget.
Running weaveO with meshO yields a oblivious monadic unfold:
it produces m () instead of m (Tree a).
(You may remember seeing another implementation of that same signature
just earlier, unfoldM_BF_G_.)
Previously, we benchmarked the function thinTreeS that outputs a tree by
forcing the tree. With an oblivious unfold, there is no tree to force.
Instead we will count the number of generated NodeF constructors:
thinTreeO::Int->IntthinTreeO= (`execState`0) .unfoldM_BF_O_ (state.f)wheref0counter= (LeafF, counter)fhcounter= (NodeF () 0 (h-1), counter+1) -- increment the counter for every NodeF
We adapt the benchmark from before to measure the complexity of
unfolding thin trees. We have to increase the baseline height from 100 to 500
because this benchmark runs so much faster than the previous ones.
$ cabal exec breadth-first-unfolds -- -p O-thin
All
O-thin
1x: OK
148 μs ± 8.3 μs, 543 KB allocated, 773 B copied, 6.0 MB peak memory
10x: OK
1.45 ms ± 113 μs, 5.4 MB allocated, 82 KB copied, 7.0 MB peak memory, 9.78x
The growth is linear, as desired:
the “10x” bench is 10x slower than the baseline “1x” bench.
Laziness for the win
The oblivious unfold avoided quadratic explosion by simplifying the problem.
Now let’s solve the original problem again,
so we can’t just get rid of fmap and liftA2.
As mentioned previously, the root cause was that (1) liftA2 calls fmap when
one of the constructors is EndS, and (2) fmap traverses the other argument.
The next solution will be to make fmap take constant time,
by storing the “mapped function” in the constructor.
Behold the “L” variant of WeaveS, which is a GADT:
The Applicative instance is… a good exercise for the reader.
The details are not immediately important—we only care about improving fmap
for now—we will come back to have a look at the Applicative instance soon.
The runner function meshL is a simple bit of type Tetris.
$ cabal exec breadth-first-unfolds -- -p "L-thin"
All
L-thin
1x: OK
14.1 μs ± 782 ns, 59 KB allocated, 5 B copied, 6.0 MB peak memory
10x: OK
140 μs ± 13 μs, 586 KB allocated, 51 B copied, 6.0 MB peak memory, 9.93x
Lazy in more ways than one
As hinted by the “L” and “S” suffixes,
WeaveL is a “lazy” variant of WeaveS: fmap for WeaveL “postpones”
work by accumulating functions in the WeftL constructor.
That work is “forced” by meshL, which is where the fmap ((<$>)) of the
underlying monad m is called, performing the work accumulated
by possibly many calls to WeaveL’s fmap.
One subtlety is that there are multiple “lazinesses” at play.
The main benefit of using WeaveL is really to delay computation,
that is a kind of laziness, but WeaveL doesn’t need to be
implemented in a lazy language.
We can rewrite all of the code we’ve seen so far in a strict language
with minor changes, and we will still observe the quadratic vs linear behavior
of WeaveS vs WeaveL on thin trees.
The “manufactured laziness” of WeaveL is a concept independent of the
“ambient laziness” in Haskell.
Nevertheless, we can still find an interesting role for that “ambient laziness”
in this story. Indeed, the function weaveL also happens to be lazier than
weaveS in the usual sense.
A concrete test case is worth a thousand words. Consider the following
tree generator which keeps unfolding left subtrees while making
every right subtree undefined:
whnfTreeS::TestTreewhnfTreeS=expectFail$testCase"S-whnf"$docasepartialTreeSofNode___->pure () -- SucceedLeaf->error"unreachable"-- definitely not a Leaf
As it turns out, this test using the “S” variant fails. (That’s
why the test is marked with expectFail.)
Forcing partialTreeS evaluates the undefined in partialTreeF.
Therefore partialTreeS is not equivalent to partialTree.
$ cabal exec breadth-first-unfolds -- -p "L-whnf"
All
L-whnf: OK
This difference can only be seen with “lazy monads”, where (>>=) is
lazy in its first argument.
(If this definition sounds not quite right, that’s probably because of seq.
It makes a precise definition of “lazy monad” more complicated.)
Examples of lazy monads from the transformers library
are Identity, Reader, lazy State, lazy Writer, and Accum.
The secret sauce is the definition of liftA2 for WeaveL:
In the third clause of liftA2, we put the function f in a lambda with a
lazy pattern (~(a, b)) directly under the topmost constructor WeftL.
Thus, we can access the result of f from the second field of WeftL
without looking at the first field. In liftS2 earlier, f was
passed as an argument to (liftA2 . liftS2), that forces us to run the
computation before we can get a hold on the result of f.
Maximizing laziness
The “L” variant of unfoldM is lazier than the “S” variant,
but there is still a gap between partialTreeL and the pure partialTree:
if we force not only the root, but also the left subtree of partialTreeL,
then we run into undefined again.
Although the unfold using WeaveL is lazier than using WeaveS,
it is not yet as lazy as it could be.
The reason is that, strictly speaking, WeaveL’s liftA2 is a strict function.
The expansion function partialTreeF produces a level with an undefined
sub-computation, which crashes the whole level.
Each level in a computation will be either completely defined or undefined.
To recap, we’ve been looking at the following trees:
It is natural to ask: can we define a breadth-first unfold that, when applied
to partialTreeF, will yield the same tree as partialTree?
More generally, the new problem is to define a breadth-first unfoldM
whose specialization with the Identity functor is equivalent to
the pure unfold even on partially-defined values. That is, it satisfies
the following equation:
unfold f = runIdentity . unfoldM (Identity . f)
Laziness without end
The strictness of liftA2 is caused by WeaveL having two constructors.
Let’s get rid of EndL.
Wait a second. I spoke too fast, GHC gives us an error:
error: [GHC-87005]
• An existential or GADT data constructor cannot be used
inside a lazy (~) pattern
• In the pattern: WeftE wa g
In the pattern: ~(WeftE wa g)
In an equation for ‘fmap’: fmap f ~(WeftE wa g) = WeftE wa (f . g)
|
641 | > fmap f ~(WeftE wa g) = WeftE wa (f . g)
| ^^^^^^^^^^
The feature we need is “first-class existentials”,
for which there is an open GHC proposal.
Not letting that stop us, there is a simple version of first-class existentials
available in the package some,
as the module Data.Some.Newtype (internally using unsafeCoerce).
That will be sufficient for our purposes.
All we need is an abstract type Some and a pattern synonym:
-- imported from Data.Some.NewtypedataSomefpatternSome::fa->Somef
And we’re back on track. Here comes the actual “E” (endless) variant:
The endless WeaveE enables an even lazier implementation of unfoldM.
When specialized to the identity monad, it lets us force the resulting
tree in any order. The forceLeftTreeE test passes (unlike forceLeftTreeL).
$ cabal exec breadth-first-unfolds -- -p "E-left"
All
E-left: OK
One can also check that forcing the left spine of partialTreeE
arbitrarily deep throws no errors.
We made it lazy, but at what cost?
First, this “Endless” variant only works for lazy monads.
With a strict monad, the runner meshE will loop forever.
It is possible to run things more incrementally by pattern-matching on
WeaveE, but you’re better off using the oblivious WeaveO anyway.
Second, when you aren’t running into an unproductive loop, the “Endless” variant of
unfoldM has quadratic time complexity Ω(height2). The reason
is essentially the same as the “Strict” variant: liftA2 keeps looping even if
one argument is a pure weave—before, that was to traverse the other
non-pure argument, now, there isn’t even a way to tell when the computation
has ended.
Thus, every leaf may create work proportional to the height of the tree.
Running the same benchmark as before, we measure even more baffling timings:
Using the previous setup comparing a baseline and a 10x run, we see a more than
700x slowdown, so much worse than the 100x predicted by a quadratic model.
Interestingly, the raw output shows that the total cumulative allocations did
grow by a 100x factor.1
But it gets weirder with more data points: it does not follow a clear power law.
If Time(n) grew as
nc for some fixed exponent c, then the ratio
Time(Mn)/Time(n) would be Mc,
a constant that does not depend on n.
In the following benchmark, we keep doubling the height (M = 2) for every
test case, and we measure the time relative to the preceding case each time.
A quadratic model predicts a 4x slowdown at every step. Instead, we
observe wildly varying factors.
Benchmark output (each time factor is relative to the preceding line,
for example, the “4x” benchmark is 9.5x slower than the “2x” benchmark):
$ cabal exec breadth-first-unfolds -- -p "E-thin-more"
All
E-thin-more
1x: OK
222 μs ± 9.3 μs, 1.2 MB allocated, 13 KB copied, 6.0 MB peak memory
2x: OK
2.43 ms ± 85 μs, 4.8 MB allocated, 236 KB copied, 7.0 MB peak memory, 10.94x
4x: OK
23.1 ms ± 1.2 ms, 19 MB allocated, 2.7 MB copied, 10 MB peak memory, 9.53x
8x: OK
126 ms ± 7.8 ms, 76 MB allocated, 18 MB copied, 24 MB peak memory, 5.44x
16x: OK
181 ms ± 7.0 ms, 119 MB allocated, 30 MB copied, 24 MB peak memory, 1.44x
I believe this benchmark is triggering some pathological behavior in the garbage
collector. I modified tasty-bench with an option to measure CPU time without GC
(mutator time). At time of writing, tasty-bench is still waiting for a new release.
We can point Cabal to an unreleased commit of tasty-bench by adding the following
lines to cabal.project.local.
For the “2x” benchmarks, we are closer the expected 4x slowdown, but there is
still a noticeable gap.
I’m going to chalk the rest to inherent measurement errors (the cost of
tasty-bench’s simplicity) exacerbated by the pathological GC behavior;
a possible explanation is that the pattern of memory usage becomes so bad that
it affects non-GC time.
Benchmark output (excluding GC time, each measurement is relative to the
preceding line):
Microbenchmarks: Queues vs Global Levels vs Weaves
So far we’ve focused on asymptotics (linear vs quadratic). Some readers
will inevitably wonder about real speed.
Among the linear-time algorithms—queues (“Q”), global levels (“G”),
and weaves (lazy “L” or oblivious “O”)—which one is faster?
tl;dr: Queues are (much) faster in these microbenchmarks (up to 25x!),
but keep in mind that these are all quite naive implementations.
There are two categories to measure separately: unfolds which produce trees,
and oblivious unfolds—which don’t produce trees. These microbenchmarks
construct full trees up to a chosen number of nodes. When there is an
output tree, we force it (using nf), otherwise we force a counter of the
number of nodes. We run on different sufficiently large sizes (500 and 5000)
to check the stability of the measured factors, ensuring that we are only
comparing the time components that dominate at scale.
The tables list times relative to the queue benchmark for each tree size.
I hope to have piqued your interest in breadth-first unfolds without
using queues.
To the best of my knowledge, this specific problem hasn’t been studied in the
literature. It is of course related to breadth-first traversals,
previously solved using the Phases applicative.2
The intersection of functional programming and breadth-first walks is a small
niche, which makes it quick to survey that corner of the world for any related
ideas to those presented here.
The paper Modular models of monoids with operations by Zhixuan Yang
and Nicolas Wu, in ICFP 2023, mentions a general construction of Phases as an
example application of their theory. Basically, Phases is defined by a
fixed-point equation:
Phases f = Day f Phases :+: Identity
We can express Phases abstractly as a least fixed-point
μx.f▫x + Id in any monoidal category with a suitable structure.
If we instantiate the monoidal product ▫ not with Day convolution,
but with functor composition (Compose), then we get Weave.
In another coincidence, the monad-coroutine package
implements a weave function which is a generalization of
liftS2—this may require some squinting.
While WeaveS as a data type coincides with the free monad Free,
monad-coroutine’s core data type Coroutine coincides
with the free monad transformer FreeT.
We can view Phases as a generalization of “zipping” from
lists to free applicatives—which are essentially lists of actions,
and Weave generalizes that further to free monads. To recap, the surprise was
that the naive data type of free monads results in a quadratic-time unfold.
That issue motivated a “lazy” variant3 which achieves a linear-time
breadth-first unfold. That in turn suggested an even “lazier” variant which
enables more control on evaluation order at the cost of efficiency.
I’ve just released the weave library which implements
the main ideas of this post. I don’t expect it to have many users, given
how much slower it is compared to queue-based solutions.
But I would be curious to find a use case for the new compositionality
afforded by this abstraction.
Recap table
Unfolds
Time
Laziness
Compositional
Phases*
No
linear†
by levels
Yes
Queue (Q)
Yes
linear†
strict
No
Global Levels (G)
Yes
linear†
by levels
No
Strict Weave (S)
Yes
quadratic‡
strict
Yes
Oblivious Weave (O)
Oblivious only
linear†
N/A
Yes
Lazy Weave (L)
Yes
linear†
by levels
Yes
Endless Weave (E)
Yes
quadratic‡E
maximally lazy◊
Yes
†Linear wrt. size: Θ(size). ‡Quadratic wrt. height: lower bound Ω(height2), upper bound O(height × size). EThe “Endless” meshE only terminates with lazy monads. *I guess there exists an “endless Phases” variant, that
would be quadratic and maximally lazy. ◊The definition of “maximally lazy” in this post actually misses a
range of possible lazy behaviors with monads other than Identity. A further
refinement seems to be another can of worms.
Note that tasty-bench also reports memory statistics
(allocated, copied, and peak memory) when certain RTS options are enabled,
which I’ve done by compiling the test executable with -with-rtsopts=-T.↩︎
The GHC developers are very pleased to announce the availability of
GHC 9.6.7. Binary distributions, source distributions, and
documentation are available on the
release page.
GHC 9.6.7 brings number of fixes, including:
GHC’s internal Unique type has been widened to 64-bits on 32-bit
architectures, avoiding potential miscompilations on large projects.
Fix a runtime crash when using the compacting GC, caused by black
holes in large objects.
Added new flags -fspec-eval and -fspec-eval-dictfun to allow
switching off speculative evaluation.
The following libraries have been updated since GHC 9.6.6:
Note about Haskell Language Server and building GHC 9.8+:
The change of Unique to 64 bit (GHC#22010)
adds the exported symbol
ghc_unique_counter64 to the RTS. Unfortunately it’s impossible to
avoid this without breaking other things. If you encounter a linker
error related to ghc_unique_counter64 when building GHC (or building a
GHC-derived package like ghc-lib-parser) with GHC 9.6.7, you probably
have to add this fix
to the program you’re building.
We would like to thank GitHub, IOG, the Zw3rk stake pool, Well-Typed,
Tweag I/O, Serokell, Equinix, SimSpace, the Haskell Foundation, and
other anonymous contributors whose on-going financial and in-kind
support has facilitated GHC maintenance and release management over
the years. Finally, this release would not have been possible without
the hundreds of open-source contributors whose work comprise this
release.
Please give this release a try and open a ticket
if you see anything amiss.
On this episode of the Haskell Interlude, Andres Löh and Mike Sperber are joined by Farhad Mehta, a professor at OST Rapperswil, and one of the organizers of ZuriHac. Fahrad tells us about formal methods, building tunnels, the importance of education, and the complicated relationship between academia and industry.
At work I sometimes need to deal with large and deep JSON objects where I'm only
interested in a few of the values. If all the interesting values are on the top
level, then aeson have functions that make it easy to implement FromJSON's
parseJSON (Constructors and accessors), but if the values are spread out then
the functions in aeson come up a bit short. That's when I reach for lens-aeson,
as lenses make it very easy to work with large structures. However, I've found
that using its lenses to implement parseJSON become a lot easier with a few
helper functions.
Many of the lenses produces results wrapped in Maybe, so the first function is
one that transforms a Maybe a to a Parser a. Here I make use of Parser
implementing MonadFail.
infixl8<!>(<!>) :: (MonadFail m) => Maybe a->String->m a(<!>) mv err = maybe (fail err) pure mv
In some code I wrote this week I used it to extract the user name out of a JWT
produced by Keycloak:
instance FromJSON OurClaimsSetwhere
parseJSON = ... $ \o ->do
cs <- parseJSON o
n <- o ^? key "preferred_username". _String <!>"preferre username missing"
...
pure $ OurClaimsSet cs n ...
Also, all the lenses start with a Value and that makes the withX functions
in aeson to not be a perfect fit. So I define variations of the withX
functions, e.g.
withObjectV :: String->(Value->Parser a)->Value->Parser awithObjectV s f = withObject s (f . Object)
That makes the full FromJSON instance for OurClaimsSet looks like this
instance FromJSON OurClaimsSetwhereparseJSON = withObjectV "OurClaimsSet"$ \o ->do
cs <- parseJSON o
n <- o ^? key "preferred_username". _String <!>"name"letrs = o ^.. key "resource_access". members . key "roles". _Array . traverse . _String
pure $ OurClaimsSet cs n rs
The GHC developers are happy to announce the release of GHC 9.12.2.
Binary distributions, source distributions, and documentation are available at
downloads.haskell.org.
We hope to have this release available via ghcup shortly. This is a small
release fixing a critical code generation bug, #25653, affecting some subword
division operations.
As always, GHC’s release status, including planned future releases, can
be found on the GHC Wiki status.
We would like to thank IOG, the Zw3rk stake pool,
Well-Typed, Tweag I/O, Serokell, Equinix, SimSpace, the Haskell
Foundation, and other anonymous contributors whose on-going financial
and in-kind support has facilitated GHC maintenance and release
management over the years. Finally, this release would not have been
possible without the hundreds of open-source contributors who
contribute their code, tickets, and energy to the GHC project.
As always, do give this release a try and open a ticket if you see
anything amiss.
I’ve created an open mirror
contest which will run in
parallel to the official contest, so if you want to grab some friends
and try solving some of the problems together using your favorite
language, be my guest!
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A few months ago I explained that one reason why this blog has become more quiet is that all my work on Lean is covered elsewhere.
This post is an exception, because it is an observation that is (arguably) interesting, but does not lead anywhere, so where else to put it than my own blog…
When defining a function recursively in Lean that has nested recursion, e.g. a recusive call that is in the argument to a higher-order function like List.map, then extra attention used to be necessary so that Lean can see that xs.map applies its argument only elements of the list xs. The usual idiom is to write xs.attach.map instead, where List.attach attaches to the list elements a proof that they are in that list. You can read more about this my Lean blog post on recursive definitions and our new shiny reference manual, look for Example “Nested Recursion in Higher-order Functions”.
To make this step less tedious I taught Lean to automatically rewrite xs.map to xs.attach.map (where suitable) within the construction of well-founded recursion, so that nested recursion just works (issue #5471). We already do such a rewriting to change if c then … else … to the dependent if h : c then … else …, but the attach-introduction is much more ambitious (the rewrites are not definitionally equal, there are higher-order arguments etc.) Rewriting the terms in a way that we can still prove the connection later when creating the equational lemmas is hairy at best. Also, we want the whole machinery to be extensible by the user, setting up their own higher order functions to add more facts to the context of the termination proof.
I implemented it like this (PR #6744) and it ships with 4.18.0, but in the course of this work I thought about a quite different and maybe better™ way to do this, and well-founded recursion in general:
WellFounded.fix : (hwf : WellFounded r) (F : (x : α) → ((y : α) → r y x → C y) → C x) (x : α) : C x
we have to rewrite the functorial of the recursive function, which naturally has type
F : ((y : α) → C y) → ((x : α) → C x)
to the one above, where all recursive calls take the termination proof r y x. This is a fairly hairy operation, mangling the type of matcher’s motives and whatnot.
so the functorial’s type is unmodified (here β will be ((x : α) → C x)), and everything else is in the propositional side-condition montone F. For this predicate we have a syntax-guided compositional tactic, and it’s easily extensible, e.g. by
theorem monotone_mapM (f : γ → α → m β) (xs : List α) (hmono : monotone f) :
monotone (fun x => xs.mapM (f x))
Once given, we don’t care about the content of that proof. In particular proving the unfolding theorem only deals with the unmodified F that closely matches the function definition as written by the user. Much simpler!
Isabelle has it easier
Isabelle also supports well-founded recursion, and has great support for nested recursion. And it’s much simpler!
There, all you have to do to make nested recursion work is to define a congruence lemma of the form, for List.map something like our List.map_congr_left
List.map_congr_left : (h : ∀ a ∈ l, f a = g a) :
List.map f l = List.map g l
This is because in Isabelle, too, the termination proofs is a side-condition that essentially states “the functorial F calls its argument f only on smaller arguments”.
Can we have it easy, too?
I had wished we could do the same in Lean for a while, but that form of congruence lemma just isn’t strong enough for us.
But maybe there is a way to do it, using an existential to give a witness that F can alternatively implemented using the more restrictive argument. The following callsOn P F predicate can express that F calls its higher-order argument only on arguments that satisfy the predicate P:
section setup
variable {α : Sort u}
variable {β : α → Sort v}
variable {γ : Sort w}
def callsOn (P : α → Prop) (F : (∀ y, β y) → γ) :=
∃ (F': (∀ y, P y → β y) → γ), ∀ f, F' (fun y _ => f y) = F f
variable (R : α → α → Prop)
variable (F : (∀ y, β y) → (∀ x, β x))
local infix:50 " ≺ " => R
def recursesVia : Prop := ∀ x, callsOn (· ≺ x) (fun f => F f x)
noncomputable def fix (wf : WellFounded R) (h : recursesVia R F) : (∀ x, β x) :=
wf.fix (fun x => (h x).choose)
def fix_eq (wf : WellFounded R) h x :
fix R F wf h x = F (fix R F wf h) x := by
unfold fix
rw [wf.fix_eq]
apply (h x).choose_spec
This allows nice compositional lemmas to discharge callsOn predicates:
theorem callsOn_base (y : α) (hy : P y) :
callsOn P (fun (f : ∀ x, β x) => f y) := by
exists fun f => f y hy
intros; rfl
@[simp]
theorem callsOn_const (x : γ) :
callsOn P (fun (_ : ∀ x, β x) => x) :=
⟨fun _ => x, fun _ => rfl⟩
theorem callsOn_app
{γ₁ : Sort uu} {γ₂ : Sort ww}
(F₁ : (∀ y, β y) → γ₂ → γ₁) -- can this also support dependent types?
(F₂ : (∀ y, β y) → γ₂)
(h₁ : callsOn P F₁)
(h₂ : callsOn P F₂) :
callsOn P (fun f => F₁ f (F₂ f)) := by
obtain ⟨F₁', h₁⟩ := h₁
obtain ⟨F₂', h₂⟩ := h₂
exists (fun f => F₁' f (F₂' f))
intros; simp_all
theorem callsOn_lam
{γ₁ : Sort uu}
(F : γ₁ → (∀ y, β y) → γ) -- can this also support dependent types?
(h : ∀ x, callsOn P (F x)) :
callsOn P (fun f x => F x f) := by
exists (fun f x => (h x).choose f)
intro f
ext x
apply (h x).choose_spec
theorem callsOn_app2
{γ₁ : Sort uu} {γ₂ : Sort ww}
(g : γ₁ → γ₂ → γ)
(F₁ : (∀ y, β y) → γ₁) -- can this also support dependent types?
(F₂ : (∀ y, β y) → γ₂)
(h₁ : callsOn P F₁)
(h₂ : callsOn P F₂) :
callsOn P (fun f => g (F₁ f) (F₂ f)) := by
apply_rules [callsOn_app, callsOn_const]
With this setup, we can have the following, possibly user-defined, lemma expressing that List.map calls its arguments only on elements of the list:
theorem callsOn_map (δ : Type uu) (γ : Type ww)
(P : α → Prop) (F : (∀ y, β y) → δ → γ) (xs : List δ)
(h : ∀ x, x ∈ xs → callsOn P (fun f => F f x)) :
callsOn P (fun f => xs.map (fun x => F f x)) := by
suffices callsOn P (fun f => xs.attach.map (fun ⟨x, h⟩ => F f x)) by
simpa
apply callsOn_app
· apply callsOn_app
· apply callsOn_const
· apply callsOn_lam
intro ⟨x', hx'⟩
dsimp
exact (h x' hx')
· apply callsOn_const
end setup
So here is the (manual) construction of a nested map for trees:
section examples
structure Tree (α : Type u) where
val : α
cs : List (Tree α)
-- essentially
-- def Tree.map (f : α → β) : Tree α → Tree β :=
-- fun t => ⟨f t.val, t.cs.map Tree.map⟩)
noncomputable def Tree.map (f : α → β) : Tree α → Tree β :=
fix (sizeOf · < sizeOf ·) (fun map t => ⟨f t.val, t.cs.map map⟩)
(InvImage.wf (sizeOf ·) WellFoundedRelation.wf) <| by
intro ⟨v, cs⟩
dsimp only
apply callsOn_app2
· apply callsOn_const
· apply callsOn_map
intro t' ht'
apply callsOn_base
-- ht' : t' ∈ cs -- !
-- ⊢ sizeOf t' < sizeOf { val := v, cs := cs }
decreasing_trivial
end examples
This makes me happy!
All details of the construction are now contained in a proof that can proceed by a syntax-driven tactic and that’s easily and (likely robustly) extensible by the user. It also means that we can share a lot of code paths (e.g. everything related to equational theorems) between well-founded recursion and partial_fixpoint.
I wonder if this construction is really as powerful as our current one, or if there are certain (likely dependently typed) functions where this doesn’t fit, but the β above is dependent, so it looks good.
With this construction, functions defined by well-founded recursion will reduce even worse in the kernel, I assume. This may be a good thing.
The cake is a lie
What unfortunately kills this idea, though, is the generation of the functional induction principles, which I believe is not (easily) possible with this construction: The functional induction principle is proved by massaging F to return a proof, but since the extra assumptions (e.g. for ite or List.map) only exist in the termination proof, they are not available in F.
Oh wey, how anticlimactic.
PS: Path dependencies
Curiously, if we didn’t have functional induction at this point yet, then very likely I’d change Lean to use this construction, and then we’d either not get functional induction, or it would be implemented very differently, maybe a more syntactic approach that would re-prove termination. I guess that’s called path dependence.
There’s yet again been a bit of functional programming-adjacent twitter drama
recently, but it’s actually sort of touched into some subtleties about sum types
that I am asked about (and think about) a lot nowadays. So, I’d like to take
this opportunity to talk a bit about the “why” and nature of sum types and how
to use them effectively, and how they contrast with other related concepts in
programming and software development and when even cases where sum types aren’t
the best option.
Sum Types at their Best
The quintessential sum type that you just can’t live without is
Maybe, now adopted in a lot of languages as
Optional:
dataMaybe a =Nothing|Just a
If you have a value of type Maybe Int, it means that its valid
values are Nothing, Just 0, Just 1,
etc.
This is also a good illustration to why we call it a “sum” type: if
a has n possible values, then Maybe a has
1 + n: we add the single new value Nothing to it.
The “benefit” of the sum type is illustrated pretty clearly here too: every
time you use a value of type Maybe Int, you are forced to
consider the fact that it could be Nothing:
showMaybeInt ::MaybeInt->StringshowMaybeInt = \caseNothing->"There's nothing here"Just i ->"Something is here: "<>show i
That’s because usually in sum type implementations, they are implemented in a
way that forces you to handle each case exhaustively. Otherwise, sum types are
much less useful.
At the most fundamental level, this behaves like a compiler-enforced null
check, but built within the language in user-space instead being compiler magic,
ad-hoc syntax1, or static analysis — and the fact that it
can live in user-space is why it’s been adopted so widely. At a higher level,
functional abstractions like Functor, Applicative, Monad, Foldable, Traversable
allow you to use a Maybe a like just a normal a with
the appropriate semantics, but that’s a
topic for another time (like 2014).
This power is very special to me on a personal level. I remember many years
ago on my first major haskell project changing a type from String
to Maybe String, and then GHC telling me every place in the
codebase where something needed to change in order for things to work still.
Coming from dynamically typed languages in the past, this sublime experience
truly altered my brain chemistry and Haskell-pilled me for the rest of my life.
I still remember the exact moment, what coffee shop I was at, what my order was,
the weather that day … it was truly the first day of the rest of my life.
It should be noted that I don’t consider sum types a “language feature” or a
compiler feature as much as I’d consider it a design pattern. Languages that
don’t have sum types built-in can usually implement them using typed unions and
an abstract visitor pattern interface (more on that later). Of course, having a
way to “check” your code before running it (like with a type system or
statically verified type annotations) does make a lot of the features much more
useful.
Anyway, this basic pattern can be extended to include more error information
in your Nothing branch, which is how you get the
Either e a type in the Haskell standard library, or the
Result<T,E> type in rust.
Along different lines, we have the common use case of defining syntax
trees:
dataExpr=LitInt|NegateExpr|AddExprExpr|SubExprExpr|MulExprExpreval ::Expr->Inteval = \caseLit i -> iNegate x ->-(eval x)Add x y -> eval x + eval ySub x y -> eval x - eval yMul x y -> eval x * eval ypretty ::Expr->Stringpretty = go 0where wrap ::Int->Int->String->String wrap prio opPrec s| prio > opPrec ="("<> s <>")"|otherwise= s go prio = \caseLit i ->show iNegate x -> wrap prio 2$"-"<> go 2 xAdd x y -> wrap prio 0$ go 0 x <>" + "<> go 1 ySub x y -> wrap prio 0$ go 0 x <>" - "<> go 1 yMul x y -> wrap prio 1$ go 1 x <>" * "<> go 2 ymain ::IO ()main =doputStrLn$ pretty myExprprint$ eval myExprwhere myExpr =Mul (Negate (Add (Lit4) (Lit5))) (Lit8)
-(4 + 5) * 8
-72
Now, if we add a new command to the sum type, the compiler enforces us to
handle it.
dataExpr=LitInt|NegateExpr|AddExprExpr|SubExprExpr|MulExprExpr|AbsExpreval ::Expr->Inteval = \caseLit i -> iNegate x ->-(eval x)Add x y -> eval x + eval ySub x y -> eval x - eval yMul x y -> eval x * eval yAbs x ->abs (eval x)pretty ::Expr->Stringpretty = go 0where wrap ::Int->Int->String->String wrap prio opPrec s| prio > opPrec ="("<> s <>")"|otherwise= s go prio = \caseLit i ->show iNegate x -> wrap prio 2$"-"<> go 2 xAdd x y -> wrap prio 0$ go 0 x <>" + "<> go 1 ySub x y -> wrap prio 0$ go 0 x <>" - "<> go 1 yMul x y -> wrap prio 1$ go 1 x <>" * "<> go 2 yAbs x -> wrap prio 2$"|"<> go 0 x <>"|"
Another example where things shine are as clearly-fined APIs between
processes. For example, we can imagine a “command” type that sends different
types of commands with different payloads. This can be interpreted as perhaps
the result of parsing command line arguments or the message in some
communication protocol.
For example, you could have a protocol that launches and controls
processes:
dataCommand a =LaunchString (Int-> a) -- ^ takes a name, returns a process ID|StopInt (Bool-> a) -- ^ takes a process ID, returns success/failurelaunch ::String->CommandIntlaunch nm =Launch nm idstop ::Int->CommandBoolstop pid =Stop pid id
This ADT is written in the “interpreter” pattern (used often with things like
free monad), where any arguments not involving a are the command
payload, any X -> a represent that the command could respond
with X.
Let’s write a sample interpreter backing the state in an IntMap in an
IORef:
importqualifiedData.IntMapasIMimportData.IntMap (IntMap)runCommand ::IORef (IntMapString) ->Command a ->IO arunCommand ref = \caseLaunch newName next ->do currMap <- readIORef reflet newId =case IM.lookupMax currMap ofNothing->0Just (i, _) -> i +1 modifyIORef ref $ IM.insert newId newNamepure (next newId)Stop procId next ->do existed <- IM.member procId <$> readIORef ref modifyIORef ref $ IM.delete procIdpure (next existed)main ::IO ()main =do ref <- newIORef IM.empty aliceId <- runCommand ref $ launch "alice"putStrLn$"Launched alice with ID "<>show aliceId bobId <- runCommand ref $ launch "bob"putStrLn$"Launched bob with ID "<>show bobId success <- runCommand ref $ stop aliceIdputStrLn$if successthen"alice succesfully stopped"else"alice unsuccesfully stopped"print=<< readIORef ref
Launched alice with ID 0
Launched bob with ID 1
alice succesfully stopped
fromList [(1, "bob")]
Let’s add a command to “query” a process id for its current status:
dataCommand a =LaunchString (Int-> a) -- ^ takes a name, returns a process ID|StopInt (Bool-> a) -- ^ takes a process ID, returns success/failure|QueryInt (String-> a) -- ^ takes a process ID, returns a status messagequery ::Int->CommandStringquery pid =Query pid idrunCommand ::IORef (IntMapString) ->Command a ->IO arunCommand ref = \case-- ...Query procId next ->do procName <- IM.lookup procId <$> readIORef refpurecase procName ofNothing->"This process doesn't exist, silly."Just n ->"Process "<> n <>" chugging along..."
Relationship with Unions
To clarify a common confusion: sum types can be described as “tagged unions”:
you have a tag to indicate which branch you are on (which can be case-matched
on), and then the rest of your data is conditionally present.
In many languages this can be implemented under the hood as a struct with a
tag and a union of data, along with some abstract visitor pattern
interface to ensure exhaustiveness.
Remember, it’s not exactly a union, because, ie, consider a type
like:
dataEntity=UserInt|PostInt
An Entity here could represent a user at a user id, or a post at
a post id. If we considered it purely as a union of Int and
Int:
union Entity {int user_id;int post_id;};
we’d lose the ability to branch on whether or not we have a user or an int.
If we have the tagged union, we recover the original tagged union semantics:
Of course, you still need an abstract interface like the visitor pattern to
actually be able to use this as a sum type with guarantees that you handle every
branch, but that’s a story for another day. Alternatively, if your language
supports dynamic dispatch nicely, that’s another underlying implementation that
would work to back a higher-level visitor pattern interface.
Subtypes Solve a Different
Problem
Now, sum types aren’t exactly a part of common programming education
curriculum, but subtypes and supertypes definitely were
drilled into every CS student’s brain and waking nightmares from their first
year.
Informally (a la Liskov), B is a subtype of A (and
A is a supertype of B) if anywhere that expects an
A, you could also provide a B.
In normal object-oriented programming, this often shows up in early lessons
as Cat and Dog being subclasses of an
Animal class, or Square and Circle being
subclasses of a Shape class.
When people first learn about sum types, there is a tendency to understand
them as similar to subtyping. This is unfortunately understandable, since a lot
of introductions to sum types often start with something like
-- | Bad Sum Type Example!dataShape=CircleDouble|RectangleDoubleDouble
While there are situations where this might be a good sum type (ie, for an
API specification or a state machine), on face-value this is a bad example on
the sum types vs. subtyping distinction.
You might notice the essential “tension” of the sum type: you declare all of
your options up-front, the functions that consume your value are open and
declared ad-hoc. And, if you add new options, all of the consuming functions
must be adjusted.
So, subtypes (and supertypes) are more effective when they lean into
the opposite end: the universe of possible options are open and declared ad-hoc,
but the consuming functions are closed. And, if you add new functions,
all of the members must be adjusted.
In typed languages with a concept of “objects” and “classes”, subtyping is
often implemented using inheritance and interfaces.
So, a function like processWidget(Widget widget) that expects a
Widget would be able to be passed a Button or
InputField or Box. And, if you had a container like
List<Widget>, you could assemble a structure using
Button, InputField, and Box. A perfect
Liskov storm.
In typical library design, you’re able to add new implementations of
Widget as an open universe easily: anyone that imports
Widget can, and they can now use it with functions taking
Widgets. But, if you ever wanted to add new functionality
to the Widget interface, that would be a breaking change to all
downstream implementations.
However, this implementation of subtyping, while prevalent, is the most
mind-numbly boring realization of the concept, and it pained my soul to even
spend time talking about it. So let’s jump into the more interesting way that
subtype and supertype relationships manifest in the only language where anything
is interesting: Haskell.
Subtyping via Parametric
Polymorphism
In Haskell, subtyping is implemented in terms of parametric polymorphism and
sometimes typeclasses. This allows for us to work nicely with the concept of
functions and APIs as subtypes and supertypes of each other.
For example, let’s look at a function that takes indexers and applies
them:
So, what functions could you pass to sumAtLocs? Can you
only pass [Double] -> Int -> Double?
Well, not quite. Look at the above where we passed (!!), which
has type forall a. [a] -> Int -> a!
In fact, what other types could we pass? Here are some examples:
fun1 :: [a] ->Int-> afun1 = (!!)fun2 :: [a] ->Int-> afun2 xs i =reverse xs !! ifun3 :: (Foldable t, Floating a) => t a ->Int-> afun3 xs i =iflength xs > i then xs !! i elsepifun4 ::Num a => [a] ->Int-> afun4 xs i =sum (take i xs)fun5 :: (Integral b, Num c) => a -> b -> cfun5 xs i =fromIntegral ifun5 :: (Foldable t, Fractional a, Integral b) => t a -> b -> afun5 xs i =sum xs /fromIntegral ifun5 :: (Foldable t, Integral b, Floating a) => t a -> b -> afun5 xs i =logBase (fromIntegral i) (sum xs)
What’s going on here? Well, the function expects a
[Double] -> Int -> Double, but there are a lot of other types
that could be passed instead.
At first this might seem like meaningless semantics or trickery, but it’s
deeper than that: remember that each of the above types actually has a very
different meaning and different possible behaviors!
forall a. [a] -> Int -> a means that the amust come from the given list. In fact, any function with that type is
guaranteed to be partial: if you pass it an empty list, there is no
a available to use.
forall a. Num a => [a] -> Int -> a means that the
result might actually come from outside of the list: the implementation could
always return 0 or 1, even if the list is empty. It
also guarantees that it will only add, subtract, multiply, or abs: it will never
divide.
forall a. Fractional a => [a] -> Int -> a means that
we could possibly do division on the result, but we can’t do anything “floating”
like square rooting or logarithms.
forall a. Floating a => [a] -> Int -> a means that we
can possibly start square rooting or taking the logarithms of our input
numbers
[Double] -> Int -> Double gives us the least guarantees
about the behavior: the result could come from thin air (and not be a part of
the list), and we can even inspect the machine representation of our
inputs.
So, we have all of these types with completely different semantics and
meanings. And yet, they can all be passed to something expecting a
[Double] -> Int -> Double. That means that they are all
subtypes of [Double] -> Int -> Double!
[Double] -> Int -> Double is a supertype that houses
multitudes of possible values, uniting all of the possible values and semantics
into one big supertype.
Through the power of parametric polymorphism and typeclasses, you can
actually create an extensible hierarchy of supertypes, not just of
subtypes.
Consider a common API for json serialization. You could have multiple
functions that serialize into JSON:
The type of toJSON :: forall a. JSON a => a -> Value is a
subtype of Foo -> Value, Bar -> Value, and
Baz -> Value, because everywhere you would want a
Foo -> Value, you could give toJSON instead. Every
time you want to serialize a Foo, you could use
toJSON.
This usage works well, as it gives you an extensible abstraction to design
code around. When you write code polymorphic over Monoid a, it
forces you to reason about your values with respect to only the aspects relating
to monoidness. If you write code polymorphic over Num a, it forces
you to reason about your values only with respect to how they can be added,
subtracted, negated, or multiplied, instead of having to worry about things like
their machine representation.
The extensibility comes from the fact that you can create even more
supertypes of forall a. ToJSON a => a -> Value easily,
just by defining a new typeclass instance. So, if you need a
MyType -> Value, you could make it a supertype of
toJSON :: ToJSON a => a -> Value by defining an instance of
the ToJSON typeclass, and now you have something you can use in its
place.
Practically this is used by many libraries. For example, ad uses it for automatic
differentiation: its diff function looks scary:
diff :: (forall s.AD s ForwardDouble->AD s ForwardDouble) ->Double->Double
But it relies on the fact that that
(forall s. AD s ForwardDouble -> AD s ForwardDuble) is a
superclass of (forall a. Floating a => a -> a),
(forall a. Num a => a -> a), etc., so you can give it
functions like \x -> x * x (which is a
forall a. Num a => a -> a) and it will work as that
AD s type:
ghci> diff (\x -> x * x) 1020-- 2*x
This “numeric overloading” method is used by libraries for GPU programming,
as well, to accept numeric functions to be optimized and compiled to GPU
code.
Another huge application is in the lens library, which
uses subtyping to unite its hierarchy of optics.
For example, an Iso is a subtype of Traversal which
is a subtype of Lens, and Lens is a supertype of
Fold and Traversal, etc. In the end the system even
allows you to use id from the Prelude as a lens or a
traversal, because the type signature of id :: a -> a is
actually a subtype of all of those types!
Subtyping using Existential
Types
What more closely matches the spirit of subtypes in OOP and other
languages is the existential type: a value that can be a value of any
type matching some interface.
For example, let’s imagine a value that could be any instance of
Num:
This is somewhat equivalent to Java’s
List<MyInterface> or List<MyClass>, or
python’s List[MyClass].
Note that to use this effectively in Haskell with superclasses and
subclasses, you need to manually wrap and unwrap:
dataSomeFrational=forall a.Fractional a =>SumFractional acastUp ::SomeFractional->SumNumcastUp (SomeFractional x) =SomeNum x
So, SomeNum is “technically” a supertype of
SomeFractional: everywhere a SomeNum is expected, a
SomeFractional can be given…but in Haskell it’s a lot less
convenient because you have to explicitly cast.
In OOP languages, you can often cast “down” using runtime reflection
(SomeNum -> Maybe SomeFractional). However, this is impossible
in Haskell the way we have written it!
That’s because of type erasure: Haskell does not (by default) couple a value
at runtime with all of its associated interface implementations. When you create
a value of type SomeNum, you are packing an untyped pointer to that
value as well as a “dictionary” of all the functions you could use it with:
dataNumDict a =NumDict { (+) :: a -> a -> a , (*) :: a -> a -> a , negate :: a -> a , abs :: a -> a , fromInteger ::Integer-> a }mkNumDict ::Num a =>NumDict amkNumDict =NumDict (+) (*) negateabsfromIntegerdataFractionalDict a =FractionalDict { numDict ::NumDict a , (/) :: a -> a -> a , fromRational ::Rational-> a }-- | Essentially equivalent to the previous 'SomeNum'dataSomeNum=forall a.SomeNum { numDict ::NumDict a , value :: a }-- | Essentially equivalent to the previous 'SomeFractional'dataSomeFractional=forall a.SomeFractional { fractionalDict ::FractionalDict a , value :: a }castUp ::SomeFractional->SomeNumcastUp (SomeFractional (FractionalDict {numDict}) x) =SomeNum d xcastDown ::SomeNum->MaybeSomeFractionalcastDown (SomeNum nd x) =error"not possible!"
All of these function pointers essentially exist at runtime inside
the SomeNum. So, SomeFractional can be “cast up” to
SomeNum by simply dropping the FractionalDict.
However, you cannot “cast down” from SomeNum because there is no
way to materialize the FractionalDict: the association from type to
instance is lost at runtime. OOP languages usually get around this by having the
value itself hold pointers to all of its interface implementations at
runtime. However, in Haskell, we have type erasure by default: there are no
tables carried around at runtime.2
In the end, existential subtyping requires explicit wrapping/unwrapping
instead of implicit or lightweight casting possible in OOP languages optimized
around this sort of behavior.3 Existential-based subtyping is just less
common in Haskell because parametric polymorphism offers a solution to most
similar problems. For more on this topic, Simon Peyton Jones has a nice lecture on the
topic.
The pattern of using existentially qualified data in a container
(like [SomeNum]) is often called the “widget pattern” because it’s
used in libraries like xmonad to allow
extensible “widgets” stored alongside the methods used to manipualte them. It’s
more common to explicitly store the handler functions (a “dictionary”) inside
the type instead of of existential typeclasses, but sometimes it can be nice to
let the compiler handle generating and passing your method tables implicitly for
you. Using existential typeclasses instead of explicit dictionaries also allows
you to bless certain methods and functions as “canonical” to your type, and the
compiler will make sure they are always coherent.
I do mention in a blog
post about different types of existential lists, however, that this
“container of instances” type is much less useful in Haskell than in other
languages for many reasons, including the up/downcasting issues mentioned above.
In addition, Haskell gives you a whole wealth of functionality to operate over
homogeneous parameters (like [a], where all items have the same
type) that jumping to heterogeneous lists gives up so much.
Aside
Let’s briefly take a moment to talk about how typeclass hierarchies give us
subtle subtype/supertype relationships.
Let’s look at the classic Num and Fractional:
classNum aclassNum a =>Fractional a
Num is a superclass of Fractional, and
Fractional is a subclass of Num. Everywhere a
Num constraint is required, you can provide a
Fractional constraint to do the same thing.
However, in these two types:
Num a => aFractional a => a
forall a. Num a => a is actually a subclass of
forall a. Fractional a => a! That’s because if you need a
forall a. Fractional a => a, you can provide a
forall a. Num a => a instead. In fact, let’s look at three
levels: Double, forall a. Fractional a => a, and
forall a. Num a => a.
-- can be used as `Double`1.0 ::Double1.0 ::Fractional a => a1 ::Num a => a-- can be used as `forall a. Fractional a => a`1.0 ::Fractional a => a1 ::Num a => a-- can be used as `forall a. Num a => a`1 ::Num a => a
So, Double is a supertype of Fractional a => a
is a supertype of Num a => a.
The general idea here is that the more super- you go, the more you “know”
about the actual term you are creating. So, with Num a => a, you
know the least (and, you have the most possible actual terms because
there are more instances of Num than of Fractional).
And, with Double, you know the most: you even know its
machine representation!
So, Num is a superclass of Fractional but
forall a. Num a => a is a subclass of
forall a. Fractional a => a. This actually follows the typical
rules of subtyping: if something appears on the “left” of an arrow
(=> in this case), it gets flipped from sub- to super-. We often
call the left side a “negative” (contravariant) position and the right side a
“positive” position, because a negative of a negative (the left side of a left
size, like a in (a -> b) -> c) is a
positive.
Also note that our “existential wrappers”:
dataSomeNum=forall a.Num a =>SomeFractional adataSomeFractional=forall a.Fractional a =>SomeFractional a
can be CPS-transformed to their equivalent types:
typeSomeNum'=forall r. (forall a.Num a => a -> r) -> rtypeSomeFractional'=forall r. (forall a.Fractional a => a -> r) -> rtoSomeNum' ::SomeNum->SomeNum'toSomeNum' (SomeNum x) f = f xtoSomeNum ::SomeNum'->SomeNumtoSomeNum sn = sn SomeNum
And in those cases, Num and Fractional again appear
in the covariant (positive) position, since they’re the negative of negative.
So, this aligns with our intuition that SomeFractional is a subtype
of SomeNum.
The Expression Problem
This tension that I described earlier is closely related to the expression
problem, and is a tension that is inherent to a lot of different aspects of
language and abstraction design. However, in the context laid out in this post,
it serves as a good general guide to decide what pattern to go down:
If you expect a canonical set of “inhabitants” and an open set of
“operations”, sum types can suit that end of the spectrum well.
If you expect a canonical set of “operations” and an open set of
“inhabitants”, consider subtyping and supertyping.
I don’t really think of the expression problem as a “problem” in the sense of
“some hindrance to deal with”. Instead, I see it in the “math problem” sort of
way: by adjusting how you approach things, you can play with the equation make
the most out of what requirements you need in your design.
Looking Forward
A lot of frustration in Haskell (and programming in general) lies in trying
to force abstraction and tools to work in a way they weren’t meant to. Hopefully
this short run-down can help you avoid going against the point of these
design patterns and start making the most of what they can offer. Happy
Haskelling!
Special Thanks
I am very humbled to be supported by an amazing community, who make it
possible for me to devote time to researching and writing these posts. Very
special thanks to my supporter at the “Amazing” level on patreon, Josh Vera! :)
Must OOP languages also have mechanisms for type erasure, but
the default is unerased, which is opposite of Haskell.↩︎
Note that there are current GHC proposals
that attempt to allow “naked” existentials without newtype wrappers, so we could
actually get the same seamless and implicit up-casting as we would get in OOP
languages. However, the jury is out on whether or not this is a good idea.↩︎
Four years ago I bought a pair of YubiKey 5s:
One YubiKey 5 Nano, which fits in my laptop’s USB slot, and another YubiKey 5 NFC as backup, which sat in my home office.
However, I kept worrying about what happens if my house burns down or something, taking both my laptop and office YubiKeys together at the same time.
On the otherhand, if I stored my YubiKey 5 NFC offsite, then whenever I needed to register a new FIDO service, I would need to go fetch the key, update it, and then return it.
Based my peronal experince, even if that were not a big pain, the "return it" step often gets delayed indefinitely because it feels so low priority.
Then I read a popular comment made on Hacker News: Get three YubiKeys.
Suddenly everything clicked!
I bought a second YubiKey 5 NFC last year.
Now, I keep a second YubiKey 5 NFC offsite, in addition to the one in my laptop and the one in my office.
If my home burns down, I still have an offsite YubiKey available.
But the best thing about having a second YubiKey 5 NFC is that it partly mitigates the offsite update problem.
In the previous scenario, we required potentially two trips offsite to update the backup YubiKey.
However, now the procedure is to register a new FIDO service is to first update the office YubiKey 5 NFC key (and the YubiKey 5 nano).
Then, at your earlist convienence, you swap the office YubiKey 5 NFC key with the offsite YubiKey 5 NFC.
When you get the offsite YubiKey home, you update it with the new FIDO service and then it becomes the new office YubiKey.
There is no need to return to the offsite location.
Part of the issue is that there is no "public FIDO key", like there is with a "public PGP key".
You need the acutual YubiKey in hand to register it with a FIDO service, no matter whether it is a discoverable credetial or not.
If you were only using the YubiKey as a OpenPGP smart card, the perhaps you could get away with just having a local key and an offsite key.
Even still, I would recommend a third YubiKey so that whenever the time comes to do some operation on your offsite key, you can perform the same swaping trick.
The title of this article says that three is the right number of YubiKeys.
However this is because I only have one nano in my laptop because that is my primary computing interface.
I do have a desktop computer that I mostly only access as a remote server.
If you have multiple computer devices that you regularly use, it would make sense to have a YubiKey nano device in each of them.
And in addition to those, have one offsite key, and one local key for swapping with the offsite key.
The GHC developers are happy to announce the availability of the first and
likely final release candidate of GHC 9.12.2. This is an important bug-fix
release resolving a significant correctness issue present in 9.12.1
(#25653).
In accordance with our under-discussion release policies this candidate
will have a two-week testing window. The final 9.12.2 release will likely come
the week of 12 March 2025.
As always, if you find anything amiss please open a ticket.
In this episode Wouter Swiestra and Niki Vazou talk with Conal Elliott. Conal discusses doing things just for the poetry, how most programs miss their purpose, and the simplest way to ask a question. Conal is currently working on a book about his ideas and actively looking for partners.
Regular, everyday stuff. But the instances for type constructors are more interesting, because they come with an instance context:
instance (Foo a, Foo b) =>Foo (a, b) where...
Then, of course, if we know both Foo a and Foo b, we can infer Foo (a, b). To make this fact overwhelmingly explicit, we can reify the usual constraint-solving logic by using the Dict type, and thus the following program will typecheck:
with the only change required coming from the type constructor instances:
instance (Foo a, Foo b) =>Foo (a, b) wheretypeEvidence (a, b) = (Foo a, Foo b)...
or, if we you want to be cute about it:
instanceEvidence (a, b) =>Foo (a, b) wheretypeEvidence (a, b) = (Foo a, Foo b)...
By sticking Evidence into the superclass constraint, GHC knows that this dictionary is always available when you’ve got a Foo dictionary around. And our earlier backwards program now typechecks as expected.
Recently I looked again at PHOAS, and once again I concluded it's nice for library APIs, but so painful to do anything with inside those libraries. So let convert to something else, like de Bruijn.
There are standalone source files if you just want to see the code:
There is always a way to cheat, though. You can turn the PHOAS ->
untyped de Bruijn machinery into the PHOAS -> typed de Bruijn
machinery by checking that future contexts indeed extend past contexts
and throwing an error otherwise (which can't happed, because future
contexts always extend past contexts, but it's a metatheorem).
In "Generic Conversions of Abstract Syntax Representation" by Steven Keuchel and Johan Jeuring, authors also "cheat" a bit. The "Parametrhic higher-order abstract syntax" section ends with a somewhat disappointing
wherepostulate whatever :_
Keuchel and Jeuring also mention "Unembedding Domain-Specific Languages" by Robert Atkey, Sam Lindley and Jeremy Yallop; where there is one unsatisfactory ⊥ (undefined in Haskell) hiding.
I think that for practical developments (say a library in Haskell), it is ok to make a small short cut; but I kept wondering isn't there is a way to make a conversion without cheating.
Well... it turns out that we cannot "cheat". Well-formedness of PHOAS representation depends on parametricity, and the conversion challenge seems to requires a theorem which there are no proof in Agda.
In unpublished (?) work Adam Chlipala shows a way to do the conversion without relying on postulates http://adam.chlipala.net/cpdt/html/Intensional.html; but that procedure requires an extra well formedness proof of given PHOAS term.
This Agda development is a translation of that developement.
Common setup
Our syntax representations will be well-typed, so we need types:
-- Typesdata Ty :Setwhere emp : Ty fun : Ty → Ty → TyCtx :SetCtx = List Tyvariable A B C : Ty Γ Δ Ω : Ctx v : Ty →Set
de Bruijn syntax
Var : Ctx → Ty →SetVar Γ A = Idx A Γ -- from agda-np, essentially membership relation.data DB (Γ : Ctx): Ty →Setwhere var : Var Γ A → DB Γ A app : DB Γ (fun A B)→ DB Γ A → DB Γ B lam : DB (A ∷ Γ) B → DB Γ (fun A B) abs : DB Γ emp → DB Γ A
Parametric Higher-order abstract syntax
data PHOAS (v : Ty →Set): Ty →Setwhere var : v A → PHOAS v A app : PHOAS v (fun A B)→ PHOAS v A → PHOAS v B lam :(v A → PHOAS v B)→ PHOAS v (fun A B) abs : PHOAS v emp → PHOAS v A-- closed "true" PHOAS terms.PHOAS° : Ty →Set₁PHOAS° A =∀{v}→ PHOAS v A
de Bruijn to PHOAS
This direction is trivial. An anecdotal evidence that de Bruijn representation is easier to transformation on.
phoasify : NP v Γ → DB Γ A → PHOAS v Aphoasify γ (var x)= var (lookup γ x)phoasify γ (app f t)= app (phoasify γ f)(phoasify γ t)phoasify γ (lam t)= lam λ x → phoasify (x ∷ γ) tphoasify γ (abs t)= abs (phoasify γ t)
Interlude: Well-formedness of PHOAS terms
dam Chlipala defines an equivalence relation between two PHOAS terms, exp_equiv in Intensional, wf in CPDT book). e only need a single term well-formedness so can do a little less
Terms like invalid cannot be values of PHOAS°, as all values of "v" inside PHOAS° have to originated from lam-constructor abstractions. We really should keep v parameter free, i.e. parametric, when constructingPHOAS terms.
The idea is then to simply to track which variables (values of v) are intoduced by lambda abstraction.
data phoasWf {v :Ty→Set} (G:List (Σ Ty v)) : {A:Ty} →PHOAS v A→Setwhere varWf :∀ {A} {x : v A}→Idx (A , x) G→ phoasWf G (var x) appWf :∀ {AB} {f :PHOAS v (fun AB)} {t :PHOAS v A}→ phoasWf G f→ phoasWf G t→ phoasWf G (app f t) lamWf :∀ {AB} {f : v A→PHOAS v B}→ (∀ (x : v A) → phoasWf ((A , x) ∷G) (f x))→ phoasWf G (lam f) absWf :∀ {A} {t :PHOAS v emp}→ phoasWf G t→ phoasWf G (abs {A=A} t)-- closed terms start with an empty GphoasWf° :PHOAS° A→Set₁phoasWf° tm =∀ {v} → phoasWf {v = v} [] tm
A meta theorem is then that all PHOASᵒ terms are well-formed, i.e.
meta-theorem-proposition :Set₁meta-theorem-proposition =∀{A}(t : PHOAS° A)→ phoasWf° t
As far as I'm aware this proposition cannot be proved nor refuted in Agda.
de Bruijn to PHOAS translation creates well-formed PHOAS terms.
As a small exercise we can show that phoasify of closed de Bruijn terms creates well-formed PHOAS terms.
toList : NP v Γ → List (Σ Ty v)toList [] = []toList (x ∷ xs)=(_ , x) ∷ toList xsphoasifyWfVar :(γ : NP v Γ)(x : Var Γ A)→ Idx (A , lookup γ x)(toList γ)phoasifyWfVar (x ∷ γ) zero = zerophoasifyWfVar (x ∷ γ)(suc i)= suc (phoasifyWfVar γ i)phoasifyWf :(γ : NP v Γ)(t : DB Γ A)→ phoasWf (toList γ)(phoasify γ t)phoasifyWf γ (var x)= varWf (phoasifyWfVar γ x)phoasifyWf γ (app f t)= appWf (phoasifyWf γ f)(phoasifyWf γ t)phoasifyWf γ (lam t)= lamWf λ x → phoasifyWf (x ∷ γ) tphoasifyWf γ (abs t)= absWf (phoasifyWf γ t)phoasifyWf° :(t : DB [] A)→ phoasWf° (phoasify [] t)phoasifyWf° t = phoasifyWf [] t
PHOAS to de Bruijn
The rest deals with the opposite direction.
In Intensional Adam Chlipala uses v = λ _ → ℕ instatiation to make the translation.
I think that in the typed setting using v = λ _ → Ctx turns out nicer.
The idea in both is that we instantiate PHOAS variables to be de Bruijn levels.
data IsSuffixOf {ℓ}{a :Set ℓ}: List a → List a →Set ℓ where refl :∀{xs}→ IsSuffixOf xs xs cons :∀{xs ys}→ IsSuffixOf xs ys →∀{y}→ IsSuffixOf xs (y ∷ ys)
We need to establish well-formedness of PHOAS expression in relation to some context Γ
Note that variables encode de Bruijn levels, thus the contexts we "remember" in variables should be the suffix of that outside context.
wf :(Γ : Ctx)→ PHOAS (λ_→ Ctx) A →Setwf {A = A} Γ (var Δ)= IsSuffixOf (A ∷ Δ) Γwf Γ (app f t)= wf Γ f × wf Γ twf Γ (lam {A = A} t)= wf (A ∷ Γ)(t Γ)wf Γ (abs t)= wf Γ t
And if (A ∷ Δ) is suffix of context Γ, we can convert the evidence to the de Bruijn index (i.e. variable):
makeVar : IsSuffixOf (A ∷ Δ) Γ → Var Γ AmakeVar refl = zeromakeVar (cons s)= suc (makeVar s)
Given the term is well-formed in relation to context Γ we can convert it to de Bruijn representation.
dbify :(t : PHOAS (λ_→ Ctx) A)→ wf Γ t → DB Γ Adbify (var x) wf = var (makeVar wf)dbify (app f t)(fʷ , tʷ)= app (dbify f fʷ)(dbify t tʷ)dbify {Γ = Γ}(lam t) wf = lam (dbify (t Γ) wf)dbify (abs t) wf = abs (dbify t wf)
What is left is to show that we can construct wf for all phoasWf-well-formed terms.
Adam Chlipala defines a helper function:
makeG′ : Ctx → List (Σ Ty (λ_→ Ctx))makeG′ [] = []makeG′ (A ∷ Γ)=(A , Γ) ∷ makeG′ Γ
However for somewhat technical reasons, we rather define
Normalization by evaluation using parametric higher order syntax. In Agda.
I couldn't find a self-contained example of PHOAS NbE, so here it is. I hope someone might find it useful.
module NbEXP.PHOAS wheredata Ty :Setwhere emp : Ty fun : Ty → Ty → Tydata Tm (v : Ty →Set): Ty →Setwhere var :∀{a}→ v a → Tm v a app :∀{a b}→ Tm v (fun a b)→ Tm v a → Tm v b lam :∀{a b}→(v a → Tm v b)→ Tm v (fun a b)data Nf (v : Ty →Set): Ty →Setdata Ne (v : Ty →Set): Ty →Setdata Ne v where nvar :∀{a}→ v a → Ne v a napp :∀{a b}→ Ne v (fun a b)→ Nf v a → Ne v bdata Nf v where neut : Ne v emp → Nf v emp nlam :∀{a b}→(v a → Nf v b)→ Nf v (fun a b)Sem :(Ty →Set)→ Ty →SetSem v emp = Ne v empSem v (fun a b)= Sem v a → Sem v blower :∀{v : Ty →Set}(a : Ty)→ Sem v a → Nf v araise :∀{v : Ty →Set}(a : Ty)→ Ne v a → Sem v alower emp s = neut slower (fun a b) s = nlam λ x → lower b (s (raise a (nvar x)))raise emp n = nraise (fun a b) n x = raise b (napp n (lower a x ))eval :{v : Ty →Set}{a : Ty}→ Tm (Sem v) a → Sem v aeval (var x)= xeval (app f t)= eval f (eval t)eval (lam t) x = eval (t x)nf :{a : Ty}→{v : Ty →Set}→ Tm (Sem v) a → Nf v anf {a} t = lower a (eval t)nf_parametric :{a : Ty}→({v : Ty →Set}→ Tm v a)->({v : Ty →Set}→ Nf v a)nf_parametric t = nf t
This last month has been fascinating. I guess LLMs have finally
resonated with me on a deeper level. It wasn’t like I woke up and
suddenly everything was different, but their impact is growing on me
non-linearly, forcing me to rewire my brain.
I've been fortunate to be nominated for a few teaching awards over my career, and even to win a couple. The nomination I just received may be the best.
As a new student at the uni, Philip Wadler was the first introductory lecture I had, and his clear passion for the subject made me feel excited to begin my journey in computer science. In particular he emphasised the importance of asking questions, which made the idea of tutorials and lectures a lot less intimidating, and went on to give really valuable advice for starting university. I enjoyed this session so much, and so was looking forward to the guest lectures he was going to do for Inf1A at the end of semester 1. They certainly did not disappoint, the content he covered was engaging, interesting, and above all very entertaining to listen to, especially when he dressed up as a superhero to cement his point. Because I found these talks so rewarding, I also attended the STMU that he spoke at about AI and ChatGPT, and everyone I talked to after the event said they had a really good time whilst also having a completely new insightful perspective on the topic. In summary, Philip Wadler has delivered the best lectures I have attended since starting university, and I have gotten a lot out of them.
President Trump has started rolling out his tariffs, something I blogged about in November. People are talking about these tariffs a lot right now, with many people (correctly) commenting on how consumers will end up with higher prices as a result of these tariffs. While that part is true, I’ve seen a lot of people taking it to the next, incorrect step: that consumers will pay the entirety of the tax. I put up a poll on X to see what people thought, and while the right answer got a lot of votes, it wasn't the winner.
Checking on people's general view of taxes. When the government imposes a tax on trade (sales tax, VAT, tariff, or even payroll tax), which party absorbs the cost of the tax?
For purposes of this blog post, our ultimate question will be the following:
Suppose apples currently sell for $1 each in the entire United States.
There are domestic sellers and foreign sellers of apples, all receiving the same price.
There are no taxes or tariffs on the purchase of apples.
The question is: if the US federal government puts a $0.50 import tariff per apple, what will be the change in the following:
Number of apples bought in the US
Price paid by buyers for apples in the US
Post-tax price received by domestic apple producers
Post-tax price received by foreign apple producers
Before we can answer that question, we need to ask an easier, first question: before instituting the tariff, why do apples cost $1?
And finally, before we dive into the details, let me provide you with the answers to the ultimate question. I recommend you try to guess these answers before reading this, and if you get it wrong, try to understand why:
The number of apples bought will go down
The buyers will pay more for each apple they buy, but not the full amount of the tariff
Domestic apple sellers will receive a higher price per apple
Foreign apple sellers will receive a lower price per apple, but not lowered by the full amount of the tariff
In other words, regardless of who sends the payment to the government, both taxed parties (domestic buyers and foreign sellers) will absorb some of the costs of the tariff, while domestic sellers will benefit from the protectionism provided by tariffs and be able to sell at a higher price per unit.
Let’s say I absolutely love apples, they’re my favorite food. How much would I be willing to pay for a single apple? You might say “$1, that’s the price in the supermarket,” and in many ways you’d be right. If I walk into supermarket A, see apples on sale for $50, and know that I can buy them at supermarket B for $1, I’ll almost certainly leave A and go buy at B.
But that’s not what I mean. What I mean is: how high would the price of apples have to go everywhere so that I’d no longer be willing to buy a single apple? This is a purely personal, subjective opinion. It’s impacted by how much money I have available, other expenses I need to cover, and how much I like apples. But let’s say the number is $5.
How much would I be willing to pay for another apple? Maybe another $5. But how much am I willing to pay for the 1,000th apple? 10,000th? At some point, I’ll get sick of apples, or run out of space to keep the apples, or not be able to eat, cook, and otherwise preserve all those apples before they rot.
The point being: I’ll be progressively willing to spend less and less money for each apple. This form of analysis is called marginal benefit: how much benefit (expressed as dollars I’m willing to spend) will I receive from each apple? This is a downward sloping function: for each additional apple I buy (quantity demanded), the price I’m willing to pay goes down. This is what gives my personal demand curve. And if we aggregate demand curves across all market participants (meaning: everyone interested in buying apples), we end up with something like this:
Assuming no changes in people’s behavior and other conditions in the market, this chart tells us how many apples will be purchased by our buyers at each price point between $0.50 and $5. And ceteris paribus (all else being equal), this will continue to be the demand curve for apples.
Marginal cost
Demand is half the story of economics. The other half is supply, or: how many apples will I sell at each price point? Supply curves are upward sloping: the higher the price, the more a person or company is willing and able to sell a product.
Let’s understand why. Suppose I have an apple orchard. It’s a large property right next to my house. With about 2 minutes of effort, I can walk out of my house, find the nearest tree, pick 5 apples off the tree, and call it a day. 5 apples for 2 minutes of effort is pretty good, right?
Yes, there was all the effort necessary to buy the land, and plant the trees, and water them… and a bunch more than I likely can’t even guess at. We’re going to ignore all of that for our analysis, because for short-term supply-and-demand movement, we can ignore these kinds of sunk costs. One other simplification: in reality, supply curves often start descending before ascending. This accounts for achieving efficiencies of scale after the first number of units purchased. But since both these topics are unneeded for understanding taxes, I won’t go any further.
Anyway, back to my apple orchard. If someone offers me $0.50 per apple, I can do 2 minutes of effort and get $2.50 in revenue, which equates to a $75/hour wage for me. I’m more than happy to pick apples at that price!
However, let’s say someone comes to buy 10,000 apples from me instead. I no longer just walk out to my nearest tree. I’m going to need to get in my truck, drive around, spend the day in the sun, pay for gas, take a day off of my day job (let’s say it pays me $70/hour). The costs go up significantly. Let’s say it takes 5 days to harvest all those apples myself, it costs me $100 in fuel and other expenses, and I lose out on my $70/hour job for 5 days. We end up with:
Total expenditure: $100 + $70 * 8 hours a day * 5 days == $2900
Total revenue: $5000 (10,000 apples at $0.50 each)
Total profit: $2100
So I’m still willing to sell the apples at this price, but it’s not as attractive as before. And as the number of apples purchased goes up, my costs keep increasing. I’ll need to spend more money on fuel to travel more of my property. At some point I won’t be able to do the work myself anymore, so I’ll need to pay others to work on the farm, and they’ll be slower at picking apples than me (less familiar with the property, less direct motivation, etc.). The point being: at some point, the number of apples can go high enough that the $0.50 price point no longer makes me any money.
This kind of analysis is called marginal cost. It refers to the additional amount of expenditure a seller has to spend in order to produce each additional unit of the good. Marginal costs go up as quantity sold goes up. And like demand curves, if you aggregate this data across all sellers, you get a supply curve like this:
Equilibrium price
We now know, for every price point, how many apples buyers will purchase, and how many apples sellers will sell. Now we find the equilibrium: where the supply and demand curves meet. This point represents where the marginal benefit a buyer would receive from the next buyer would be less than the cost it would take the next seller to make it. Let’s see it in a chart:
You’ll notice that these two graphs cross at the $1 price point, where 63 apples are both demanded (bought by consumers) and supplied (sold by producers). This is our equilibrium price. We also have a visualization of the surplus created by these trades. Everything to the left of the equilibrium point and between the supply and demand curves represents surplus: an area where someone is receiving something of more value than they give. For example:
When I bought my first apple for $1, but I was willing to spend $5, I made $4 of consumer surplus. The consumer portion of the surplus is everything to the left of the equilibrium point, between the supply and demand curves, and above the equilibrium price point.
When a seller sells his first apple for $1, but it only cost $0.50 to produce it, the seller made $0.50 of producer surplus. The producer portion of the surplus is everything to the left of the equilibrium point, between the supply and demand curves, and below the equilibrium price point.
Another way of thinking of surplus is “every time someone got a better price than they would have been willing to take.”
OK, with this in place, we now have enough information to figure out how to price in the tariff, which we’ll treat as a negative externality.
Modeling taxes
Alright, the government has now instituted a $0.50 tariff on every apple sold within the US by a foreign producer. We can generally model taxes by either increasing the marginal cost of each unit sold (shifting the supply curve up), or by decreasing the marginal benefit of each unit bought (shifting the demand curve down). In this case, since only some of the producers will pay the tax, it makes more sense to modify the supply curve.
First, let’s see what happens to the foreign seller-only supply curve when you add in the tariff:
With the tariff in place, for each quantity level, the price at which the seller will sell is $0.50 higher than before the tariff. That makes sense: if I was previously willing to sell my 82nd apple for $3, I would now need to charge $3.50 for that apple to cover the cost of the tariff. We see this as the tariff “pushing up” or “pushing left” the original supply curve.
We can add this new supply curve to our existing (unchanged) supply curve for domestic-only sellers, and we end up with a result like this:
The total supply curve adds up the individual foreign and domestic supply curves. At each price point, we add up the total quantity each group would be willing to sell to determine the total quantity supplied for each price point. Once we have that cumulative supply curve defined, we can produce an updated supply-and-demand chart including the tariff:
As we can see, the equilibrium has shifted:
The equilibrium price paid by consumers has risen from $1 to $1.20.
The total number of apples purchased has dropped from 63 apples to 60 apples.
Consumers therefore received 3 less apples. They spent $72 for these 60 apples, whereas previously they spent $63 for 3 more apples, a definite decrease in consumer surplus.
Foreign producers sold 36 of those apples (see the raw data in the linked Google Sheet), for a gross revenue of $43.20. However, they also need to pay the tariff to the US government, which accounts for $18, meaning they only receive $25.20 post-tariff. Previously, they sold 42 apples at $1 each with no tariff to be paid, meaning they took home $42.
Domestic producers sold the remaining 24 apples at $1.20, giving them a revenue of $28.80. Since they don’t pay the tariff, they take home all of that money. By contrast, previously, they sold 21 apples at $1, for a take-home of $21.
The government receives $0.50 for each of the 60 apples sold, or in other words receives $30 in revenue it wouldn’t have received otherwise.
We could be more specific about the surpluses, and calculate the actual areas for consumer surplus, producer surplus, inefficiency from the tariff, and government revenue from the tariff. But I won’t bother, as those calculations get slightly more involved. Instead, let’s just look at the aggregate outcomes:
Consumers were unquestionably hurt. Their price paid went up by $0.20 per apple, and received less apples.
Foreign producers were also hurt. Their price received went down from the original $1 to the new post-tariff price of $1.20, minus the $0.50 tariff. In other words: foreign producers only receive $0.70 per apple now. This hurt can be mitigated by shifting sales to other countries without a tariff, but the pain will exist regardless.
Domestic producers scored. They can sell less apples and make more revenue doing it.
And the government walked away with an extra $30.
Hopefully you now see the answer to the original questions. Importantly, while the government imposed a $0.50 tariff, neither side fully absorbed that cost. Consumers paid a bit more, foreign producers received a bit less. The exact details of how that tariff was split across the groups is mediated by the relevant supply and demand curves of each group. If you want to learn more about this, the relevant search term is “price elasticity,” or how much a group’s quantity supplied or demanded will change based on changes in the price.
Other taxes
Most taxes are some kind of a tax on trade. Tariffs on apples is an obvious one. But the same applies to income tax (taxing the worker for the trade of labor for money) or payroll tax (same thing, just taxing the employer instead). Interestingly, you can use the same model for analyzing things like tax incentives. For example, if the government decided to subsidize domestic apple production by giving the domestic producers a $0.50 bonus for each apple they sell, we would end up with a similar kind of analysis, except instead of the foreign supply curve shifting up, we’d see the domestic supply curve shifting down.
And generally speaking, this is what you’ll always see with government involvement in the economy. It will result in disrupting an existing equilibrium, letting the market readjust to a new equilibrium, and incentivization of some behavior, causing some people to benefit and others to lose out. We saw with the apple tariff, domestic producers and the government benefited while others lost.
You can see the reverse though with tax incentives. If I give a tax incentive of providing a deduction (not paying income tax) for preschool, we would end up with:
Government needs to make up the difference in tax revenue, either by raising taxes on others or printing more money (leading to inflation). Either way, those paying the tax or those holding government debased currency will pay a price.
Those people who don’t use the preschool deduction will receive no benefit, so they simply pay a cost.
Those who do use the preschool deduction will end up paying less on tax+preschool than they would have otherwise.
This analysis is fully amoral. It’s not saying whether providing subsidized preschool is a good thing or not, it simply tells you where the costs will be felt, and points out that such government interference in free economic choice does result in inefficiencies in the system. Once you have that knowledge, you’re more well educated on making a decision about whether the costs of government intervention are worth the benefits.
Then from this we can subtract the triangle 
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