Monday, December 21, 2015

Solving the GCHQ puzzle "by hand"

The GCHQ 2015 Christmas puzzle is a Nonogram puzzle, which involves filling in squares on a grid to reveal a picture, guided by constraints on the rows and columns. For a computer, a nice way to solve this problem is using a SAT solver. But humans aren't great at SAT solving, and I was given a print-out of this puzzle while on holiday, with no computer. I'd never encountered such a puzzle before, so working with a friend (and some wine) we came up with an approach, and set about applying it. Alas, robustly applying an algorithm with many steps is not easy for a human, and we eventually ended up with contradictions. On returning from holiday, I automated our approach, and tested it. Our approach worked, and the code is below.

The Problem

The puzzle is:

It comprises a 25x25 grid, some filled in squares, and alongside each row/column are the number of consecutive squares that must be filled in each line. For example, the 8th row down must have two runs of three filled squares, with a gap in between, and potentially gaps before or after.

Our Approach

Our approach was to take each line and compute the number of "free" gaps - how many spaces could be inserted with choice. For one row (4 from the bottom) the entire grid is constrained, with no free gaps. Starting with the most constrained lines, we tried to figure out where the pieces could go, based on the existing squares. We quickly realised that negative information was important, so tagged each square with "don't know" (left blank), must be filled (we shaded it in) or must be unfilled (we drew a circle in it). For each line in isolation, looking at the constraints, we inferred squares to be filled or unfilled by examining the possible locations of each run.

The Code

I implemented our approach in Haskell, complete code is available here.

Our constraint system works over a grid where each square is in one of three states. We can encode the grid as [[Maybe Bool]]. The [[.]] is a list of lists, where the outer list is a list of rows, and the inner list is a list of squares. Each of the inner lists must be the same length, and for the GCHQ puzzle, they must all be 25 elements long. For the squares we use Maybe Bool, with Nothing for unknown and Just for known, using True as filled and False as unfilled.

Given the [[Maybe Bool]] grid and the constraints, our approach was to pick a single line, and try to layout the runs, identifying squares that must be True/False. To replicate that process on a computer, I wrote a function tile that given the constraints and the existing line, produces all possible lines that fit. The code is:

tile :: [Int] -> [Maybe Bool] -> [[Bool]]
tile [] xs = maybeToList \$ xs ~> replicate (length xs) False
tile (c:cs) xs = concat [map (\r -> a ++ b ++ c ++ r) \$ tile cs xs
| gap <- [0 .. length xs - (c + sum cs + length cs)]
, (false,xs) <- [splitAt gap xs]
, (true,xs) <- [splitAt c xs]
, (space,xs) <- [splitAt 1 xs]
, Just a <- [false ~> replicate gap False]
, Just b <- [true ~> replicate c True]
, Just c <- [space ~> replicate (length space) False]]

The first equation (second line) says that if there are no remaining constraints we set all remaining elements to False. We use the ~> operator to check our desired assignment is consistent with the information already in the line:

(~>) :: [Maybe Bool] -> [Bool] -> Maybe [Bool]
(~>) xs ys | length xs == length ys &&
and (zipWith (\x y -> maybe True (== y) x) xs ys)
= Just ys
(~>) _ _ = Nothing

This function takes a line of the grid (which may have unknowns), and a possible line (which is entirely concrete), and either returns Nothing (inconsistent) or Just the proposed line. We first check the sizes are consistent, then that everything which is concrete (not a Nothing) matches the proposed value.

Returning to the second equation in tile, the idea is to compute how many spaces could occur at this point. Taking the example of a line 25 long, with two runs of size 3, we could have anywhere between 0 and 18 (25-3-3-1) spaces first. For each possible size of gap, we split the line up (the splitAt calls), then constrain each piece to match the existing grid (using ~>).

Given a way of returning all possible lines, we then collapse that into a single line, by marking all squares which could be either True or False as Nothing:

constrainLine :: [Int] -> [Maybe Bool] -> Maybe [Maybe Bool]
constrainLine cs xs = if null xs2 then Nothing else mapM f \$ transpose xs2
where xs2 = tile cs xs
f (x:xs) = Just \$ if not x `elem` xs then Nothing else Just x

If there are no satisfying assignments for the line, we return Nothing - that implies the constraints are unsatisfiable. Next, we scale up to a side of constraints, by combining all the constraints and lines:

constrainSide :: [[Int]] -> [[Maybe Bool]] -> Maybe [[Maybe Bool]]
constrainSide cs xs = sequence \$ zipWith constrainLine cs xs

Finally, to constrain the entire grid, we constrain one side, then the other. To simplify the code, we just transpose the grid in between, so we can treat the rows and columns identically:

constrainGrid :: [[Int]] -> [[Int]] -> [[Maybe Bool]] -> Maybe [[Maybe Bool]]
constrainGrid rows cols xs =
fmap transpose . constrainSide cols .
transpose =<< constrainSide rows xs

To constrain the whole problem we apply constrainGrid repeatedly, until it returns Nothing (the problem is unsatisfiable), we have a complete solution (problem solved), or nothing changes. If nothing changes then there might be two solutions, or our approach might not be powerful enough without using search.

The Result

After four iterations we end up with a fully constrained answer. To see the progress, after one iteration we have:

..XXXXX...X.OO..X.XXXXX..
..OOOOX.X.O.....O.XOOOO..
..XXX.X....O...OX.X.XX...
X.XXX.X....XXXXXX.X.XX...
X.XXX.X..XXXX..XX.X.XX..X
X.OOOOX...XO...OO.XOOO..X
XXXXXXXOXOXOXOXOXOXXXXXXX
..OOO.OO..XOOOX.XOOOOO.O.
..XX..XX.OXOXOXXXOXO...X.
..XO..OO....OXX.O.O....X.
..X...X......X..X......O.
..O...O......XO.X........
..XX..X.X....O.OO.X......
..OXX.O.X....XXXX.X......
..XX..XXXXX..O.OO........
..X...O.X..O..O.X...O....
..X...X.X.OXO.O.X...X....
..OOO.O.X..O..O.X...X..X.
X.XXXXX.......O.X...X..X.
X.OOO.X.....XOO.X...X..X.
X.XXX.X.....XXOOX...X...O
XOXXXOXOXXXOXXXXXXXXXXOXX
..XXX.X.....XXXXX..XXXX.O
..OOOOX......OOOO...O.X..
..XXXXX......XOXX.O.X.X..

Here a . stands for Nothing. After four iterations we reach the answer in a total of 0.28s:

XXXXXXXOXXXOOOXOXOXXXXXXX
XOOOOOXOXXOXXOOOOOXOOOOOX
XOXXXOXOOOOOXXXOXOXOXXXOX
XOXXXOXOXOOXXXXXXOXOXXXOX
XOXXXOXOOXXXXXOXXOXOXXXOX
XOOOOOXOOXXOOOOOOOXOOOOOX
XXXXXXXOXOXOXOXOXOXXXXXXX
OOOOOOOOXXXOOOXXXOOOOOOOO
XOXXOXXXOOXOXOXXXOXOOXOXX
XOXOOOOOOXXXOXXOOOOXOOOXO
OXXXXOXOXXXXOXXOXOOOOXXOO
OXOXOOOXOOOXOXOXXXXOXOXXX
OOXXOOXOXOXOOOOOOXXOXXXXX
OOOXXXOXXOXXOXXXXXXOXXXOX
XOXXXXXXXXXOXOXOOXXOOOOXO
OXXOXOOXXOOOXXOXXXOOOOOXO
XXXOXOXOXOOXOOOOXXXXXOXOO
OOOOOOOOXOOOXXOXXOOOXXXXX
XXXXXXXOXOOXXOOOXOXOXOXXX
XOOOOOXOXXOOXOOXXOOOXXOXO
XOXXXOXOOOXXXXOOXXXXXOOXO
XOXXXOXOXXXOXXXXXXXXXXOXX
XOXXXOXOXOOXXXXXXOXXXXXXO
XOOOOOXOOXXOOOOOOXOXOXXOO
XXXXXXXOXXOOOXOXXOOOXXXXX

Update: On the third attempt, my friend managed to solve it manually using our technique, showing it does work for humans too.

Wednesday, December 09, 2015

MinGHC is Dead, Long Live Stack

Summary: The MinGHC project has now finished.

The MinGHC project was started to produce a minimal Windows installer which didn't contain many packages, but which could install many packages - in particular the network package. But time has moved on, and Stack now offers everything MinGHC does, but cross-platform and better. To install GHC using Stack, just do stack setup, then stack exec -- my command. Even if you prefer to use Cabal, stack exec -- cabal install hlint is a reasonable approach.

A few final remarks on MinGHC:

• MinGHC was an experiment started by Michael Snoyman, which myself (Neil Mitchell) and Elliot Cameron joined in with. I had fun working with those guys.
• The ideas and approaches behind MinGHC got reused in Stack, so the knowledge learnt has transferred.
• The MinGHC project involved a Shake build system coupled to an NSIS EDSL installer. I think the technologies for building the installer worked out quite nicely.
• The existing MinGHC installers are not going away, but we won't be making any changes, and probably won't upload new versions for GHC 7.10.3.
• It's possible to build Windows installers on a Linux box using a Wine version of NSIS. Very cool.
• I find maintaining such fundamental pieces of an ecosystem, involving installation and system configuration, to be much less fun than just writing Haskell code. Kudos to the Stack and Cabal guys.

Tuesday, December 08, 2015

What's the point of Stackage LTS?

Summary: Stackage LTS is mostly pointless.

Stackage provides a set of precise versions of a subset of Hackage which all place nicely together. At any one time, there are two such version sets:

• Nightly, which aims to be the latest version of all packages.
• LTS (Long Term Support), which with every major release updates every package, and every minor release updates packages that only changed their minor version.

Stack currently defaults to LTS. I don't think LTS fulfils any need, and does cause harm (albeit very mild harm), so should be abandoned. Instead, people should use a specific nightly that suits them, and upgrade to later nightlies on whatever schedule suits them. I share these somewhat half-baked thoughts to hopefully guide how I think Stackage should evolve, and there may be good counterarguments I haven't thought of.

Why is Nightly better?

Nightly always has newer versions of packages. I believe that newer versions of packages are better.

• As a package author, with each new release, I fix bugs and evolve the library in ways which I think are beneficial. I make mistakes, and new versions are how I fix them, so people using old versions of my software are suffering for my past mistakes.
• As a package user, if I find a bug, I will endeavour to report it to the author. If I'm not using the latest version, it's highly likely the author will ask me to upgrade first. If a feature I want gets added, it's going to be to the latest version. The latest version of a package always has better support.
• If the latest version of a package does not suit my needs, that's an important alarm bell, and I need to take action. In some cases, that requires finding a fork or alternative package to do the same thing. In some cases, that involves talking to the author to make them aware of my particular use case. In either case, doing this sooner rather than later makes it easier to find a good solution.

What's the benefit of LTS?

As far as I am aware, LTS major releases are just Nightly at a particular point in time - so if you only ever use x.0 LTS, you might as well just use Nightly. The main benefit of LTS comes from picking a major LTS version, then upgrading to the subsequent minor LTS versions, to access new minor versions of your dependencies.

Upgrading packages is always a risk, as LTS Haskell says. What LTS minor releases do are minimize the risk of having to fix compilation errors, at the cost of not upgrading to the latest packages. However, the reasons I do upgrades are:

• Sometimes, if one package has a known bug that impacts me, I specifically upgrade just that one package. I don't want to upgrade other packages (it introduces unnecessary risk), so I take my package set (be it LTS or Nightly) and replace one constraint.
• Every so often, I want to upgrade all my packages, so that I'm not missing out on improvements and so I'm not falling behind the latest versions - reducing the time to upgrade in future. I do that by picking the latest version of all packages, fixing breakages, and upgrading.
• When upgrading, compile-time errors are a minor issue at worst. Before Stackage, my major headache was finding a compatible set of versions, but now that is trivial. Fixing compile-time errors is usually a little work, but fairly easy and predictable. Checking for regressions is more time consuming, and running the risk of regressions that escape the test suite has a significant cost. Tracking down if there are any resulting regressions is very time-consuming.

I don't see a use case for upgrading "a little bit", since I get a lot less benefit for only marginally less work.

Why is LTS better?

When I asked this question on Twitter, I got two reasons that did seem plausible:

• Matt Parsons suggested that LTS provided a higher likelihood of precompiled binaries shared between projects. That does make sense, but a similar benefit could be achieved by preferring Nightly on the 1st on the month.
• Gabriel Gonzalez suggested that when including a resolver in tutorials, LTS looks much better than a Nightly. I agree, but if there only was Nightly, then it wouldn't look quite as bad.

If Haskell packages had bug fixes that were regularly backported to minor releases, then the case for an LTS version would be stronger. However, my experience is that fixes are rarely backported, even for the rare security issues.

Is this a problem?

Currently everyone developing Haskell has two choices, and thanks to Stack, might not even be aware which one they've ended up making (Stack will pick nightly on its own if it thinks it needs to). Reducing the choices and simplifying the story removes work for the Stackage maintainers, and cognitive burden from the Stackage users.

Stackage was a radical experiment in doing things in an entirely different way. I think it's fair to say it has succeeded. Now, after experience using the features, I think it's right to critically look at tweaks to the model.