I've started writing solvers for games of *Simon Tatham's Portable Puzzle Collection*.
Those Problems can usually be reduced to some NP-complete problem. This is quite
a fun exercise.

We will be starting with the puzzle "Pattern",
also known as Nonograms that we **reduce to SAT**.

First to explain the puzzle: The cells of the puzzle can be filled with
black or white squares. The numbers on the sides of the rows and colums denote
the number of consecutive black squares. Every run of black squares
has to be seperated from another run by white squares.

First we want to define a type for our problem.

type PSize = (Int, Int)
type Run = [[Int]]
type Runs = (Run, Run)
data Problem = Problem PSize Runs deriving Show

The puzzle on the side is defined in the Tatham format by the string
`10x10:2/3.2/1.1.3/2.5/2.2.4/1.1.4/1.3/1.1.2/4/4.1/5.2/2.3.3/1.2/6/3/1/4/6/7/4.1`.
It has a size information and then the list of runs. Another format we want to
support is the Haskell `Runs` format which would be
`([[2],[3,2],[1,1,3],[2,5],[2,2,4],[1,1,4],[1,3],[1,1,2],[4],[4,1]],[[5,2],[2,3,3],[1,2],[6],[3],[1],[4],[6],[7],[4,1]])`.
This format seperates the runs for rows and columns and ommits the size information.

For the second format we can use Haskell's own `read` function. For Tatham's
format we have to define a **parser**. I've added type information for most functions
which is often needed for code involving `read`.

import Text.ParserCombinators.ReadP
number' = many1 (choice (map char ['0'..'9']))
r_size :: ReadP PSize
r_size = do l <- number'
char 'x'
r <- number'
return (read l, read r)
r_run :: ReadP [Int]
r_run = do { li <- sepBy number' (char '.'); return $ map read li }
r_runs :: ReadP [[Int]]
r_runs = sepBy r_run (char '/')
r_problem :: ReadP Problem
r_problem = do size <- r_size; char ':'
runs <- r_runs
skipSpaces; eof
return $ Problem size (take (fst size) runs, drop (fst size) runs)

What do we now do with such a problem? First we want to check it for **consistency**.
If we got size information it should fit the runs and the numbers in the runs
(and the spaces between them) shouldn't add up to more than the size.

import qualified Prelude as P (all, (&&), not)
consistent :: Problem -> Bool
consistent (Problem size (xin, yin)) = consistent' size xin P.&& consistent' (swap size) yin
consistent' (w, h) xs = (w == length xs) P.&& P.all (\x -> (sum x) + (length x - 1) <= h) xs

But now on to solving the problem. We will **reduce this problem to SAT** using
the Ersatz library. The **encoding of a problem**, I guess, is obvious. We say
that every cell within the Nonogram is a Variable `Bit` and a bit being `True`
has the semantics of that field being colored black while `False` would mean
white.

We want to build constraints of single rows or columns and then **solve** the conjunction
of all those **row/column-constraints for satisfiability**. Therefore we define a function
of the type `[Int] -> [Bit] -> Bit` that takes a list of Integers (the numbers of
one column or row) and the list of remaining Bits that do not yet are constrained.

First we begin with the end of the recursion. If no number is left or the remaining
number is zero (for empty columns or rows), we want all remaining variables to be
`False`. (we take `for` to be defined as `flip map`)

build_runs :: [Int] -> [Bit] -> Bit
build_runs nums vars = case nums of
[] -> all not vars
[0] -> all not vars
x:xs -> let max_d = length vars - (length xs + sum xs) - x
run = [not] ++ replicate x id ++ [not]
v = [false] ++ vars ++ [false]
in any id $ for [0..max_d] $ \ d ->
(all not $ take d vars)
&& (all id $ zipWith id run $ take (x + 2) $ drop d v)
&& (build_runs xs $ drop (x + d + 1) vars)

The recursion is something to think about a bit. We're given a number of consecutive
black squares `x` and all the variables in the column/row `vars`. The `x`
Trues can be put anywhere from 0 to `max_d` where `max_d` is all the rest of
the variables take the minimum necessary space of the remaining number of numbers
`xs`, while all variables before that number of black squares and one after need
to be white. For every possible position the rest of the variables need to fulfill
the same condition with the rest of the numbers.

Now we have to combine all the constraints from a given problem.

nono :: (MonadState s m, HasSAT s) => Problem -> m (Map (Int,Int) Bit)
nono (Problem (sizex, sizey) (xin, yin)) = do
let indices = [(x, y) | x <- [1..sizex], y <- [1..sizey] ]
vars' <- replicateM (length indices) exists
vars <- return $ Map.fromList $ zip indices vars'
let getx x = for [1..sizey] $ \y -> (vars ! (x,y))
gety y = for [1..sizex] $ \x -> (vars ! (x,y))
xs = zipWith build_runs xin (map getx [1..])
ys = zipWith build_runs yin (map gety [1..])
assert (all id xs && all id ys)
return vars

Internally we let Ersatz identify variables by their position in the grid `Map (Int,Int) Bit`.
Now we can solve our problem easily using minisat with Ersatz's native minisat
binding. Given a solution, we just format it correctly and give it to the user.

main = do
input <- getContents
let parsed = parse_tat input
problem = case parsed of
Just p -> p
Nothing -> let (xin, yin) = read input :: ([[Int]],[[Int]])
in Problem (length xin, length yin) (xin, yin)
putStrLn $ "Size: " ++ (show $ problem_size problem)
unless (consistent problem) $ do {putStrLn "Problem inconsitent"; exitFailure}
(Satisfied, Just solution) <- solveWith minisat (nono problem)
let sizey = snd $ problem_size problem
lines = for [1..sizey] $ \i -> filter (\ ((x, y), b) -> y == i) $ toList solution
conv = map $ \((x,y), b)-> if b then 'X' else ' '
join = foldr (:) ""
forM_ (map (join . conv) lines) print

And we're done.

**This post is part of the series "Tatham's Puzzles":**