# Haskell Quiz/Verbal Arithmetic/Solution Dolio

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This solution doesn't do anything particularly special. It parses the equations using a simple Parsec parser, and then searches for solutions using backtracking search via the list monad. However, to make things a bit simpler, I wrapped up the searching into what I called the CSP monad (for constraint satisfaction problem), that automatically handles the remembering of already assigned variables, and branching when attempting to use a previously unassigned variable. It's not a clever solution, algorithmically, but it gets the job done for such a small problem as this. |
This solution doesn't do anything particularly special. It parses the equations using a simple Parsec parser, and then searches for solutions using backtracking search via the list monad. However, to make things a bit simpler, I wrapped up the searching into what I called the CSP monad (for constraint satisfaction problem), that automatically handles the remembering of already assigned variables, and branching when attempting to use a previously unassigned variable. It's not a clever solution, algorithmically, but it gets the job done for such a small problem as this. |
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## Revision as of 20:25, 18 June 2007

This solution doesn't do anything particularly special. It parses the equations using a simple Parsec parser, and then searches for solutions using backtracking search via the list monad. However, to make things a bit simpler, I wrapped up the searching into what I called the CSP monad (for constraint satisfaction problem), that automatically handles the remembering of already assigned variables, and branching when attempting to use a previously unassigned variable. It's not a clever solution, algorithmically, but it gets the job done for such a small problem as this.

The problem specific code:

module Main(main) where import Control.Monad import Data.Char import Data.List import Data.Maybe import Data.Map (Map) import qualified Data.Map as M import Text.ParserCombinators.Parsec import CSP data Equation = Equation { summands :: [String], result :: String } deriving Show -- A quick parser for the equations in use. I didn't bother trying to cover all -- eventualities, but it will parse 'word1 + word2 + word3 ... = result' add = char '+' >> spaces >> return (++) equ = char '=' >> spaces vnum = many1 lower >>= \a -> spaces >> return a equation = do l <- chainl1 (liftM return vnum) add equ r <- vnum return (Equation l r) -- Constraints on individual digits. Digits in a column, plus the carry-in, mod 10 -- must equal the result digit. Returns the carry-out digitcsp :: [Char] -> Int -> Char -> CSP Int Int digitcsp vs carry result = do l <- mapM byName vs let (d, m) = divMod (carry + sum l) 10 byName result >>= guard . (m ==) return d -- Constraint to make sure the left-most digit on any term is not zero. We don't want -- 'two = 012', for instance. sigfig :: Equation -> CSP Int () sigfig (Equation ss r) = chk r >> foldM_ (const chk) () ss where chk (h:_) = byName h >>= guard . (0 /=) -- The overall constraint for solving equations eqncsp :: Equation -> CSP Int () eqncsp e@(Equation s r@(hr:_)) = go (columns s) (reverse r) 0 >> sigfig e where go (vs:vss) (r:rs) carry = digitcsp vs carry r >>= go vss rs go [] [r] carry = byName r >>= guard . (carry ==) go [] [] 0 = return () go _ _ _ = mzero -- turns a list of terms into a list of letters in each column, to be summed. columns :: [String] -> [[Char]] columns = transpose . map reverse -- Given an equation, and mappings for all the letters, constructs the numerals showEqn :: Equation -> Map Char Int -> String showEqn (Equation ss r) m = intercalate " + " (map showWord ss) ++ " = " ++ showWord r where showWord = foldr (\d s -> intToDigit (fromJust (M.lookup d m)) : s) [] main = mapM_ display . map (fmap solve . parse equation "foo") . lines =<< getContents where display (Left _) = putStrLn "Bad parse." display (Right ss) = mapM_ putStrLn ss solve :: Equation -> [String] solve e = map (showEqn e) ms where ms = execCSP (eqncsp e) [0..9]

And the CSP monad. There is an auxiliary 'SupplyT' monad which encapsulates selecting non-deterministically from a stored supply.

{-# OPTIONS_GHC -fglasgow-exts #-} module CSP where import Data.Map (Map) import qualified Data.Map as M import Control.Arrow import Control.Monad import Control.Monad.State import Control.Monad.Reader import Control.Monad.Trans newtype CSP b a = CSP { unCSP :: StateT (Map Char b) (SupplyT b []) a } deriving (Monad, MonadPlus, MonadState (Map Char b), MonadSupply b) newtype SupplyT s m a = SupplyT { unSupplyT :: StateT [s] m a } deriving (Monad, MonadPlus, MonadState [s]) class MonadPlus m => MonadSupply s m where getSupply :: m s instance MonadPlus m => MonadSupply s (SupplyT s m) where getSupply = do l <- get (v, l') <- choose (pick l) put l' return v where pick (h:t) = (h, t) : map (second (h:)) (pick t) pick [] = mzero instance MonadSupply s m => MonadSupply s (StateT r m) where getSupply = lift getSupply runSupplyT :: Monad m => SupplyT s m a -> [s] -> m a runSupplyT st s = flip evalStateT s . unSupplyT $ st runCSP :: CSP b a -> [b] -> [(a, Map Char b)] runCSP c l = flip runSupplyT l . flip runStateT M.empty . unCSP $ c evalCSP :: CSP b a -> [b] -> [a] evalCSP c l = flip runSupplyT l . flip evalStateT M.empty . unCSP $ c execCSP :: CSP b a -> [b] -> [Map Char b] execCSP c l = flip runSupplyT l . flip execStateT M.empty . unCSP $ c byName :: Char -> CSP b b byName k = gets (M.lookup k) >>= maybe c return where c = getSupply >>= assign k assign :: Char -> a -> CSP a a assign k a = modify (M.insert k a) >> return a choose :: MonadPlus m => [a] -> m a choose = msum . map return