{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE FlexibleInstances #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE StandaloneDeriving #-}
{-# LANGUAGE TypeFamilies #-}
{-# LANGUAGE TypeSynonymInstances #-}
{-# OPTIONS -fno-warn-incomplete-patterns #-}
-- |
-- Module : Data.Boolean
-- Copyright : Sebastian Fischer
-- License : BSD3
--
-- Maintainer : Sebastian Fischer (sebf@informatik.uni-kiel.de)
-- Stability : experimental
-- Portability : portable
--
-- This library provides a representation of boolean formulas that is
-- used by the solver in "Data.Boolean.SatSolver".
--
-- We also define a function to simplify formulas, a type for
-- conjunctive normalforms, and a function that creates them from
-- boolean formulas.
--
module Data.Boolean (
Boolean(..),
Literal(..), literalVar, invLiteral, isPositiveLiteral,
CNF, Clause, booleanToCNF
) where
import Control.Monad ( guard, liftM )
import Data.Generics (Data, Typeable)
import qualified Data.IntMap as IM
import Data.Logic.ATP.Formulas (IsAtom, IsFormula(..))
import Data.Logic.ATP.Lit (IsLiteral(..))
import Data.Logic.ATP.Prop (IsPropositional(..), JustPropositional)
import Data.Logic.ATP.Pretty (HasFixity(..), Pretty(pPrint), text)
import Data.Maybe ( mapMaybe )
-- | Boolean formulas are represented as values of type @Boolean@.
--
data Boolean
-- | Variables are labeled with an @Int@,
= Var Int
-- | @Yes@ represents /true/,
| Yes
-- | @No@ represents /false/,
| No
-- | @Not@ constructs negated formulas,
| Not Boolean
-- | and finally we provide conjunction
| Boolean :&&: Boolean
-- | and disjunction of boolean formulas.
| Boolean :||: Boolean
deriving Show
-- | Literals are variables that occur either positively or negatively.
--
data Literal = Pos Int | Neg Int deriving (Eq, Show)
instance Ord Literal where
compare (Neg _) (Pos _) = LT
compare (Pos _) (Neg _) = GT
compare (Pos m) (Pos n) = compare m n
compare (Neg m) (Neg n) = compare m n
deriving instance Data Literal
deriving instance Typeable Literal
-- | This function returns the name of the variable in a literal.
--
literalVar :: Literal -> Int
literalVar (Pos n) = n
literalVar (Neg n) = n
-- | This function negates a literal.
--
invLiteral :: Literal -> Literal
invLiteral (Pos n) = Neg n
invLiteral (Neg n) = Pos n
-- | This predicate checks whether the given literal is positive.
--
isPositiveLiteral :: Literal -> Bool
isPositiveLiteral (Pos _) = True
isPositiveLiteral _ = False
-- | Conjunctive normalforms are lists of lists of literals.
--
type CNF = [Clause]
type Clause = [Literal]
instance JustPropositional CNF
instance HasFixity Int
instance IsAtom Int
instance IsFormula CNF where
type AtomOf CNF = Int
atomic = error "FIXME: IsFormula CNF MyAtom"
overatoms = error "FIXME: IsFormula CNF MyAtom"
onatoms = error "FIXME: IsFormula CNF MyAtom"
asBool = error "FIXME: HasBoolean CNF"
true = error "FIXME: HasBoolean CNF"
false = error "FIXME: HasBoolean CNF"
instance Pretty Literal where
pPrint = text . show
instance IsPropositional CNF where
foldPropositional' = error "FIXME: IsPropositional CNF MyAtom"
foldCombination = error "FIXME: IsCombinable CNF"
_ .|. _ = error "FIXME: IsCombinable CNF"
_ .&. _ = error "FIXME: IsCombinable CNF"
_ .=>. _ = error "FIXME: IsCombinable CNF"
_ .<=>. _ = error "FIXME: IsCombinable CNF"
instance HasFixity CNF where
precedence _ = error "FIXME: HasFixity CNF"
associativity _ = error "FIXME: HasFixity CNF"
instance IsLiteral CNF where
foldLiteral' = error "FIXME: IsLiteral CNF MyAtom"
naiveNegate = error "FIXME: IsNegatable CNF"
foldNegation = error "FIXME: IsNegatable CNF"
-- |
-- We convert boolean formulas to conjunctive normal form by pushing
-- negations down to variables and repeatedly applying the
-- distributive laws.
--
booleanToCNF :: Boolean -> CNF
booleanToCNF
= mapMaybe (simpleClause . map literal . disjunction)
. conjunction
. asLongAsPossible distribute
. asLongAsPossible pushNots
. asLongAsPossible elim
where
elim (Not Yes) = Just No
elim (Not No) = Just Yes
elim (No :&&: _) = Just No
elim (Yes :&&: x) = Just x
elim (_ :&&: No) = Just No
elim (x :&&: Yes) = Just x
elim (Yes :||: _) = Just Yes
elim (No :||: x) = Just x
elim (_ :||: Yes) = Just Yes
elim (x :||: No) = Just x
elim _ = Nothing
pushNots (Not (Not x)) = Just x
pushNots (Not (x:&&:y)) = Just (Not x :||: Not y)
pushNots (Not (x:||:y)) = Just (Not x :&&: Not y)
pushNots _ = Nothing
distribute (x:||:(y:&&:z)) = Just ((x:||:y):&&:(x:||:z))
distribute ((x:&&:y):||:z) = Just ((x:||:z):&&:(y:||:z))
distribute _ = Nothing
literal (Var x) = Pos x
literal (Not (Var x)) = Neg x
-- private helper functions
-- remove duplicate literals from clauses and drop clauses that
-- contain one literal both positively and negatively.
--
simpleClause :: Clause -> Maybe Clause
simpleClause = liftM (map lit . IM.toList) . foldl add (Just IM.empty)
where
lit (x,True) = Pos x
lit (x,False) = Neg x
add mm l = do
m <- mm
let x = literalVar l; kind = isPositiveLiteral l
maybe (Just (IM.insert x kind m))
(\b -> guard (b==kind) >> Just m)
(IM.lookup x m)
conjunction :: Boolean -> [Boolean]
conjunction b = flat b []
where flat Yes = id
flat (x:&&:y) = flat x . flat y
flat x = (x:)
disjunction :: Boolean -> [Boolean]
disjunction b = flat b []
where flat No = id
flat (x:||:y) = flat x . flat y
flat x = (x:)
asLongAsPossible :: (Boolean -> Maybe Boolean) -> Boolean -> Boolean
asLongAsPossible f = everywhere g
where g x = maybe x (everywhere g) (f x)
everywhere :: (Boolean -> Boolean) -> Boolean -> Boolean
everywhere f = f . atChildren (everywhere f)
atChildren :: (Boolean -> Boolean) -> Boolean -> Boolean
atChildren f (Not x) = Not (f x)
atChildren f (x:&&:y) = f x :&&: f y
atChildren f (x:||:y) = f x :||: f y
atChildren _ x = x