{-# 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, booleanToCNF
) where
-- | 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
-- | Literals are variables that occur either positively or negatively.
--
data Literal = Pos Int | Neg Int deriving (Eq, Show)
-- | 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]
-- |
-- We convert boolean formulas to conjunctive normal form by pushing
-- negations down to variables and repeatedly applying the
-- distributive laws.
--
booleanToCNF :: Boolean -> CNF
booleanToCNF
= map (map literal . flatDisjunction)
. flatConjunction
. 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
flatConjunction :: Boolean -> [Boolean]
flatConjunction b = flat b []
where flat (x:&&:y) = flat x . flat y
flat x = (x:)
flatDisjunction :: Boolean -> [Boolean]
flatDisjunction b = flat b []
where 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