{-# 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 Data.Maybe ( mapMaybe )
import qualified Data.IntMap as IM

import Control.Monad ( guard, liftM )

-- | 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)

-- | 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
  = 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