path: root/compiler/utils/BooleanFormula.hs

{-# LANGUAGE DeriveDataTypeable, DeriveFunctor, DeriveFoldable,
             DeriveTraversable #-}

--------------------------------------------------------------------------------
-- | Boolean formulas without quantifiers and without negation.
-- Such a formula consists of variables, conjunctions (and), and disjunctions (or).
--
-- This module is used to represent minimal complete definitions for classes.
--
module BooleanFormula (
        BooleanFormula(..), LBooleanFormula,
        mkFalse, mkTrue, mkAnd, mkOr, mkVar,
        isFalse, isTrue,
        eval, simplify, isUnsatisfied,
        implies, impliesAtom,
        pprBooleanFormula, pprBooleanFormulaNice
  ) where

import GhcPrelude

import Data.List ( nub, intersperse )
import Data.Data

import MonadUtils
import Outputable
import Binary
import GHC.Types.SrcLoc
import GHC.Types.Unique
import GHC.Types.Unique.Set

----------------------------------------------------------------------
-- Boolean formula type and smart constructors
----------------------------------------------------------------------

type LBooleanFormula a = Located (BooleanFormula a)

data BooleanFormula a = Var a | And [LBooleanFormula a] | Or [LBooleanFormula a]
                      | Parens (LBooleanFormula a)
  deriving (Eq, Data, Functor, Foldable, Traversable)

mkVar :: a -> BooleanFormula a
mkVar = Var

mkFalse, mkTrue :: BooleanFormula a
mkFalse = Or []
mkTrue = And []

-- Convert a Bool to a BooleanFormula
mkBool :: Bool -> BooleanFormula a
mkBool False = mkFalse
mkBool True  = mkTrue

-- Make a conjunction, and try to simplify
mkAnd :: Eq a => [LBooleanFormula a] -> BooleanFormula a
mkAnd = maybe mkFalse (mkAnd' . nub) . concatMapM fromAnd
  where
  -- See Note [Simplification of BooleanFormulas]
  fromAnd :: LBooleanFormula a -> Maybe [LBooleanFormula a]
  fromAnd (L _ (And xs)) = Just xs
     -- assume that xs are already simplified
     -- otherwise we would need: fromAnd (And xs) = concat <$> traverse fromAnd xs
  fromAnd (L _ (Or [])) = Nothing
     -- in case of False we bail out, And [..,mkFalse,..] == mkFalse
  fromAnd x = Just [x]
  mkAnd' [x] = unLoc x
  mkAnd' xs = And xs

mkOr :: Eq a => [LBooleanFormula a] -> BooleanFormula a
mkOr = maybe mkTrue (mkOr' . nub) . concatMapM fromOr
  where
  -- See Note [Simplification of BooleanFormulas]
  fromOr (L _ (Or xs)) = Just xs
  fromOr (L _ (And [])) = Nothing
  fromOr x = Just [x]
  mkOr' [x] = unLoc x
  mkOr' xs = Or xs


{-
Note [Simplification of BooleanFormulas]
~~~~~~~~~~~~~~~~~~~~~~
The smart constructors (`mkAnd` and `mkOr`) do some attempt to simplify expressions. In particular,
 1. Collapsing nested ands and ors, so
     `(mkAnd [x, And [y,z]]`
    is represented as
     `And [x,y,z]`
    Implemented by `fromAnd`/`fromOr`
 2. Collapsing trivial ands and ors, so
     `mkAnd [x]` becomes just `x`.
    Implemented by mkAnd' / mkOr'
 3. Conjunction with false, disjunction with true is simplified, i.e.
     `mkAnd [mkFalse,x]` becomes `mkFalse`.
 4. Common subexpression elimination:
     `mkAnd [x,x,y]` is reduced to just `mkAnd [x,y]`.

This simplification is not exhaustive, in the sense that it will not produce
the smallest possible equivalent expression. For example,
`Or [And [x,y], And [x]]` could be simplified to `And [x]`, but it currently
is not. A general simplifier would need to use something like BDDs.

The reason behind the (crude) simplifier is to make for more user friendly
error messages. E.g. for the code
  > class Foo a where
  >     {-# MINIMAL bar, (foo, baq | foo, quux) #-}
  > instance Foo Int where
  >     bar = ...
  >     baz = ...
  >     quux = ...
We don't show a ridiculous error message like
    Implement () and (either (`foo' and ()) or (`foo' and ()))
-}

----------------------------------------------------------------------
-- Evaluation and simplification
----------------------------------------------------------------------

isFalse :: BooleanFormula a -> Bool
isFalse (Or []) = True
isFalse _ = False

isTrue :: BooleanFormula a -> Bool
isTrue (And []) = True
isTrue _ = False

eval :: (a -> Bool) -> BooleanFormula a -> Bool
eval f (Var x)  = f x
eval f (And xs) = all (eval f . unLoc) xs
eval f (Or xs)  = any (eval f . unLoc) xs
eval f (Parens x) = eval f (unLoc x)

-- Simplify a boolean formula.
-- The argument function should give the truth of the atoms, or Nothing if undecided.
simplify :: Eq a => (a -> Maybe Bool) -> BooleanFormula a -> BooleanFormula a
simplify f (Var a) = case f a of
  Nothing -> Var a
  Just b  -> mkBool b
simplify f (And xs) = mkAnd (map (\(L l x) -> L l (simplify f x)) xs)
simplify f (Or xs) = mkOr (map (\(L l x) -> L l (simplify f x)) xs)
simplify f (Parens x) = simplify f (unLoc x)

-- Test if a boolean formula is satisfied when the given values are assigned to the atoms
-- if it is, returns Nothing
-- if it is not, return (Just remainder)
isUnsatisfied :: Eq a => (a -> Bool) -> BooleanFormula a -> Maybe (BooleanFormula a)
isUnsatisfied f bf
    | isTrue bf' = Nothing
    | otherwise  = Just bf'
  where
  f' x = if f x then Just True else Nothing
  bf' = simplify f' bf

-- prop_simplify:
--   eval f x == True   <==>  isTrue  (simplify (Just . f) x)
--   eval f x == False  <==>  isFalse (simplify (Just . f) x)

-- If the boolean formula holds, does that mean that the given atom is always true?
impliesAtom :: Eq a => BooleanFormula a -> a -> Bool
Var x  `impliesAtom` y = x == y
And xs `impliesAtom` y = any (\x -> (unLoc x) `impliesAtom` y) xs
           -- we have all of xs, so one of them implying y is enough
Or  xs `impliesAtom` y = all (\x -> (unLoc x) `impliesAtom` y) xs
Parens x `impliesAtom` y = (unLoc x) `impliesAtom` y

implies :: Uniquable a => BooleanFormula a -> BooleanFormula a -> Bool
implies e1 e2 = go (Clause emptyUniqSet [e1]) (Clause emptyUniqSet [e2])
  where
    go :: Uniquable a => Clause a -> Clause a -> Bool
    go l@Clause{ clauseExprs = hyp:hyps } r =
        case hyp of
            Var x | memberClauseAtoms x r -> True
                  | otherwise -> go (extendClauseAtoms l x) { clauseExprs = hyps } r
            Parens hyp' -> go l { clauseExprs = unLoc hyp':hyps }     r
            And hyps'  -> go l { clauseExprs = map unLoc hyps' ++ hyps } r
            Or hyps'   -> all (\hyp' -> go l { clauseExprs = unLoc hyp':hyps } r) hyps'
    go l r@Clause{ clauseExprs = con:cons } =
        case con of
            Var x | memberClauseAtoms x l -> True
                  | otherwise -> go l (extendClauseAtoms r x) { clauseExprs = cons }
            Parens con' -> go l r { clauseExprs = unLoc con':cons }
            And cons'   -> all (\con' -> go l r { clauseExprs = unLoc con':cons }) cons'
            Or cons'    -> go l r { clauseExprs = map unLoc cons' ++ cons }
    go _ _ = False

-- A small sequent calculus proof engine.
data Clause a = Clause {
        clauseAtoms :: UniqSet a,
        clauseExprs :: [BooleanFormula a]
    }
extendClauseAtoms :: Uniquable a => Clause a -> a -> Clause a
extendClauseAtoms c x = c { clauseAtoms = addOneToUniqSet (clauseAtoms c) x }

memberClauseAtoms :: Uniquable a => a -> Clause a -> Bool
memberClauseAtoms x c = x `elementOfUniqSet` clauseAtoms c

----------------------------------------------------------------------
-- Pretty printing
----------------------------------------------------------------------

-- Pretty print a BooleanFormula,
-- using the arguments as pretty printers for Var, And and Or respectively
pprBooleanFormula' :: (Rational -> a -> SDoc)
                   -> (Rational -> [SDoc] -> SDoc)
                   -> (Rational -> [SDoc] -> SDoc)
                   -> Rational -> BooleanFormula a -> SDoc
pprBooleanFormula' pprVar pprAnd pprOr = go
  where
  go p (Var x)  = pprVar p x
  go p (And []) = cparen (p > 0) $ empty
  go p (And xs) = pprAnd p (map (go 3 . unLoc) xs)
  go _ (Or  []) = keyword $ text "FALSE"
  go p (Or  xs) = pprOr p (map (go 2 . unLoc) xs)
  go p (Parens x) = go p (unLoc x)

-- Pretty print in source syntax, "a | b | c,d,e"
pprBooleanFormula :: (Rational -> a -> SDoc) -> Rational -> BooleanFormula a -> SDoc
pprBooleanFormula pprVar = pprBooleanFormula' pprVar pprAnd pprOr
  where
  pprAnd p = cparen (p > 3) . fsep . punctuate comma
  pprOr  p = cparen (p > 2) . fsep . intersperse vbar

-- Pretty print human in readable format, "either `a' or `b' or (`c', `d' and `e')"?
pprBooleanFormulaNice :: Outputable a => BooleanFormula a -> SDoc
pprBooleanFormulaNice = pprBooleanFormula' pprVar pprAnd pprOr 0
  where
  pprVar _ = quotes . ppr
  pprAnd p = cparen (p > 1) . pprAnd'
  pprAnd' [] = empty
  pprAnd' [x,y] = x <+> text "and" <+> y
  pprAnd' xs@(_:_) = fsep (punctuate comma (init xs)) <> text ", and" <+> last xs
  pprOr p xs = cparen (p > 1) $ text "either" <+> sep (intersperse (text "or") xs)

instance (OutputableBndr a) => Outputable (BooleanFormula a) where
  ppr = pprBooleanFormulaNormal

pprBooleanFormulaNormal :: (OutputableBndr a)
                        => BooleanFormula a -> SDoc
pprBooleanFormulaNormal = go
  where
    go (Var x)    = pprPrefixOcc x
    go (And xs)   = fsep $ punctuate comma (map (go . unLoc) xs)
    go (Or [])    = keyword $ text "FALSE"
    go (Or xs)    = fsep $ intersperse vbar (map (go . unLoc) xs)
    go (Parens x) = parens (go $ unLoc x)


----------------------------------------------------------------------
-- Binary
----------------------------------------------------------------------

instance Binary a => Binary (BooleanFormula a) where
  put_ bh (Var x)    = putByte bh 0 >> put_ bh x
  put_ bh (And xs)   = putByte bh 1 >> put_ bh xs
  put_ bh (Or  xs)   = putByte bh 2 >> put_ bh xs
  put_ bh (Parens x) = putByte bh 3 >> put_ bh x

  get bh = do
    h <- getByte bh
    case h of
      0 -> Var    <$> get bh
      1 -> And    <$> get bh
      2 -> Or     <$> get bh
      _ -> Parens <$> get bh