logic-classes-0.44: Data/Logic/Classes/Logic.hs

-- | Class Logic defines the basic boolean logic operations,
-- AND, OR, NOT, and so on.  Definitions which pertain to both
-- propositional and first order logic are here.
module Data.Logic.Classes.Logic where

import Data.Logic.Classes.Negatable

-- |A type class for logical formulas.  Minimal implementation:
-- @
--  (.|.), (.~.)
-- @
class (Negatable formula, Ord formula) => Logic formula where
    -- | Disjunction/OR
    (.|.) :: formula -> formula -> formula

    -- | Derived formula combinators.  These could (and should!) be
    -- overridden with expressions native to the instance.
    --
    -- | Conjunction/AND
    (.&.) :: formula -> formula -> formula
    x .&. y = (.~.) ((.~.) x .|. (.~.) y)
    -- | Formula combinators: Equivalence
    (.<=>.) :: formula -> formula -> formula
    x .<=>. y = (x .=>. y) .&. (y .=>. x)
    -- | Implication
    (.=>.) :: formula -> formula -> formula
    x .=>. y = ((.~.) x .|. y)
    -- | Reverse implication:
    (.<=.) :: formula -> formula -> formula
    x .<=. y = y .=>. x
    -- | Exclusive or
    (.<~>.) :: formula -> formula -> formula
    x .<~>. y = ((.~.) x .&. y) .|. (x .&. (.~.) y)
    -- | Nor
    (.~|.) :: formula -> formula -> formula
    x .~|. y = (.~.) (x .|. y)
    -- | Nand
    (.~&.) :: formula -> formula -> formula
    x .~&. y = (.~.) (x .&. y)

infixl 2  .<=>. ,  .=>. ,  .<~>.
infixl 3  .&.
infixl 4  .|.  -- a & b | c means a & (b | c), which in cnf would be [[a], [b, c]]