{-# OPTIONS -fallow-undecidable-instances #-}

{- |

Module : $Header$

Description : Instance of class Logic for propositional logic

Copyright : (c) Dominik Luecke, Uni Bremen 2007

License : similar to LGPL, see HetCATS/LICENSE.txt or LIZENZ.txt

Maintainer : luecke@tzi.de

Stability : experimental

Portability : portable

Definition of morphisms for propositional logic

-}

{-

Ref.

Till Mossakowski, Joseph Goguen, Razvan Diaconescu, Andrzej Tarlecki.

What is a Logic?.

In Jean-Yves Beziau (Ed.), Logica Universalis, pp. 113-@133. Birkhäuser.

2005.

-}

module Propositional.Morphism

(

Morphism (..) -- datatype for Morphisms

,pretty -- pretty printing

,idMor -- identity morphism

,isLegalMorphism -- check if morhpism is ok

,composeMor -- composition

) where

import qualified Data.Map as Map

import qualified Data.Set as Set

import Propositional.Sign as Sign

import Common.Id

import Common.Result

import Common.Doc

import Common.DocUtils

-- Maps are simple maps between elements of sets

-- By the definition of maps in Data.Map

-- these maps are injective

type PropMap = Map.Map Id Id

-- | The datatype for morphisms in propositional logic as

-- | simple injective maps of sets

data Morphism = Morphism

{

source :: Sign

, target :: Sign

, propMap :: PropMap

} deriving (Eq, Show)

instance Pretty Morphism where

pretty = printMorphism

-- | Constructs an id-morphism as the diagonal

idMor :: Sign -> Morphism

idMor a = Morphism

{ source = a

, target = a

, propMap = makeIdMor $ items a

}

where

makeIdMor :: (Ord b) => Set.Set b -> Map.Map b b

makeIdMor b = Set.fold (\x -> Map.insert x x) Map.empty b

-- | Determines whether a morphism is valid

isLegalMorphism :: Morphism -> Bool

isLegalMorphism pmor =

let

psource = items $ source pmor

ptarget = items $ target pmor

pdom = Map.keysSet $ propMap pmor

pcodom = Set.map (applyMorphism pmor) $ psource

in

Set.isSubsetOf pcodom ptarget && Set.isSubsetOf pdom psource

-- | Application funtion for morphisms

applyMorphism :: Morphism -> Id -> Id

applyMorphism mor idt = Map.findWithDefault idt idt $ propMap mor

-- | Application function for propMaps

applyMap :: PropMap -> Id -> Id

applyMap pmap idt = Map.findWithDefault idt idt pmap

-- | Composition of morphisms in propositional Logic

composeMor :: Morphism -> Morphism -> Result Morphism

composeMor f g

| fTarget /= gSource = fail "Morphisms are not composable"

| otherwise = return Morphism

{

source = fSource

, target = gTarget

, propMap = if Map.null gMap then fMap else

Set.fold ( \ i ->

let j = applyMap gMap (applyMap fMap i) in

if i == j then id else Map.insert i j)

Map.empty $ items fSource

}

where

fSource = source f

fTarget = target f

gSource = source g

gTarget = target g

fMap = propMap f

gMap = propMap g

-- | Pretty printing for Morphisms

printMorphism :: Morphism -> Doc

printMorphism m = pretty (source m) <> text "-->" <> pretty (target m)

<> vcat (map ( \ (x, y) -> lparen <> pretty x <> text ","

<> pretty y <> rparen) $ Map.assocs $ propMap m)