{-# LANGUAGE DeriveDataTypeable #-}
{- |
Module      :  ./QBF/Morphism.hs
Description :  Morphisms in Propositional logic extended with QBFs
Copyright   :  (c) Jonathan von Schroeder, DFKI GmbH 2010
License     :  GPLv2 or higher, see LICENSE.txt

Maintainer  :  <jonathan.von_schroeder@dfki.de>
Stability   :  experimental
Portability :  portable

Definition of morphisms for propositional logic
copied to "Temporal.Morphism"
  Ref.

  Till Mossakowski, Joseph Goguen, Razvan Diaconescu, Andrzej Tarlecki.
  What is a Logic?.
  In Jean-Yves Beziau (Ed.), Logica Universalis, pp. 113-133. Birkhaeuser.
  2005.
-}

module QBF.Morphism
  ( Morphism (..)               -- datatype for Morphisms
  , pretty                      -- pretty printing
  , idMor                       -- identity morphism
  , isLegalMorphism             -- check if morhpism is ok
  , composeMor                  -- composition
  , inclusionMap                -- inclusion map
  , mapSentence                 -- map of sentences
  , mapSentenceH                -- map of sentences, without Result type
  , applyMap                    -- application function for maps
  , applyMorphism               -- application function for morphism
  , morphismUnion
  ) where

import Data.Data
import qualified Data.Map as Map
import qualified Data.Set as Set

import Propositional.Sign as Sign

import qualified QBF.AS_BASIC_QBF as AS_BASIC

import Common.Id as Id
import Common.Result
import Common.Doc
import Common.DocUtils
import qualified Common.Result as Result

import Control.Monad (unless)
import qualified Control.Monad.Fail as Fail

{- | The datatype for morphisms in propositional logic as
maps of sets -}
data Morphism = Morphism
  { Morphism -> Sign
source :: Sign
  , Morphism -> Sign
target :: Sign
  , Morphism -> Map Id Id
propMap :: Map.Map Id Id
  } deriving (Int -> Morphism -> ShowS
[Morphism] -> ShowS
Morphism -> String
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showsPrec :: Int -> Morphism -> ShowS
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Data)

instance Pretty Morphism where
    pretty :: Morphism -> Doc
pretty = Morphism -> Doc
printMorphism

-- | Constructs an id-morphism
idMor :: Sign -> Morphism
idMor :: Sign -> Morphism
idMor a :: Sign
a = Sign -> Sign -> Morphism
inclusionMap Sign
a Sign
a

-- | Determines whether a morphism is valid
isLegalMorphism :: Morphism -> Result ()
isLegalMorphism :: Morphism -> Result ()
isLegalMorphism pmor :: Morphism
pmor =
    let psource :: Set Id
psource = Sign -> Set Id
items (Sign -> Set Id) -> Sign -> Set Id
forall a b. (a -> b) -> a -> b
$ Morphism -> Sign
source Morphism
pmor
        ptarget :: Set Id
ptarget = Sign -> Set Id
items (Sign -> Set Id) -> Sign -> Set Id
forall a b. (a -> b) -> a -> b
$ Morphism -> Sign
target Morphism
pmor
        pdom :: Set Id
pdom = Map Id Id -> Set Id
forall k a. Map k a -> Set k
Map.keysSet (Map Id Id -> Set Id) -> Map Id Id -> Set Id
forall a b. (a -> b) -> a -> b
$ Morphism -> Map Id Id
propMap Morphism
pmor
        pcodom :: Set Id
pcodom = (Id -> Id) -> Set Id -> Set Id
forall b a. Ord b => (a -> b) -> Set a -> Set b
Set.map (Morphism -> Id -> Id
applyMorphism Morphism
pmor) Set Id
psource
    in Bool -> Result () -> Result ()
forall (f :: * -> *). Applicative f => Bool -> f () -> f ()
unless (Set Id -> Set Id -> Bool
forall a. Ord a => Set a -> Set a -> Bool
Set.isSubsetOf Set Id
pcodom Set Id
ptarget Bool -> Bool -> Bool
&& Set Id -> Set Id -> Bool
forall a. Ord a => Set a -> Set a -> Bool
Set.isSubsetOf Set Id
pdom Set Id
psource)
        (Result () -> Result ()) -> Result () -> Result ()
forall a b. (a -> b) -> a -> b
$ String -> Result ()
forall (m :: * -> *) a. MonadFail m => String -> m a
Fail.fail "illegal QBF morphism"

-- | Application funtion for morphisms
applyMorphism :: Morphism -> Id -> Id
applyMorphism :: Morphism -> Id -> Id
applyMorphism mor :: Morphism
mor idt :: Id
idt = Id -> Id -> Map Id Id -> Id
forall k a. Ord k => a -> k -> Map k a -> a
Map.findWithDefault Id
idt Id
idt (Map Id Id -> Id) -> Map Id Id -> Id
forall a b. (a -> b) -> a -> b
$ Morphism -> Map Id Id
propMap Morphism
mor

-- | Application function for propMaps
applyMap :: Map.Map Id Id -> Id -> Id
applyMap :: Map Id Id -> Id -> Id
applyMap pmap :: Map Id Id
pmap idt :: Id
idt = Id -> Id -> Map Id Id -> Id
forall k a. Ord k => a -> k -> Map k a -> a
Map.findWithDefault Id
idt Id
idt Map Id Id
pmap

-- | Composition of morphisms in propositional Logic
composeMor :: Morphism -> Morphism -> Result Morphism
composeMor :: Morphism -> Morphism -> Result Morphism
composeMor f :: Morphism
f g :: Morphism
g =
  let fSource :: Sign
fSource = Morphism -> Sign
source Morphism
f
      gTarget :: Sign
gTarget = Morphism -> Sign
target Morphism
g
      fMap :: Map Id Id
fMap = Morphism -> Map Id Id
propMap Morphism
f
      gMap :: Map Id Id
gMap = Morphism -> Map Id Id
propMap Morphism
g
  in Morphism -> Result Morphism
forall (m :: * -> *) a. Monad m => a -> m a
return Morphism :: Sign -> Sign -> Map Id Id -> Morphism
Morphism
  { source :: Sign
source = Sign
fSource
  , target :: Sign
target = Sign
gTarget
  , propMap :: Map Id Id
propMap = if Map Id Id -> Bool
forall k a. Map k a -> Bool
Map.null Map Id Id
gMap then Map Id Id
fMap else
      (Id -> Map Id Id -> Map Id Id) -> Map Id Id -> Set Id -> Map Id Id
forall a b. (a -> b -> b) -> b -> Set a -> b
Set.fold ( \ i :: Id
i -> let j :: Id
j = Map Id Id -> Id -> Id
applyMap Map Id Id
gMap (Map Id Id -> Id -> Id
applyMap Map Id Id
fMap Id
i) in
                        if Id
i Id -> Id -> Bool
forall a. Eq a => a -> a -> Bool
== Id
j then Map Id Id -> Map Id Id
forall a. a -> a
id else Id -> Id -> Map Id Id -> Map Id Id
forall k a. Ord k => k -> a -> Map k a -> Map k a
Map.insert Id
i Id
j)
                                  Map Id Id
forall k a. Map k a
Map.empty (Set Id -> Map Id Id) -> Set Id -> Map Id Id
forall a b. (a -> b) -> a -> b
$ Sign -> Set Id
items Sign
fSource }

-- | Pretty printing for Morphisms
printMorphism :: Morphism -> Doc
printMorphism :: Morphism -> Doc
printMorphism m :: Morphism
m = Sign -> Doc
forall a. Pretty a => a -> Doc
pretty (Morphism -> Sign
source Morphism
m) Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> String -> Doc
text "-->" Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> Sign -> Doc
forall a. Pretty a => a -> Doc
pretty (Morphism -> Sign
target Morphism
m)
  Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> [Doc] -> Doc
vcat (((Id, Id) -> Doc) -> [(Id, Id)] -> [Doc]
forall a b. (a -> b) -> [a] -> [b]
map ( \ (x :: Id
x, y :: Id
y) -> Doc
lparen Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> Id -> Doc
forall a. Pretty a => a -> Doc
pretty Id
x Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> String -> Doc
text ","
  Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> Id -> Doc
forall a. Pretty a => a -> Doc
pretty Id
y Doc -> Doc -> Doc
forall a. Semigroup a => a -> a -> a
<> Doc
rparen) ([(Id, Id)] -> [Doc]) -> [(Id, Id)] -> [Doc]
forall a b. (a -> b) -> a -> b
$ Map Id Id -> [(Id, Id)]
forall k a. Map k a -> [(k, a)]
Map.assocs (Map Id Id -> [(Id, Id)]) -> Map Id Id -> [(Id, Id)]
forall a b. (a -> b) -> a -> b
$ Morphism -> Map Id Id
propMap Morphism
m)

-- | Inclusion map of a subsig into a supersig
inclusionMap :: Sign.Sign -> Sign.Sign -> Morphism
inclusionMap :: Sign -> Sign -> Morphism
inclusionMap s1 :: Sign
s1 s2 :: Sign
s2 = Morphism :: Sign -> Sign -> Map Id Id -> Morphism
Morphism
  { source :: Sign
source = Sign
s1
  , target :: Sign
target = Sign
s2
  , propMap :: Map Id Id
propMap = Map Id Id
forall k a. Map k a
Map.empty }

{- | sentence translation along signature morphism
here just the renaming of formulae -}
mapSentence :: Morphism -> AS_BASIC.FORMULA -> Result.Result AS_BASIC.FORMULA
mapSentence :: Morphism -> FORMULA -> Result FORMULA
mapSentence mor :: Morphism
mor = FORMULA -> Result FORMULA
forall (m :: * -> *) a. Monad m => a -> m a
return (FORMULA -> Result FORMULA)
-> (FORMULA -> FORMULA) -> FORMULA -> Result FORMULA
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor

mapSentenceH :: Morphism -> AS_BASIC.FORMULA -> AS_BASIC.FORMULA
mapSentenceH :: Morphism -> FORMULA -> FORMULA
mapSentenceH mor :: Morphism
mor frm :: FORMULA
frm = case FORMULA
frm of
  AS_BASIC.Negation form :: FORMULA
form rn :: Range
rn -> FORMULA -> Range -> FORMULA
AS_BASIC.Negation (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form) Range
rn
  AS_BASIC.Conjunction form :: [FORMULA]
form rn :: Range
rn ->
      [FORMULA] -> Range -> FORMULA
AS_BASIC.Conjunction ((FORMULA -> FORMULA) -> [FORMULA] -> [FORMULA]
forall a b. (a -> b) -> [a] -> [b]
map (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor) [FORMULA]
form) Range
rn
  AS_BASIC.Disjunction form :: [FORMULA]
form rn :: Range
rn ->
      [FORMULA] -> Range -> FORMULA
AS_BASIC.Disjunction ((FORMULA -> FORMULA) -> [FORMULA] -> [FORMULA]
forall a b. (a -> b) -> [a] -> [b]
map (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor) [FORMULA]
form) Range
rn
  AS_BASIC.Implication form1 :: FORMULA
form1 form2 :: FORMULA
form2 rn :: Range
rn -> FORMULA -> FORMULA -> Range -> FORMULA
AS_BASIC.Implication
      (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form1) (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form2) Range
rn
  AS_BASIC.Equivalence form1 :: FORMULA
form1 form2 :: FORMULA
form2 rn :: Range
rn -> FORMULA -> FORMULA -> Range -> FORMULA
AS_BASIC.Equivalence
      (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form1) (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form2) Range
rn
  AS_BASIC.TrueAtom rn :: Range
rn -> Range -> FORMULA
AS_BASIC.TrueAtom Range
rn
  AS_BASIC.FalseAtom rn :: Range
rn -> Range -> FORMULA
AS_BASIC.FalseAtom Range
rn
  AS_BASIC.Predication predH :: Token
predH -> Token -> FORMULA
AS_BASIC.Predication
      (Token -> FORMULA) -> Token -> FORMULA
forall a b. (a -> b) -> a -> b
$ Id -> Token
id2SimpleId (Id -> Token) -> Id -> Token
forall a b. (a -> b) -> a -> b
$ Morphism -> Id -> Id
applyMorphism Morphism
mor (Id -> Id) -> Id -> Id
forall a b. (a -> b) -> a -> b
$ Token -> Id
Id.simpleIdToId Token
predH
  AS_BASIC.ForAll xs :: [Token]
xs form :: FORMULA
form rn :: Range
rn -> [Token] -> FORMULA -> Range -> FORMULA
AS_BASIC.ForAll ((Token -> Token) -> [Token] -> [Token]
forall a b. (a -> b) -> [a] -> [b]
map
    (Id -> Token
id2SimpleId (Id -> Token) -> (Token -> Id) -> Token -> Token
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Morphism -> Id -> Id
applyMorphism Morphism
mor (Id -> Id) -> (Token -> Id) -> Token -> Id
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Token -> Id
Id.simpleIdToId) [Token]
xs)
    (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form) Range
rn
  AS_BASIC.Exists xs :: [Token]
xs form :: FORMULA
form rn :: Range
rn -> [Token] -> FORMULA -> Range -> FORMULA
AS_BASIC.Exists
    ((Token -> Token) -> [Token] -> [Token]
forall a b. (a -> b) -> [a] -> [b]
map (Id -> Token
id2SimpleId (Id -> Token) -> (Token -> Id) -> Token -> Token
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Morphism -> Id -> Id
applyMorphism Morphism
mor (Id -> Id) -> (Token -> Id) -> Token -> Id
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Token -> Id
Id.simpleIdToId) [Token]
xs)
    (Morphism -> FORMULA -> FORMULA
mapSentenceH Morphism
mor FORMULA
form) Range
rn

morphismUnion :: Morphism -> Morphism -> Result.Result Morphism
morphismUnion :: Morphism -> Morphism -> Result Morphism
morphismUnion mor1 :: Morphism
mor1 mor2 :: Morphism
mor2 =
  let pmap1 :: Map Id Id
pmap1 = Morphism -> Map Id Id
propMap Morphism
mor1
      pmap2 :: Map Id Id
pmap2 = Morphism -> Map Id Id
propMap Morphism
mor2
      p1 :: Sign
p1 = Morphism -> Sign
source Morphism
mor1
      p2 :: Sign
p2 = Morphism -> Sign
source Morphism
mor2
      up1 :: Set Id
up1 = Set Id -> Set Id -> Set Id
forall a. Ord a => Set a -> Set a -> Set a
Set.difference (Sign -> Set Id
items Sign
p1) (Set Id -> Set Id) -> Set Id -> Set Id
forall a b. (a -> b) -> a -> b
$ Map Id Id -> Set Id
forall k a. Map k a -> Set k
Map.keysSet Map Id Id
pmap1
      up2 :: Set Id
up2 = Set Id -> Set Id -> Set Id
forall a. Ord a => Set a -> Set a -> Set a
Set.difference (Sign -> Set Id
items Sign
p2) (Set Id -> Set Id) -> Set Id -> Set Id
forall a b. (a -> b) -> a -> b
$ Map Id Id -> Set Id
forall k a. Map k a -> Set k
Map.keysSet Map Id Id
pmap2
      (pds :: [Diagnosis]
pds, pmap :: Map Id Id
pmap) = ((Id, Id) -> ([Diagnosis], Map Id Id) -> ([Diagnosis], Map Id Id))
-> ([Diagnosis], Map Id Id)
-> [(Id, Id)]
-> ([Diagnosis], Map Id Id)
forall (t :: * -> *) a b.
Foldable t =>
(a -> b -> b) -> b -> t a -> b
foldr ( \ (i :: Id
i, j :: Id
j) (ds :: [Diagnosis]
ds, m :: Map Id Id
m) -> case Id -> Map Id Id -> Maybe Id
forall k a. Ord k => k -> Map k a -> Maybe a
Map.lookup Id
i Map Id Id
m of
          Nothing -> ([Diagnosis]
ds, Id -> Id -> Map Id Id -> Map Id Id
forall k a. Ord k => k -> a -> Map k a -> Map k a
Map.insert Id
i Id
j Map Id Id
m)
          Just k :: Id
k -> if Id
j Id -> Id -> Bool
forall a. Eq a => a -> a -> Bool
== Id
k then ([Diagnosis]
ds, Map Id Id
m) else
              (DiagKind -> String -> Range -> Diagnosis
Diag DiagKind
Error
               ("incompatible mapping of prop " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Id -> ShowS
showId Id
i " to "
                String -> ShowS
forall a. [a] -> [a] -> [a]
++ Id -> ShowS
showId Id
j " and " String -> ShowS
forall a. [a] -> [a] -> [a]
++ Id -> ShowS
showId Id
k "")
               Range
nullRange Diagnosis -> [Diagnosis] -> [Diagnosis]
forall a. a -> [a] -> [a]
: [Diagnosis]
ds, Map Id Id
m)) ([], Map Id Id
pmap1)
          (Map Id Id -> [(Id, Id)]
forall k a. Map k a -> [(k, a)]
Map.toList Map Id Id
pmap2 [(Id, Id)] -> [(Id, Id)] -> [(Id, Id)]
forall a. [a] -> [a] -> [a]
++ (Id -> (Id, Id)) -> [Id] -> [(Id, Id)]
forall a b. (a -> b) -> [a] -> [b]
map (\ a :: Id
a -> (Id
a, Id
a))
                      (Set Id -> [Id]
forall a. Set a -> [a]
Set.toList (Set Id -> [Id]) -> Set Id -> [Id]
forall a b. (a -> b) -> a -> b
$ Set Id -> Set Id -> Set Id
forall a. Ord a => Set a -> Set a -> Set a
Set.union Set Id
up1 Set Id
up2))
   in if [Diagnosis] -> Bool
forall (t :: * -> *) a. Foldable t => t a -> Bool
null [Diagnosis]
pds then Morphism -> Result Morphism
forall (m :: * -> *) a. Monad m => a -> m a
return Morphism :: Sign -> Sign -> Map Id Id -> Morphism
Morphism
      { source :: Sign
source = Sign -> Sign -> Sign
unite Sign
p1 Sign
p2
      , target :: Sign
target = Sign -> Sign -> Sign
unite (Morphism -> Sign
target Morphism
mor1) (Sign -> Sign) -> Sign -> Sign
forall a b. (a -> b) -> a -> b
$ Morphism -> Sign
target Morphism
mor2
      , propMap :: Map Id Id
propMap = Map Id Id
pmap } else [Diagnosis] -> Maybe Morphism -> Result Morphism
forall a. [Diagnosis] -> Maybe a -> Result a
Result [Diagnosis]
pds Maybe Morphism
forall a. Maybe a
Nothing