Sublogic.hs revision 1a38107941725211e7c3f051f7a8f5e12199f03a
{-# LANGUAGE DeriveDataTypeable #-}
{- |
Module : $Header$
Description : Sublogics for propositional logic
Copyright : (c) Dominik Luecke, Uni Bremen 2007
License : GPLv2 or higher, see LICENSE.txt
Maintainer : luecke@informatik.uni-bremen.de
Stability : experimental
Portability : non-portable (imports Logic.Logic)
Sublogics 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. Birkhaeuser.
2005.
-}
module Propositional.Sublogic
(
sl_basic_spec -- determine sublogic for basic spec
, PropFormulae (..) -- types of propositional formulae
, PropSL (..) -- sublogics for propositional logic
, sublogics_max -- join of sublogics
, top -- Propositional
, bottom -- Horn
, sublogics_all -- all sublogics
, sublogics_name -- name of sublogics
, sl_sig -- sublogic for a signature
, sl_form -- sublogic for a formula
, sl_sym -- sublogic for symbols
, sl_symit -- sublogic for symbol items
, sl_mor -- sublogic for morphisms
, sl_symmap -- sublogic for symbol map items
, prSymbolM -- projection of symbols
, prSig -- projections of signatures
, prMor -- projections of morphisms
, prSymMapM -- projections of symbol maps
, prSymM -- projections of SYMB_ITEMS
, prFormulaM -- projections of formulae
, prBasicSpec -- projections of basic specs
, isProp
, isHC
)
where
import Data.Data
import qualified Propositional.Tools as Tools
import qualified Propositional.AS_BASIC_Propositional as AS_BASIC
import qualified Propositional.Sign as Sign
import qualified Propositional.Symbol as Symbol
import qualified Propositional.Morphism as Morphism
import qualified Common.Lib.State as State
import qualified Common.AS_Annotation as AS_Anno
{- -----------------------------------------------------------------------------
datatypes --
----------------------------------------------------------------------------- -}
-- | types of propositional formulae
data PropFormulae = PlainFormula -- Formula without structural constraints
| HornClause -- Horn Clause Formulae
deriving (Show, Eq, Ord, Typeable, Data)
-- | sublogics for propositional logic
data PropSL = PropSL
{
format :: PropFormulae -- Structural restrictions
} deriving (Show, Eq, Ord, Typeable, Data)
isProp :: PropSL -> Bool
isProp sl = format sl == PlainFormula
isHC :: PropSL -> Bool
isHC sl = format sl == HornClause
-- | comparison of sublogics
compareLE :: PropSL -> PropSL -> Bool
compareLE p1 p2 =
let f1 = format p1
f2 = format p2
in
case (f1, f2) of
(_, PlainFormula) -> True
(HornClause, HornClause) -> True
(_, HornClause) -> False
{- ----------------------------------------------------------------------------
Special elements in the Lattice of logics --
---------------------------------------------------------------------------- -}
top :: PropSL
top = PropSL PlainFormula
bottom :: PropSL
bottom = PropSL HornClause
need_PF :: PropSL
need_PF = bottom { format = PlainFormula }
{- -----------------------------------------------------------------------------
join and max --
----------------------------------------------------------------------------- -}
sublogics_join :: (PropFormulae -> PropFormulae -> PropFormulae)
-> PropSL -> PropSL -> PropSL
sublogics_join pfF a b = PropSL
{
format = pfF (format a) (format b)
}
joinType :: PropFormulae -> PropFormulae -> PropFormulae
joinType HornClause HornClause = HornClause
joinType _ _ = PlainFormula
sublogics_max :: PropSL -> PropSL -> PropSL
sublogics_max = sublogics_join joinType
{- -----------------------------------------------------------------------------
Helpers --
----------------------------------------------------------------------------- -}
-- compute sublogics from a list of sublogics
--
comp_list :: [PropSL] -> PropSL
comp_list = foldl sublogics_max bottom
{- ----------------------------------------------------------------------------
Functions to compute minimal sublogic for a given element, these work --
by recursing into all subelements --
---------------------------------------------------------------------------- -}
-- | determines the sublogic for symbol map items
sl_symmap :: PropSL -> AS_BASIC.SYMB_MAP_ITEMS -> PropSL
sl_symmap ps _ = ps
-- | determines the sublogic for a morphism
sl_mor :: PropSL -> Morphism.Morphism -> PropSL
sl_mor ps _ = ps
-- | determines the sublogic for a Signature
sl_sig :: PropSL -> Sign.Sign -> PropSL
sl_sig ps _ = ps
-- | determines the sublogic for Symbol items
sl_symit :: PropSL -> AS_BASIC.SYMB_ITEMS -> PropSL
sl_symit ps _ = ps
-- | determines the sublogic for symbols
sl_sym :: PropSL -> Symbol.Symbol -> PropSL
sl_sym ps _ = ps
-- | determines sublogic for formula
sl_form :: PropSL -> AS_BASIC.FORMULA -> PropSL
sl_form ps f = sl_fl_form ps $ Tools.flatten f
-- | determines sublogic for flattened formula
sl_fl_form :: PropSL -> [AS_BASIC.FORMULA] -> PropSL
sl_fl_form ps f = foldl sublogics_max ps
$ map (\ x -> State.evalState (ana_form ps x) 0) f
-- analysis of single "clauses"
ana_form :: PropSL -> AS_BASIC.FORMULA -> State.State Int PropSL
ana_form ps f =
case f of
AS_BASIC.Conjunction l _ ->
do
st <- State.get
return $ sublogics_max need_PF $ comp_list $ map
(\ x -> State.evalState (ana_form ps x) (st + 1)) l
AS_BASIC.Implication l m _ ->
do
st <- State.get
let analyze = sublogics_max need_PF $ sublogics_max
((\ x -> State.evalState (ana_form ps x) (st + 1)) l)
((\ x -> State.evalState (ana_form ps x) (st + 1)) m)
return $
if st < 1 && format ps == HornClause && checkHornPos l m
then ps else analyze
AS_BASIC.Equivalence l m _ ->
do
st <- State.get
return $ sublogics_max need_PF $ sublogics_max
((\ x -> State.evalState (ana_form ps x) (st + 1)) l)
((\ x -> State.evalState (ana_form ps x) (st + 1)) m)
AS_BASIC.Negation l _ ->
if isLiteral l
then return ps
else
do
st <- State.get
return $ (\ x -> State.evalState (ana_form ps x) (st + 1)) l
AS_BASIC.Disjunction l _ ->
let lprime = concatMap Tools.flattenDis l in
if all isLiteral lprime
then return $
if moreThanNLit lprime 1
then sublogics_max need_PF ps else ps
else
do
st <- State.get
return $ sublogics_max need_PF $ comp_list $ map
(\ x -> State.evalState (ana_form ps x) (st + 1))
lprime
AS_BASIC.True_atom _ -> return ps
AS_BASIC.False_atom _ -> return ps
AS_BASIC.Predication _ -> return ps
moreThanNLit :: [AS_BASIC.FORMULA] -> Int -> Bool
moreThanNLit form n = foldl (\ y x -> if x then y + 1 else y) 0
(map isPosLiteral form) > n
-- determines wheter a Formula is a literal
isLiteral :: AS_BASIC.FORMULA -> Bool
isLiteral (AS_BASIC.Predication _) = True
isLiteral (AS_BASIC.Negation (AS_BASIC.Predication _) _ ) = True
isLiteral (AS_BASIC.Negation _ _) = False
isLiteral (AS_BASIC.Conjunction _ _) = False
isLiteral (AS_BASIC.Implication {}) = False
isLiteral (AS_BASIC.Equivalence {}) = False
isLiteral (AS_BASIC.Disjunction _ _) = False
isLiteral (AS_BASIC.True_atom _ ) = True
isLiteral (AS_BASIC.False_atom _) = True
-- determines wheter a Formula is a positive literal
isPosLiteral :: AS_BASIC.FORMULA -> Bool
isPosLiteral (AS_BASIC.Predication _) = True
isPosLiteral (AS_BASIC.Negation _ _) = False
isPosLiteral (AS_BASIC.Conjunction _ _) = False
isPosLiteral (AS_BASIC.Implication {}) = False
isPosLiteral (AS_BASIC.Equivalence {}) = False
isPosLiteral (AS_BASIC.Disjunction _ _) = False
isPosLiteral (AS_BASIC.True_atom _ ) = True
isPosLiteral (AS_BASIC.False_atom _) = True
-- | determines subloig for basic items
sl_basic_items :: PropSL -> AS_BASIC.BASIC_ITEMS -> PropSL
sl_basic_items ps bi =
case bi of
AS_BASIC.Pred_decl _ -> ps
AS_BASIC.Axiom_items xs -> comp_list $ map (sl_form ps . AS_Anno.item) xs
-- | determines sublogic for basic spec
sl_basic_spec :: PropSL -> AS_BASIC.BASIC_SPEC -> PropSL
sl_basic_spec ps (AS_BASIC.Basic_spec spec) =
comp_list $ map (sl_basic_items ps . AS_Anno.item) spec
-- | all sublogics
sublogics_all :: [PropSL]
sublogics_all =
[PropSL
{
format = HornClause
}
, PropSL
{
format = PlainFormula
}
]
{- -----------------------------------------------------------------------------
Conversion functions to String --
----------------------------------------------------------------------------- -}
sublogics_name :: PropSL -> String
sublogics_name f = case format f of
HornClause -> "HornClause"
PlainFormula -> "Prop"
{- -----------------------------------------------------------------------------
Projections to sublogics --
----------------------------------------------------------------------------- -}
prSymbolM :: PropSL -> Symbol.Symbol -> Maybe Symbol.Symbol
prSymbolM _ = Just
prSig _ sig = sig
prMor :: PropSL -> Morphism.Morphism -> Morphism.Morphism
prMor _ mor = mor
prSymMapM :: PropSL
-> Maybe AS_BASIC.SYMB_MAP_ITEMS
prSymMapM _ = Just
prSymM :: PropSL -> AS_BASIC.SYMB_ITEMS -> Maybe AS_BASIC.SYMB_ITEMS
prSymM _ = Just
-- keep an element if its computed sublogic is in the given sublogic
--
prFormulaM :: PropSL -> AS_BASIC.FORMULA -> Maybe AS_BASIC.FORMULA
prFormulaM sl form
| compareLE (sl_form bottom form) sl = Just form
| otherwise = Nothing
prPredItem :: PropSL -> AS_BASIC.PRED_ITEM -> AS_BASIC.PRED_ITEM
prPredItem _ prI = prI
prBASIC_items :: PropSL -> AS_BASIC.BASIC_ITEMS -> AS_BASIC.BASIC_ITEMS
prBASIC_items pSL bI =
case bI of
AS_BASIC.Pred_decl pI -> AS_BASIC.Pred_decl $ prPredItem pSL pI
AS_BASIC.Axiom_items aIS -> AS_BASIC.Axiom_items $ concatMap mapH aIS
where
mapH :: AS_Anno.Annoted AS_BASIC.FORMULA
mapH annoForm = let formP = prFormulaM pSL $ AS_Anno.item annoForm in
case formP of
Nothing -> []
Just f -> [ AS_Anno.Annoted
{
AS_Anno.item = f
, AS_Anno.opt_pos = AS_Anno.opt_pos annoForm
, AS_Anno.l_annos = AS_Anno.l_annos annoForm
, AS_Anno.r_annos = AS_Anno.r_annos annoForm
}
]
prBasicSpec :: PropSL -> AS_BASIC.BASIC_SPEC -> AS_BASIC.BASIC_SPEC
prBasicSpec pSL (AS_BASIC.Basic_spec bS) =
AS_BASIC.Basic_spec $ map mapH bS
where
mapH :: AS_Anno.Annoted AS_BASIC.BASIC_ITEMS
mapH aBI = AS_Anno.Annoted
{
AS_Anno.item = prBASIC_items pSL $ AS_Anno.item aBI
, AS_Anno.opt_pos = AS_Anno.opt_pos aBI
, AS_Anno.l_annos = AS_Anno.l_annos aBI
, AS_Anno.r_annos = AS_Anno.r_annos aBI
}
checkHornPos :: AS_BASIC.FORMULA -> AS_BASIC.FORMULA -> Bool
checkHornPos fc fl =
case fc of
AS_BASIC.Conjunction _ _ -> all isPosLiteral $ Tools.flatten fc
_ -> False
&&
isPosLiteral fl