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(* Title: HOL/Tools/prop_logic.ML
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ID: $Id$
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Author: Tjark Weber
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Copyright 2004
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Formulas of propositional logic.
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*)
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signature PROP_LOGIC =
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sig
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datatype prop_formula =
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True
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| False
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| BoolVar of int (* NOTE: only use indices >= 1 *)
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| Not of prop_formula
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| Or of prop_formula * prop_formula
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| And of prop_formula * prop_formula
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val SNot : prop_formula -> prop_formula
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val SOr : prop_formula * prop_formula -> prop_formula
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val SAnd : prop_formula * prop_formula -> prop_formula
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val indices : prop_formula -> int list (* set of all variable indices *)
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val maxidx : prop_formula -> int (* maximal variable index *)
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val nnf : prop_formula -> prop_formula (* negation normal form *)
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val cnf : prop_formula -> prop_formula (* conjunctive normal form *)
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val defcnf : prop_formula -> prop_formula (* definitional cnf *)
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val exists : prop_formula list -> prop_formula (* finite disjunction *)
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val all : prop_formula list -> prop_formula (* finite conjunction *)
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val dot_product : prop_formula list * prop_formula list -> prop_formula
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val eval : (int -> bool) -> prop_formula -> bool (* semantics *)
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end;
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structure PropLogic : PROP_LOGIC =
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struct
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(* ------------------------------------------------------------------------- *)
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(* prop_formula: formulas of propositional logic, built from Boolean *)
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(* variables (referred to by index) and True/False using *)
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(* not/or/and *)
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(* ------------------------------------------------------------------------- *)
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datatype prop_formula =
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True
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| False
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| BoolVar of int (* NOTE: only use indices >= 1 *)
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| Not of prop_formula
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| Or of prop_formula * prop_formula
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| And of prop_formula * prop_formula;
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(* ------------------------------------------------------------------------- *)
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(* The following constructor functions make sure that True and False do not *)
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(* occur within any of the other connectives (i.e. Not, Or, And), and *)
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(* perform double-negation elimination. *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> prop_formula *)
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fun SNot True = False
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| SNot False = True
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| SNot (Not fm) = fm
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| SNot fm = Not fm;
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(* prop_formula * prop_formula -> prop_formula *)
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fun SOr (True, _) = True
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| SOr (_, True) = True
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| SOr (False, fm) = fm
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| SOr (fm, False) = fm
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| SOr (fm1, fm2) = Or (fm1, fm2);
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(* prop_formula * prop_formula -> prop_formula *)
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fun SAnd (True, fm) = fm
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| SAnd (fm, True) = fm
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| SAnd (False, _) = False
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| SAnd (_, False) = False
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| SAnd (fm1, fm2) = And (fm1, fm2);
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(* ------------------------------------------------------------------------- *)
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(* indices: collects all indices of Boolean variables that occur in a *)
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(* propositional formula 'fm'; no duplicates *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> int list *)
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fun indices True = []
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| indices False = []
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| indices (BoolVar i) = [i]
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| indices (Not fm) = indices fm
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| indices (Or (fm1,fm2)) = (indices fm1) union_int (indices fm2)
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| indices (And (fm1,fm2)) = (indices fm1) union_int (indices fm2);
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(* ------------------------------------------------------------------------- *)
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(* maxidx: computes the maximal variable index occuring in a formula of *)
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(* propositional logic 'fm'; 0 if 'fm' contains no variable *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> int *)
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fun maxidx True = 0
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| maxidx False = 0
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| maxidx (BoolVar i) = i
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| maxidx (Not fm) = maxidx fm
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| maxidx (Or (fm1,fm2)) = Int.max (maxidx fm1, maxidx fm2)
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| maxidx (And (fm1,fm2)) = Int.max (maxidx fm1, maxidx fm2);
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(* ------------------------------------------------------------------------- *)
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(* nnf: computes the negation normal form of a formula 'fm' of propositional *)
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(* logic (i.e. only variables may be negated, but not subformulas) *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> prop_formula *)
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fun
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(* constants *)
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nnf True = True
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| nnf False = False
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(* variables *)
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| nnf (BoolVar i) = BoolVar i
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(* 'or' and 'and' as outermost connectives are left untouched *)
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| nnf (Or (fm1,fm2)) = SOr (nnf fm1, nnf fm2)
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| nnf (And (fm1,fm2)) = SAnd (nnf fm1, nnf fm2)
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(* 'not' + constant *)
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| nnf (Not True) = False
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| nnf (Not False) = True
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(* 'not' + variable *)
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| nnf (Not (BoolVar i)) = Not (BoolVar i)
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(* pushing 'not' inside of 'or'/'and' using de Morgan's laws *)
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| nnf (Not (Or (fm1,fm2))) = SAnd (nnf (SNot fm1), nnf (SNot fm2))
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| nnf (Not (And (fm1,fm2))) = SOr (nnf (SNot fm1), nnf (SNot fm2))
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(* double-negation elimination *)
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| nnf (Not (Not fm)) = nnf fm;
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(* ------------------------------------------------------------------------- *)
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(* cnf: computes the conjunctive normal form (i.e. a conjunction of *)
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(* disjunctions) of a formula 'fm' of propositional logic. The result *)
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(* formula may be exponentially longer than 'fm'. *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> prop_formula *)
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fun cnf fm =
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let
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fun
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(* constants *)
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cnf_from_nnf True = True
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| cnf_from_nnf False = False
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(* literals *)
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| cnf_from_nnf (BoolVar i) = BoolVar i
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| cnf_from_nnf (Not (BoolVar i)) = Not (BoolVar i)
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(* pushing 'or' inside of 'and' using distributive laws *)
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| cnf_from_nnf (Or (fm1,fm2)) =
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let
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val fm1' = cnf_from_nnf fm1
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val fm2' = cnf_from_nnf fm2
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in
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case fm1' of
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And (fm11,fm12) => cnf_from_nnf (SAnd (SOr(fm11,fm2'),SOr(fm12,fm2')))
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| _ =>
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(case fm2' of
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And (fm21,fm22) => cnf_from_nnf (SAnd (SOr(fm1',fm21),SOr(fm1',fm22)))
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(* neither subformula contains 'and' *)
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| _ => Or (fm1,fm2))
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end
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(* 'and' as outermost connective is left untouched *)
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| cnf_from_nnf (And (fm1,fm2)) = SAnd (cnf_from_nnf fm1, cnf_from_nnf fm2)
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(* 'not' + something other than a variable: formula is not in negation normal form *)
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| cnf_from_nnf _ = raise ERROR
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in
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(cnf_from_nnf o nnf) fm
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end;
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(* ------------------------------------------------------------------------- *)
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(* defcnf: computes the definitional conjunctive normal form of a formula *)
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(* 'fm' of propositional logic, introducing auxiliary variables if *)
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(* necessary to avoid an exponential blowup of the formula. The result *)
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(* formula is satisfiable if and only if 'fm' is satisfiable. *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula -> prop_formula *)
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fun defcnf fm =
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let
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(* prop_formula * int -> prop_formula * int *)
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(* 'new' specifies the next index that is available to introduce an auxiliary variable *)
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fun
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(* constants *)
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defcnf_from_nnf (True,new) = (True, new)
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| defcnf_from_nnf (False,new) = (False, new)
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(* literals *)
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| defcnf_from_nnf (BoolVar i,new) = (BoolVar i, new)
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| defcnf_from_nnf (Not (BoolVar i),new) = (Not (BoolVar i), new)
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(* pushing 'or' inside of 'and' using distributive laws *)
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| defcnf_from_nnf (Or (fm1,fm2),new) =
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let
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val (fm1',new') = defcnf_from_nnf (fm1, new)
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val (fm2',new'') = defcnf_from_nnf (fm2, new')
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in
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case fm1' of
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And (fm11,fm12) =>
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let
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val aux = BoolVar new''
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in
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(* '(fm11 AND fm12) OR fm2' is SAT-equivalent to '(fm11 OR aux) AND (fm12 OR aux) AND (fm2 OR NOT aux)' *)
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defcnf_from_nnf (SAnd (SAnd (SOr (fm11,aux), SOr (fm12,aux)), SOr(fm2', Not aux)), new''+1)
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end
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| _ =>
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(case fm2' of
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And (fm21,fm22) =>
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let
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val aux = BoolVar new''
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in
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(* 'fm1 OR (fm21 AND fm22)' is SAT-equivalent to '(fm1 OR NOT aux) AND (fm21 OR aux) AND (fm22 OR NOT aux)' *)
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defcnf_from_nnf (SAnd (SOr (fm1', Not aux), SAnd (SOr (fm21,aux), SOr (fm22,aux))), new''+1)
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end
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(* neither subformula contains 'and' *)
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| _ => (Or (fm1,fm2),new))
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end
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(* 'and' as outermost connective is left untouched *)
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| defcnf_from_nnf (And (fm1,fm2),new) =
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let
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val (fm1',new') = defcnf_from_nnf (fm1, new)
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val (fm2',new'') = defcnf_from_nnf (fm2, new')
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in
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(SAnd (fm1', fm2'), new'')
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end
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(* 'not' + something other than a variable: formula is not in negation normal form *)
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| defcnf_from_nnf (_,_) = raise ERROR
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in
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(fst o defcnf_from_nnf) (nnf fm, (maxidx fm)+1)
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end;
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(* ------------------------------------------------------------------------- *)
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(* exists: computes the disjunction over a list 'xs' of propositional *)
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(* formulas *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula list -> prop_formula *)
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fun exists xs = foldl SOr (False, xs);
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(* ------------------------------------------------------------------------- *)
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(* all: computes the conjunction over a list 'xs' of propositional formulas *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula list -> prop_formula *)
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fun all xs = foldl SAnd (True, xs);
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(* ------------------------------------------------------------------------- *)
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(* dot_product: ([x1,...,xn], [y1,...,yn]) -> x1*y1+...+xn*yn *)
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(* ------------------------------------------------------------------------- *)
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(* prop_formula list * prop_formula list -> prop_formula *)
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fun dot_product (xs,ys) = exists (map SAnd (xs~~ys));
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(* ------------------------------------------------------------------------- *)
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(* eval: given an assignment 'a' of Boolean values to variable indices, the *)
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(* truth value of a propositional formula 'fm' is computed *)
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(* ------------------------------------------------------------------------- *)
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(* (int -> bool) -> prop_formula -> bool *)
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fun eval a True = true
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| eval a False = false
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| eval a (BoolVar i) = (a i)
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| eval a (Not fm) = not (eval a fm)
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| eval a (Or (fm1,fm2)) = (eval a fm1) orelse (eval a fm2)
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| eval a (And (fm1,fm2)) = (eval a fm1) andalso (eval a fm2);
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end;
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