src/HOL/Tools/prop_logic.ML

term_of_prop_formula added

(*  Title:      HOL/Tools/prop_logic.ML
    ID:         $Id$
    Author:     Tjark Weber
    Copyright   2004-2005

Formulas of propositional logic.
*)

signature PROP_LOGIC =
sig
	datatype prop_formula =
		  True
		| False
		| BoolVar of int  (* NOTE: only use indices >= 1 *)
		| Not of prop_formula
		| Or of prop_formula * prop_formula
		| And of prop_formula * prop_formula

	val SNot     : prop_formula -> prop_formula
	val SOr      : prop_formula * prop_formula -> prop_formula
	val SAnd     : prop_formula * prop_formula -> prop_formula
	val simplify : prop_formula -> prop_formula  (* eliminates True/False and double-negation *)

	val indices : prop_formula -> int list  (* set of all variable indices *)
	val maxidx  : prop_formula -> int       (* maximal variable index *)

	val exists      : prop_formula list -> prop_formula  (* finite disjunction *)
	val all         : prop_formula list -> prop_formula  (* finite conjunction *)
	val dot_product : prop_formula list * prop_formula list -> prop_formula

	val nnf    : prop_formula -> prop_formula  (* negation normal form *)
	val cnf    : prop_formula -> prop_formula  (* conjunctive normal form *)
	val auxcnf : prop_formula -> prop_formula  (* cnf with auxiliary variables *)
	val defcnf : prop_formula -> prop_formula  (* definitional cnf *)

	val eval : (int -> bool) -> prop_formula -> bool  (* semantics *)

	(* propositional representation of HOL terms *)
	val prop_formula_of_term : Term.term -> int Termtab.table -> prop_formula * int Termtab.table
	(* HOL term representation of propositional formulae *)
	val term_of_prop_formula : prop_formula -> Term.term
end;

structure PropLogic : PROP_LOGIC =
struct

(* ------------------------------------------------------------------------- *)
(* prop_formula: formulas of propositional logic, built from Boolean         *)
(*               variables (referred to by index) and True/False using       *)
(*               not/or/and                                                  *)
(* ------------------------------------------------------------------------- *)

	datatype prop_formula =
		  True
		| False
		| BoolVar of int  (* NOTE: only use indices >= 1 *)
		| Not of prop_formula
		| Or of prop_formula * prop_formula
		| And of prop_formula * prop_formula;

(* ------------------------------------------------------------------------- *)
(* The following constructor functions make sure that True and False do not  *)
(* occur within any of the other connectives (i.e. Not, Or, And), and        *)
(* perform double-negation elimination.                                      *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun SNot True     = False
	  | SNot False    = True
	  | SNot (Not fm) = fm
	  | SNot fm       = Not fm;

	(* prop_formula * prop_formula -> prop_formula *)

	fun SOr (True, _)   = True
	  | SOr (_, True)   = True
	  | SOr (False, fm) = fm
	  | SOr (fm, False) = fm
	  | SOr (fm1, fm2)  = Or (fm1, fm2);

	(* prop_formula * prop_formula -> prop_formula *)

	fun SAnd (True, fm) = fm
	  | SAnd (fm, True) = fm
	  | SAnd (False, _) = False
	  | SAnd (_, False) = False
	  | SAnd (fm1, fm2) = And (fm1, fm2);

(* ------------------------------------------------------------------------- *)
(* simplify: eliminates True/False below other connectives, and double-      *)
(*      negation                                                             *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun simplify (Not fm)         = SNot (simplify fm)
	  | simplify (Or (fm1, fm2))  = SOr (simplify fm1, simplify fm2)
	  | simplify (And (fm1, fm2)) = SAnd (simplify fm1, simplify fm2)
	  | simplify fm               = fm;

(* ------------------------------------------------------------------------- *)
(* indices: collects all indices of Boolean variables that occur in a        *)
(*      propositional formula 'fm'; no duplicates                            *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> int list *)

	fun indices True             = []
	  | indices False            = []
	  | indices (BoolVar i)      = [i]
	  | indices (Not fm)         = indices fm
	  | indices (Or (fm1, fm2))  = (indices fm1) union_int (indices fm2)
	  | indices (And (fm1, fm2)) = (indices fm1) union_int (indices fm2);

(* ------------------------------------------------------------------------- *)
(* maxidx: computes the maximal variable index occuring in a formula of      *)
(*      propositional logic 'fm'; 0 if 'fm' contains no variable             *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> int *)

	fun maxidx True             = 0
	  | maxidx False            = 0
	  | maxidx (BoolVar i)      = i
	  | maxidx (Not fm)         = maxidx fm
	  | maxidx (Or (fm1, fm2))  = Int.max (maxidx fm1, maxidx fm2)
	  | maxidx (And (fm1, fm2)) = Int.max (maxidx fm1, maxidx fm2);

(* ------------------------------------------------------------------------- *)
(* exists: computes the disjunction over a list 'xs' of propositional        *)
(*      formulas                                                             *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula list -> prop_formula *)

	fun exists xs = Library.foldl SOr (False, xs);

(* ------------------------------------------------------------------------- *)
(* all: computes the conjunction over a list 'xs' of propositional formulas  *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula list -> prop_formula *)

	fun all xs = Library.foldl SAnd (True, xs);

(* ------------------------------------------------------------------------- *)
(* dot_product: ([x1,...,xn], [y1,...,yn]) -> x1*y1+...+xn*yn                *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula list * prop_formula list -> prop_formula *)

	fun dot_product (xs,ys) = exists (map SAnd (xs~~ys));

(* ------------------------------------------------------------------------- *)
(* nnf: computes the negation normal form of a formula 'fm' of propositional *)
(*      logic (i.e. only variables may be negated, but not subformulas).     *)
(*      Simplification (c.f. 'simplify') is performed as well.               *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun
	(* constants *)
	    nnf True                   = True
	  | nnf False                  = False
	(* variables *)
	  | nnf (BoolVar i)            = (BoolVar i)
	(* 'or' and 'and' as outermost connectives are left untouched *)
	  | nnf (Or  (fm1, fm2))       = SOr  (nnf fm1, nnf fm2)
	  | nnf (And (fm1, fm2))       = SAnd (nnf fm1, nnf fm2)
	(* 'not' + constant *)
	  | nnf (Not True)             = False
	  | nnf (Not False)            = True
	(* 'not' + variable *)
	  | nnf (Not (BoolVar i))      = Not (BoolVar i)
	(* pushing 'not' inside of 'or'/'and' using de Morgan's laws *)
	  | nnf (Not (Or  (fm1, fm2))) = SAnd (nnf (SNot fm1), nnf (SNot fm2))
	  | nnf (Not (And (fm1, fm2))) = SOr  (nnf (SNot fm1), nnf (SNot fm2))
	(* double-negation elimination *)
	  | nnf (Not (Not fm))         = nnf fm;

(* ------------------------------------------------------------------------- *)
(* cnf: computes the conjunctive normal form (i.e. a conjunction of          *)
(*      disjunctions) of a formula 'fm' of propositional logic.  The result  *)
(*      formula may be exponentially longer than 'fm'.                       *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun cnf fm =
	let
		fun
		(* constants *)
		    cnf_from_nnf True             = True
		  | cnf_from_nnf False            = False
		(* literals *)
		  | cnf_from_nnf (BoolVar i)      = BoolVar i
		  | cnf_from_nnf (Not fm1)        = Not fm1  (* 'fm1' must be a variable since the formula is in NNF *)
		(* pushing 'or' inside of 'and' using distributive laws *)
		  | cnf_from_nnf (Or (fm1, fm2))  =
			let
				fun cnf_or (And (fm11, fm12), fm2) =
					And (cnf_or (fm11, fm2), cnf_or (fm12, fm2))
				  | cnf_or (fm1, And (fm21, fm22)) =
					And (cnf_or (fm1, fm21), cnf_or (fm1, fm22))
				(* neither subformula contains 'and' *)
				  | cnf_or (fm1, fm2) =
					Or (fm1, fm2)
			in
				cnf_or (cnf_from_nnf fm1, cnf_from_nnf fm2)
			end
		(* 'and' as outermost connective is left untouched *)
		  | cnf_from_nnf (And (fm1, fm2)) = And (cnf_from_nnf fm1, cnf_from_nnf fm2)
	in
		(cnf_from_nnf o nnf) fm
	end;

(* ------------------------------------------------------------------------- *)
(* auxcnf: computes the definitional conjunctive normal form of a formula    *)
(*      'fm' of propositional logic, introducing auxiliary variables if      *)
(*      necessary to avoid an exponential blowup of the formula.  The result *)
(*      formula is satisfiable if and only if 'fm' is satisfiable.           *)
(*      Auxiliary variables are introduced as switches for OR-nodes, based   *)
(*      on the observation that e.g. "fm1 OR (fm21 AND fm22)" is             *)
(*      equisatisfiable with "(fm1 OR ~aux) AND (fm21 OR aux) AND (fm22 OR   *)
(*      aux)".                                                               *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* Note: 'auxcnf' tends to use fewer variables and fewer clauses than        *)
(*      'defcnf' below, but sometimes generates much larger SAT problems     *)
(*      overall (hence it must sometimes generate longer clauses than        *)
(*      'defcnf' does).  It is currently not quite clear to me if one of the *)
(*      algorithms is clearly superior to the other, but I suggest using     *)
(*      'defcnf' instead.                                                    *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun auxcnf fm =
	let
		(* convert formula to NNF first *)
		val fm' = nnf fm
		(* 'new' specifies the next index that is available to introduce an auxiliary variable *)
		(* int ref *)
		val new = ref (maxidx fm' + 1)
		(* unit -> int *)
		fun new_idx () = let val idx = !new in new := idx+1; idx end
		(* prop_formula -> prop_formula *)
		fun
		(* constants *)
		    auxcnf_from_nnf True  = True
		  | auxcnf_from_nnf False = False
		(* literals *)
		  | auxcnf_from_nnf (BoolVar i) = BoolVar i
		  | auxcnf_from_nnf (Not fm1)   = Not fm1  (* 'fm1' must be a variable since the formula is in NNF *)
		(* pushing 'or' inside of 'and' using auxiliary variables *)
		  | auxcnf_from_nnf (Or (fm1, fm2)) =
			let
				val fm1' = auxcnf_from_nnf fm1
				val fm2' = auxcnf_from_nnf fm2
				(* prop_formula * prop_formula -> prop_formula *)
				fun auxcnf_or (And (fm11, fm12), fm2) =
					(case fm2 of
					(* do not introduce an auxiliary variable for literals *)
					  BoolVar _ =>
							And (auxcnf_or (fm11, fm2), auxcnf_or (fm12, fm2))
					| Not _ =>
							And (auxcnf_or (fm11, fm2), auxcnf_or (fm12, fm2))
					| _ =>
						let
							val aux = BoolVar (new_idx ())
						in
							And (And (auxcnf_or (fm11, aux), auxcnf_or (fm12, aux)), auxcnf_or (fm2, Not aux))
						end)
				  | auxcnf_or (fm1, And (fm21, fm22)) =
					(case fm1 of
					(* do not introduce an auxiliary variable for literals *)
					  BoolVar _ =>
							And (auxcnf_or (fm1, fm21), auxcnf_or (fm1, fm22))
					| Not _ =>
							And (auxcnf_or (fm1, fm21), auxcnf_or (fm1, fm22))
					| _ =>
						let
							val aux = BoolVar (new_idx ())
						in
							And (auxcnf_or (fm1, Not aux), And (auxcnf_or (fm21, aux), auxcnf_or (fm22, aux)))
						end)
				(* neither subformula contains 'and' *)
				  | auxcnf_or (fm1, fm2) =
					Or (fm1, fm2)
			in
				auxcnf_or (fm1', fm2')
			end
		(* 'and' as outermost connective is left untouched *)
		  | auxcnf_from_nnf (And (fm1, fm2)) =
				And (auxcnf_from_nnf fm1, auxcnf_from_nnf fm2)
	in
		auxcnf_from_nnf fm'
	end;

(* ------------------------------------------------------------------------- *)
(* defcnf: computes the definitional conjunctive normal form of a formula    *)
(*      'fm' of propositional logic, introducing auxiliary variables to      *)
(*      avoid an exponential blowup of the formula.  The result formula is   *)
(*      satisfiable if and only if 'fm' is satisfiable.  Auxiliary variables *)
(*      are introduced as abbreviations for AND-, OR-, and NOT-nodes, based  *)
(*      on the following equisatisfiabilities (+/- indicates polarity):      *)
(*      LITERAL+       == LITERAL                                            *)
(*      LITERAL-       == NOT LITERAL                                        *)
(*      (NOT fm1)+     == aux AND (NOT aux OR fm1-)                          *)
(*      (fm1 OR fm2)+  == aux AND (NOT aux OR fm1+ OR fm2+)                  *)
(*      (fm1 AND fm2)+ == aux AND (NOT aux OR fm1+) AND (NOT aux OR fm2+)    *)
(*      (NOT fm1)-     == aux AND (NOT aux OR fm1+)                          *)
(*      (fm1 OR fm2)-  == aux AND (NOT aux OR fm1-) AND (NOT aux OR fm2-)    *)
(*      (fm1 AND fm2)- == aux AND (NOT aux OR fm1- OR fm2-)                  *)
(*      Example:                                                             *)
(*      NOT (a AND b) == aux1 AND (NOT aux1 OR aux2)                         *)
(*                            AND (NOT aux2 OR NOT a OR NOT b)               *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun defcnf fm =
	let
		(* simplify formula first *)
		val fm' = simplify fm
		(* 'new' specifies the next index that is available to introduce an auxiliary variable *)
		(* int ref *)
		val new = ref (maxidx fm' + 1)
		(* unit -> int *)
		fun new_idx () = let val idx = !new in new := idx+1; idx end
		(* optimization for n-ary disjunction/conjunction *)
		(* prop_formula -> prop_formula list *)
		fun disjuncts (Or (fm1, fm2)) = (disjuncts fm1) @ (disjuncts fm2)
		  | disjuncts fm1             = [fm1]
		(* prop_formula -> prop_formula list *)
		fun conjuncts (And (fm1, fm2)) = (conjuncts fm1) @ (conjuncts fm2)
		  | conjuncts fm1              = [fm1]
		(* polarity -> formula -> (defining clauses, literal) *)
		(* bool -> prop_formula -> prop_formula * prop_formula *)
		fun
		(* constants *)
		    defcnf' true  True  = (True, True)
		  | defcnf' false True  = (*(True, False)*) error "formula is not simplified, True occurs with negative polarity"
		  | defcnf' true  False = (True, False)
		  | defcnf' false False = (*(True, True)*) error "formula is not simplified, False occurs with negative polarity"
		(* literals *)
		  | defcnf' true  (BoolVar i)       = (True, BoolVar i)
		  | defcnf' false (BoolVar i)       = (True, Not (BoolVar i))
		  | defcnf' true  (Not (BoolVar i)) = (True, Not (BoolVar i))
		  | defcnf' false (Not (BoolVar i)) = (True, BoolVar i)
		(* 'not' *)
		  | defcnf' polarity (Not fm1) =
			let
				val (def1, aux1) = defcnf' (not polarity) fm1
				val aux          = BoolVar (new_idx ())
				val def          = Or (Not aux, aux1)
			in
				(SAnd (def1, def), aux)
			end
		(* 'or' *)
		  | defcnf' polarity (Or (fm1, fm2)) =
			let
				val fms          = disjuncts (Or (fm1, fm2))
				val (defs, auxs) = split_list (map (defcnf' polarity) fms)
				val aux          = BoolVar (new_idx ())
				val def          = if polarity then Or (Not aux, exists auxs) else all (map (fn a => Or (Not aux, a)) auxs)
			in
				(SAnd (all defs, def), aux)
			end
		(* 'and' *)
		  | defcnf' polarity (And (fm1, fm2)) =
			let
				val fms          = conjuncts (And (fm1, fm2))
				val (defs, auxs) = split_list (map (defcnf' polarity) fms)
				val aux          = BoolVar (new_idx ())
				val def          = if not polarity then Or (Not aux, exists auxs) else all (map (fn a => Or (Not aux, a)) auxs)
			in
				(SAnd (all defs, def), aux)
			end
		(* optimization: do not introduce auxiliary variables for parts of the formula that are in CNF already *)
		(* prop_formula -> prop_formula * prop_formula *)
		fun defcnf_or (Or (fm1, fm2)) =
			let
				val (def1, aux1) = defcnf_or fm1
				val (def2, aux2) = defcnf_or fm2
			in
				(SAnd (def1, def2), Or (aux1, aux2))
			end
		  | defcnf_or fm =
			defcnf' true fm
		(* prop_formula -> prop_formula * prop_formula *)
		fun defcnf_and (And (fm1, fm2)) =
			let
				val (def1, aux1) = defcnf_and fm1
				val (def2, aux2) = defcnf_and fm2
			in
				(SAnd (def1, def2), And (aux1, aux2))
			end
		  | defcnf_and (Or (fm1, fm2)) =
			let
				val (def1, aux1) = defcnf_or fm1
				val (def2, aux2) = defcnf_or fm2
			in
				(SAnd (def1, def2), Or (aux1, aux2))
			end
		  | defcnf_and fm =
			defcnf' true fm
	in
		SAnd (defcnf_and fm')
	end;

(* ------------------------------------------------------------------------- *)
(* eval: given an assignment 'a' of Boolean values to variable indices, the  *)
(*      truth value of a propositional formula 'fm' is computed              *)
(* ------------------------------------------------------------------------- *)

	(* (int -> bool) -> prop_formula -> bool *)

	fun eval a True            = true
	  | eval a False           = false
	  | eval a (BoolVar i)     = (a i)
	  | eval a (Not fm)        = not (eval a fm)
	  | eval a (Or (fm1,fm2))  = (eval a fm1) orelse (eval a fm2)
	  | eval a (And (fm1,fm2)) = (eval a fm1) andalso (eval a fm2);

(* ------------------------------------------------------------------------- *)
(* prop_formula_of_term: returns the propositional structure of a HOL term,  *)
(*      with subterms replaced by Boolean variables.  Also returns a table   *)
(*      of terms and corresponding variables that extends the table that was *)
(*      given as an argument.  Usually, you'll just want to use              *)
(*      'Termtab.empty' as value for 'table'.                                *)
(* ------------------------------------------------------------------------- *)

(* Note: The implementation is somewhat optimized; the next index to be used *)
(*       is computed only when it is actually needed.  However, when         *)
(*       'prop_formula_of_term' is invoked many times, it might be more      *)
(*       efficient to pass and return this value as an additional parameter, *)
(*       so that it does not have to be recomputed (by folding over the      *)
(*       table) for each invocation.                                         *)

	(* Term.term -> int Termtab.table -> prop_formula * int Termtab.table *)
	fun prop_formula_of_term t table =
	let
		val next_idx_is_valid = ref false
		val next_idx          = ref 0
		fun get_next_idx () =
			if !next_idx_is_valid then
				inc next_idx
			else (
				next_idx          := Termtab.fold (curry Int.max o snd) table 0;
				next_idx_is_valid := true;
				inc next_idx
			)
		fun aux (Const ("True", _))         table =
			(True, table)
		  | aux (Const ("False", _))        table =
			(False, table)
		  | aux (Const ("Not", _) $ x)      table =
			apfst Not (aux x table)
		  | aux (Const ("op |", _) $ x $ y) table =
			let
				val (fm1, table1) = aux x table
				val (fm2, table2) = aux y table1
			in
				(Or (fm1, fm2), table2)
			end
		  | aux (Const ("op &", _) $ x $ y) table =
			let
				val (fm1, table1) = aux x table
				val (fm2, table2) = aux y table1
			in
				(And (fm1, fm2), table2)
			end
		  | aux x                           table =
			(case Termtab.lookup table x of
			  SOME i =>
				(BoolVar i, table)
			| NONE   =>
				let
					val i = get_next_idx ()
				in
					(BoolVar i, Termtab.update (x, i) table)
				end)
	in
		aux t table
	end;

(* ------------------------------------------------------------------------- *)
(* term_of_prop_formula: returns a HOL term that corresponds to a            *)
(*      propositional formula, with Boolean variables replaced by Free's     *)
(* ------------------------------------------------------------------------- *)

(* Note: A more generic implementation should take another argument of type  *)
(*       Term.term Inttab.table (or so) that specifies HOL terms for some    *)
(*       Boolean variables in the formula, similar to 'prop_formula_of_term' *)
(*       (but the other way round).                                          *)

	(* prop_formula -> Term.term *)
	fun term_of_prop_formula True             =
			HOLogic.true_const
		| term_of_prop_formula False            =
			HOLogic.false_const
		| term_of_prop_formula (BoolVar i)      =
			Free ("v" ^ Int.toString i, HOLogic.boolT)
		| term_of_prop_formula (Not fm)         =
			HOLogic.mk_not (term_of_prop_formula fm)
		| term_of_prop_formula (Or (fm1, fm2))  =
			HOLogic.mk_disj (term_of_prop_formula fm1, term_of_prop_formula fm2)
		| term_of_prop_formula (And (fm1, fm2)) =
			HOLogic.mk_conj (term_of_prop_formula fm1, term_of_prop_formula fm2);

end;