src/HOL/Tools/prop_logic.ML
author haftmann
Thu, 08 Jul 2010 16:19:24 +0200
changeset 37744 3daaf23b9ab4
parent 33243 17014b1b9353
child 38549 d0385f2764d8
permissions -rw-r--r--
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(*  Title:      HOL/Tools/prop_logic.ML
    Author:     Tjark Weber
    Copyright   2004-2009

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 is_nnf : prop_formula -> bool  (* returns true iff the formula is in negation normal form *)
	val is_cnf : prop_formula -> bool  (* returns true iff the formula is in conjunctive normal form *)

	val nnf    : prop_formula -> prop_formula  (* negation normal form *)
	val cnf    : prop_formula -> prop_formula  (* conjunctive normal form *)
	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 -> int Termtab.table -> prop_formula * int Termtab.table
	(* HOL term representation of propositional formulae *)
	val term_of_prop_formula : prop_formula -> 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))  = union (op =) (indices fm1) (indices fm2)
	  | indices (And (fm1, fm2)) = union (op =) (indices fm1) (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));

(* ------------------------------------------------------------------------- *)
(* is_nnf: returns 'true' iff the formula is in negation normal form (i.e.,  *)
(*         only variables may be negated, but not subformulas).              *)
(* ------------------------------------------------------------------------- *)

	local
		fun is_literal (BoolVar _)       = true
		  | is_literal (Not (BoolVar _)) = true
		  | is_literal _                 = false
		fun is_conj_disj (Or (fm1, fm2))  =
			is_conj_disj fm1 andalso is_conj_disj fm2
		  | is_conj_disj (And (fm1, fm2)) =
			is_conj_disj fm1 andalso is_conj_disj fm2
		  | is_conj_disj fm               =
			is_literal fm
	in
		fun is_nnf True  = true
		  | is_nnf False = true
		  | is_nnf fm    = is_conj_disj fm
	end;

(* ------------------------------------------------------------------------- *)
(* is_cnf: returns 'true' iff the formula is in conjunctive normal form      *)
(*         (i.e., a conjunction of disjunctions of literals). 'is_cnf'       *)
(*         implies 'is_nnf'.                                                 *)
(* ------------------------------------------------------------------------- *)

	local
		fun is_literal (BoolVar _)       = true
		  | is_literal (Not (BoolVar _)) = true
		  | is_literal _                 = false
		fun is_disj (Or (fm1, fm2)) = is_disj fm1 andalso is_disj fm2
		  | is_disj fm              = is_literal fm
		fun is_conj (And (fm1, fm2)) = is_conj fm1 andalso is_conj fm2
		  | is_conj fm               = is_disj fm
	in
		fun is_cnf True             = true
		  | is_cnf False            = true
		  | is_cnf fm               = is_conj fm
	end;

(* ------------------------------------------------------------------------- *)
(* nnf: computes the negation normal form of a formula 'fm' of propositional *)
(*      logic (i.e., only variables may be negated, but not subformulas).    *)
(*      Simplification (cf. 'simplify') is performed as well. Not            *)
(*      surprisingly, 'is_nnf o nnf' always returns 'true'. 'nnf fm' returns *)
(*      'fm' if (and only if) 'is_nnf fm' returns 'true'.                    *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

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

(* ------------------------------------------------------------------------- *)
(* cnf: computes the conjunctive normal form (i.e., a conjunction of         *)
(*      disjunctions of literals) of a formula 'fm' of propositional logic.  *)
(*      Simplification (cf. 'simplify') is performed as well. The result     *)
(*      is equivalent to 'fm', but may be exponentially longer. Not          *)
(*      surprisingly, 'is_cnf o cnf' always returns 'true'. 'cnf fm' returns *)
(*      'fm' if (and only if) 'is_cnf fm' returns 'true'.                    *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun cnf fm =
	let
		(* function to push an 'Or' below 'And's, using distributive laws *)
		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)
		fun cnf_from_nnf True             = True
		  | cnf_from_nnf False            = False
		  | cnf_from_nnf (BoolVar i)      = BoolVar i
		(* 'fm' must be a variable since the formula is in NNF *)
		  | cnf_from_nnf (Not fm)         = Not fm
		(* 'Or' may need to be pushed below 'And' *)
		  | cnf_from_nnf (Or (fm1, fm2))  =
		    cnf_or (cnf_from_nnf fm1, cnf_from_nnf fm2)
		(* 'And' as outermost connective is left untouched *)
		  | cnf_from_nnf (And (fm1, fm2)) =
		    And (cnf_from_nnf fm1, cnf_from_nnf fm2)
	in
		if is_cnf fm then
			fm
		else
			(cnf_from_nnf o nnf) fm
	end;

(* ------------------------------------------------------------------------- *)
(* defcnf: computes a definitional conjunctive normal form of a formula 'fm' *)
(*      of propositional logic. Simplification (cf. 'simplify') is performed *)
(*      as well. 'defcnf' may introduce auxiliary Boolean variables to avoid *)
(*      an exponential blowup of the formula.  The result is equisatisfiable *)
(*      (i.e., satisfiable if and only if 'fm' is satisfiable), but not      *)
(*      necessarily equivalent to 'fm'. Not surprisingly, 'is_cnf o defcnf'  *)
(*      always returns 'true'. 'defcnf fm' returns 'fm' if (and only if)     *)
(*      'is_cnf fm' returns 'true'.                                          *)
(* ------------------------------------------------------------------------- *)

	(* prop_formula -> prop_formula *)

	fun defcnf fm =
		if is_cnf fm then
			fm
		else
		let
			val fm' = nnf fm
			(* 'new' specifies the next index that is available to introduce an auxiliary variable *)
			(* int ref *)
			val new = Unsynchronized.ref (maxidx fm' + 1)
			(* unit -> int *)
			fun new_idx () = let val idx = !new in new := idx+1; idx end
			(* replaces 'And' by an auxiliary variable (and its definition) *)
			(* prop_formula -> prop_formula * prop_formula list *)
			fun defcnf_or (And x) =
				let
					val i = new_idx ()
				in
					(* Note that definitions are in NNF, but not CNF. *)
					(BoolVar i, [Or (Not (BoolVar i), And x)])
				end
			  | defcnf_or (Or (fm1, fm2)) =
				let
					val (fm1', defs1) = defcnf_or fm1
					val (fm2', defs2) = defcnf_or fm2
				in
					(Or (fm1', fm2'), defs1 @ defs2)
				end
			  | defcnf_or fm =
				(fm, [])
			(* prop_formula -> prop_formula *)
			fun defcnf_from_nnf True             = True
			  | defcnf_from_nnf False            = False
			  | defcnf_from_nnf (BoolVar i)      = BoolVar i
			(* 'fm' must be a variable since the formula is in NNF *)
			  | defcnf_from_nnf (Not fm)         = Not fm
			(* 'Or' may need to be pushed below 'And' *)
			(* 'Or' of literal and 'And': use distributivity *)
			  | defcnf_from_nnf (Or (BoolVar i, And (fm1, fm2))) =
				And (defcnf_from_nnf (Or (BoolVar i, fm1)),
				     defcnf_from_nnf (Or (BoolVar i, fm2)))
			  | defcnf_from_nnf (Or (Not (BoolVar i), And (fm1, fm2))) =
				And (defcnf_from_nnf (Or (Not (BoolVar i), fm1)),
				     defcnf_from_nnf (Or (Not (BoolVar i), fm2)))
			  | defcnf_from_nnf (Or (And (fm1, fm2), BoolVar i)) =
				And (defcnf_from_nnf (Or (fm1, BoolVar i)),
				     defcnf_from_nnf (Or (fm2, BoolVar i)))
			  | defcnf_from_nnf (Or (And (fm1, fm2), Not (BoolVar i))) =
				And (defcnf_from_nnf (Or (fm1, Not (BoolVar i))),
				     defcnf_from_nnf (Or (fm2, Not (BoolVar i))))
			(* all other cases: turn the formula into a disjunction of literals, *)
			(*                  adding definitions as necessary                  *)
			  | defcnf_from_nnf (Or x) =
				let
					val (fm, defs) = defcnf_or (Or x)
					val cnf_defs   = map defcnf_from_nnf defs
				in
					all (fm :: cnf_defs)
				end
			(* 'And' as outermost connective is left untouched *)
			  | defcnf_from_nnf (And (fm1, fm2)) =
				And (defcnf_from_nnf fm1, defcnf_from_nnf fm2)
		in
			defcnf_from_nnf 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 = Unsynchronized.ref false
		val next_idx          = Unsynchronized.ref 0
		fun get_next_idx () =
			if !next_idx_is_valid then
				Unsynchronized.inc next_idx
			else (
				next_idx := Termtab.fold (Integer.max o snd) table 0;
				next_idx_is_valid := true;
				Unsynchronized.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;