src/HOL/Tools/sat_solver.ML

fixed dpll solver (now uses NNF)

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

Interface to external SAT solvers, and (simple) built-in SAT solvers.
*)

signature SAT_SOLVER =
sig
	type solver  (* PropLogic.prop_formula -> (int -> bool) option option *)

	(* external SAT solvers *)
	val write_dimacs_cnf_file : Path.T -> PropLogic.prop_formula -> unit
	val write_dimacs_sat_file : Path.T -> PropLogic.prop_formula -> unit
	val parse_std_result_file : Path.T -> string -> (int -> bool) option
	val make_external_solver  : string -> (PropLogic.prop_formula -> unit) -> (unit -> (int -> bool) option)
			-> PropLogic.prop_formula -> (int -> bool) option

	(* generic interface *)
	val solvers       : (string * solver) list ref
	val add_solver    : string * solver -> unit
	val invoke_solver : string -> solver  (* exception OPTION *)
end;

structure SatSolver : SAT_SOLVER =
struct

	open PropLogic;

(* ------------------------------------------------------------------------- *)
(* Type of SAT solvers: given a propositional formula, a satisfying          *)
(*      assignment may be returned                                           *)
(*      - 'Some None' means that no satisfying assignment was found          *)
(*      - 'None' means that the solver was not configured/installed properly *)
(* ------------------------------------------------------------------------- *)

	type solver = prop_formula -> (int -> bool) option option;

(* ------------------------------------------------------------------------- *)
(* write_dimacs_cnf_file: serializes a formula 'fm' of propositional logic   *)
(*      to a file in DIMACS CNF format (see "Satisfiability Suggested        *)
(*      Format", May 8 1993, Section 2.1)                                    *)
(* Note: 'fm' must not contain a variable index less than 1.                 *)
(* Note: 'fm' is converted into (definitional) CNF.                          *)
(* ------------------------------------------------------------------------- *)

	(* Path.T -> prop_formula -> unit *)

	fun write_dimacs_cnf_file path fm =
	let
		(* prop_formula -> prop_formula *)
		fun cnf_True_False_elim True =
			Or (BoolVar 1, Not (BoolVar 1))
		  | cnf_True_False_elim False =
			And (BoolVar 1, Not (BoolVar 1))
		  | cnf_True_False_elim fm =
			fm  (* since 'fm' is in CNF, either 'fm'='True'/'False', or 'fm' does not contain 'True'/'False' at all *)
		(* prop_formula -> int *)
		fun cnf_number_of_clauses (And (fm1,fm2)) =
			(cnf_number_of_clauses fm1) + (cnf_number_of_clauses fm2)
		  | cnf_number_of_clauses _ =
			1
		(* prop_formula -> string *)
		fun cnf_string True =
			error "formula is not in CNF"
		  | cnf_string False =
			error "formula is not in CNF"
		  | cnf_string (BoolVar i) =
			(assert (i>=1) "formula contains a variable index less than 1";
			string_of_int i)
		  | cnf_string (Not fm) =
			"-" ^ (cnf_string fm)
		  | cnf_string (Or (fm1,fm2)) =
			(cnf_string fm1) ^ " " ^ (cnf_string fm2)
		  | cnf_string (And (fm1,fm2)) =
			(cnf_string fm1) ^ " 0\n" ^ (cnf_string fm2)
	in
		File.write path
			(let
				val cnf               = (cnf_True_False_elim o defcnf) fm  (* conversion into def. CNF *)
				val number_of_vars    = maxidx cnf
				val number_of_clauses = cnf_number_of_clauses cnf
			in
				"c This file was generated by SatSolver.write_dimacs_cnf_file\n" ^
				"c (c) Tjark Weber\n" ^
				"p cnf " ^ (string_of_int number_of_vars) ^ " " ^ (string_of_int number_of_clauses) ^ "\n" ^
				(cnf_string cnf) ^ "\n"
			end)
	end;

(* ------------------------------------------------------------------------- *)
(* write_dimacs_sat_file: serializes a formula 'fm' of propositional logic   *)
(*      to a file in DIMACS SAT format (see "Satisfiability Suggested        *)
(*      Format", May 8 1993, Section 2.2)                                    *)
(* Note: 'fm' must not contain a variable index less than 1.                 *)
(* ------------------------------------------------------------------------- *)

	(* Path.T -> prop_formula -> unit *)

	fun write_dimacs_sat_file path fm =
	let
		(* prop_formula -> string *)
		fun sat_string True =
			"*()"
		  | sat_string False =
			"+()"
		  | sat_string (BoolVar i) =
			(assert (i>=1) "formula contains a variable index less than 1";
			string_of_int i)
		  | sat_string (Not fm) =
			"-(" ^ (sat_string fm) ^ ")"
		  | sat_string (Or (fm1,fm2)) =
			"+(" ^ (sat_string fm1) ^ " " ^ (sat_string fm2) ^ ")"
		  | sat_string (And (fm1,fm2)) =
			"*(" ^ (sat_string fm1) ^ " " ^ (sat_string fm2) ^ ")"
	in
		File.write path
			(let
				val number_of_vars = Int.max (maxidx fm, 1)
			in
				"c This file was generated by SatSolver.write_dimacs_sat_file\n" ^
				"c (c) Tjark Weber\n" ^
				"p sat " ^ (string_of_int number_of_vars) ^ "\n" ^
				"(" ^ (sat_string fm) ^ ")\n"
			end)
	end;

(* ------------------------------------------------------------------------- *)
(* parse_std_result_file: scans a SAT solver's output file for a satisfying  *)
(*      variable assignment.  Returns the assignment, or 'None' if the SAT   *)
(*      solver did not find one.  The file format must be as follows:        *)
(*      ,-----                                                               *)
(*      | 0 or more lines not containing 'success'                           *)
(*      | A line containing 'success' as a substring                         *)
(*      | A line ASSIGNMENT containing integers, separated by " "            *)
(*      | 0 or more lines                                                    *)
(*      `-----                                                               *)
(*      If variable i is contained in ASSIGNMENT, then i is interpreted as   *)
(*      'true'.  If ~i is contained in ASSIGNMENT, then i is interpreted as  *)
(*      'false'.                                                             *)
(* ------------------------------------------------------------------------- *)

	(* Path.T -> string -> (int -> bool) option *)

	fun parse_std_result_file path success =
	let
		(* 'a option -> 'a Library.option *)
		fun option (SOME a) =
			Some a
		  | option NONE =
			None
		(* string -> int list *)
		fun int_list_from_string s =
			mapfilter (option o Int.fromString) (space_explode " " s)
		(* int list -> int -> bool *)
		fun assignment_from_list [] i =
			false  (* could be 'true' just as well; the SAT solver didn't provide a value for this variable *)
		  | assignment_from_list (x::xs) i =
			if x=i then true
			else if x=(~i) then false
			else assignment_from_list xs i
		(* string -> string -> bool *)
		fun is_substring needle haystack =
		let
			val length1 = String.size needle
			val length2 = String.size haystack
		in
			if length2 < length1 then
				false
			else if needle = String.substring (haystack, 0, length1) then
				true
			else is_substring needle (String.substring (haystack, 1, length2-1))
		end
		(* string list -> int list option *)
		fun parse_lines [] =
			None
		  | parse_lines (line::lines) =
			if is_substring success line then
				(* the next line must be a list of integers *)
				(Some o assignment_from_list o int_list_from_string o hd) lines
			else
				parse_lines lines
	in
		(parse_lines o split_lines o File.read) path
	end;

(* ------------------------------------------------------------------------- *)
(* make_external_solver: call 'writefn', execute 'cmd', call 'readfn'        *)
(* ------------------------------------------------------------------------- *)

	(* string -> (prop_formula -> unit) -> (unit -> (int -> bool) option) -> prop_formula -> (int -> bool) option *)

	fun make_external_solver cmd writefn readfn fm =
		(writefn fm;
		assert ((system cmd)=0) ("system command " ^ quote cmd ^ " failed (make sure the SAT solver is installed)");
		readfn ());

(* ------------------------------------------------------------------------- *)
(* solvers: a (reference to a) table of all registered SAT solvers           *)
(* ------------------------------------------------------------------------- *)

	val solvers = ref ([] : (string * solver) list);

(* ------------------------------------------------------------------------- *)
(* add_solver: updates 'solvers' by adding a new solver                      *)
(* ------------------------------------------------------------------------- *)

	(* string * solver -> unit *)

	fun add_solver (name, new_solver) =
		(solvers := overwrite_warn (!solvers, (name, new_solver)) ("SAT solver " ^ quote name ^ " was defined before"));

(* ------------------------------------------------------------------------- *)
(* invoke_solver: returns the solver associated with the given 'name'        *)
(* Note: If no solver is associated with 'name', exception 'OPTION' will be  *)
(*       raised.                                                             *)
(* ------------------------------------------------------------------------- *)

	(* string -> solver *)

	fun invoke_solver name =
		(the o assoc) (!solvers, name);

end;  (* SatSolver *)


(* ------------------------------------------------------------------------- *)
(* Predefined SAT solvers                                                    *)
(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)
(* Internal SAT solver, available as 'SatSolver.solve "enumerate"' -- simply *)
(* enumerates assignments until a satisfying assignment is found             *)
(* ------------------------------------------------------------------------- *)

let
	fun enum_solver fm =
	let
		(* int list *)
		val indices = PropLogic.indices fm
		(* int list -> int list -> int list option *)
		(* binary increment: list 'xs' of current bits, list 'ys' of all bits (lower bits first) *)
		fun next_list _ ([]:int list) =
			None
		  | next_list [] (y::ys) =
			Some [y]
		  | next_list (x::xs) (y::ys) =
			if x=y then
				(* reset the bit, continue *)
				next_list xs ys
			else
				(* set the lowest bit that wasn't set before, keep the higher bits *)
				Some (y::x::xs)
		(* int list -> int -> bool *)
		fun assignment_from_list xs i =
			i mem xs
		(* int list -> (int -> bool) option *)
		fun solver_loop xs =
			if PropLogic.eval (assignment_from_list xs) fm then
				Some (assignment_from_list xs)
			else
				(case next_list xs indices of
				  Some xs' => solver_loop xs'
				| None     => None)
	in
		(* start with the "first" assignment (all variables are interpreted as 'False') *)
		solver_loop []
	end
in
	SatSolver.add_solver ("enumerate", Some o enum_solver)
end;

(* ------------------------------------------------------------------------- *)
(* Internal SAT solver, available as 'SatSolver.solve "dpll"' -- a simple    *)
(* implementation of the DPLL algorithm (cf. L. Zhang, S. Malik: "The Quest  *)
(* for Efficient Boolean Satisfiability Solvers", July 2002, Fig. 1).        *)
(* ------------------------------------------------------------------------- *)

let
	local
		open PropLogic
	in
		fun dpll_solver fm =
		let
			(* We could use 'PropLogic.defcnf fm' instead of 'nnf fm', but that *)
			(* sometimes introduces more boolean variables than we can handle   *)
			(* efficiently.                                                     *)
			val fm' = PropLogic.nnf fm
			(* int list *)
			val indices = PropLogic.indices fm'
			(* int list -> int -> prop_formula *)
			fun partial_var_eval []      i = BoolVar i
			  | partial_var_eval (x::xs) i = if x=i then True else if x=(~i) then False else partial_var_eval xs i
			(* int list -> prop_formula -> prop_formula *)
			fun partial_eval xs True             = True
			  | partial_eval xs False            = False
			  | partial_eval xs (BoolVar i)      = partial_var_eval xs i
			  | partial_eval xs (Not fm)         = SNot (partial_eval xs fm)
			  | partial_eval xs (Or (fm1, fm2))  = SOr (partial_eval xs fm1, partial_eval xs fm2)
			  | partial_eval xs (And (fm1, fm2)) = SAnd (partial_eval xs fm1, partial_eval xs fm2)
			  | partial_eval xs _                = raise ERROR  (* formula is not in negation normal form *)
			(* prop_formula -> int list *)
			fun forced_vars True              = []
			  | forced_vars False             = []
			  | forced_vars (BoolVar i)       = [i]
			  | forced_vars (Not (BoolVar i)) = [~i]
			  | forced_vars (Or (fm1, fm2))   = (forced_vars fm1) inter_int (forced_vars fm2)
			  | forced_vars (And (fm1, fm2))  = (forced_vars fm1) union_int (forced_vars fm2)
			  (* Above, i *and* ~i may be forced.  In this case the first occurrence takes   *)
			  (* precedence, and the next partial evaluation of the formula returns 'False'. *)
			  | forced_vars _                 = raise ERROR  (* formula is not in negation normal form *)
			(* int list -> prop_formula -> (int list * prop_formula) *)
			fun eval_and_force xs fm =
			let
				val fm' = partial_eval xs fm
				val xs' = forced_vars fm'
			in
				if null xs' then
					(xs, fm')
				else
					eval_and_force (xs@xs') fm'  (* xs and xs' should be distinct, so '@' here should have *)
					                             (* the same effect as 'union_int'                         *)
			end
			(* int list -> int option *)
			fun fresh_var xs =
				Library.find_first (fn i => not (i mem_int xs) andalso not ((~i) mem_int xs)) indices
			(* int list -> prop_formula -> int list option *)
			(* partial assignment 'xs' *)
			fun dpll xs fm =
			let
				val (xs', fm') = eval_and_force xs fm
			in
				case fm' of
				  True  => Some xs'
				| False => None
				| _     =>
					let
						val x = the (fresh_var xs')  (* a fresh variable must exist since 'fm' did not evaluate to 'True'/'False' *)
					in
						case dpll (x::xs') fm' of  (* passing fm' rather than fm should save some simplification work *)
						  Some xs'' => Some xs''
						| None      => dpll ((~x)::xs') fm'  (* now try interpreting 'x' as 'False' *)
					end
			end
			(* int list -> int -> bool *)
			fun assignment_from_list [] i =
				false  (* could be 'true' just as well; the DPLL procedure didn't provide a value for this variable *)
			  | assignment_from_list (x::xs) i =
				if x=i then true
				else if x=(~i) then false
				else assignment_from_list xs i
		in
			(* initially, no variable is interpreted yet *)
			apsome assignment_from_list (dpll [] fm')
		end
	end  (* local *)
in
	SatSolver.add_solver ("dpll", Some o dpll_solver)
end;

(* ------------------------------------------------------------------------- *)
(* Internal SAT solver, available as 'SatSolver.solve "auto"': uses the      *)
(* first installed solver (other than "auto" itself)                         *)
(* ------------------------------------------------------------------------- *)

let
	fun auto_solver fm =
		get_first (fn (name,solver) =>
			if name="auto" then
				None
			else
				((if name="dpll" then
					warning ("Using SAT solver \"dpll\"; for better performance, consider using an external solver.")
				else
					());
				solver fm)) (rev (!SatSolver.solvers))
in
	SatSolver.add_solver ("auto", auto_solver)
end;

(* ------------------------------------------------------------------------- *)
(* ZChaff, Version 2003.12.04                                                *)
(* ------------------------------------------------------------------------- *)

let
	fun zchaff fm =
	let
		val inname     = "isabelle.cnf"
		val outname    = "result"
		val inpath     = File.tmp_path (Path.unpack inname)
		val outpath    = File.tmp_path (Path.unpack outname)
		val cmd        = (getenv "ZCHAFF_HOME") ^ "/zchaff " ^ (Path.pack inpath) ^ " > " ^ (Path.pack outpath)
		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath fm
		fun readfn ()  = SatSolver.parse_std_result_file outpath "Verify Solution successful. Instance satisfiable"
		val _          = assert (not (File.exists inpath)) ("file " ^ quote (Path.pack inpath) ^ " exists, please delete (will not overwrite)")
		val _          = assert (not (File.exists outpath)) ("file " ^ quote (Path.pack outpath) ^ " exists, please delete (will not overwrite)")
		val assignment = SatSolver.make_external_solver cmd writefn readfn fm
		val _          = (File.rm inpath handle _ => ())
		val _          = (File.rm outpath handle _ => ())
	in
		assignment
	end
in
		SatSolver.add_solver ("zchaff", (fn fm => if getenv "ZCHAFF_HOME" <> "" then Some (zchaff fm) else None))
end;

(* ------------------------------------------------------------------------- *)
(* Add code for other SAT solvers below.                                     *)
(* ------------------------------------------------------------------------- *)

(*
let
	fun mysolver fm = ...
in
	SatSolver.add_solver ("myname", (fn fm => if mysolver_is_installed then Some (mysolver fm) else None));  -- register the solver
end;

-- the solver is now available as SatSolver.invoke_solver "myname"
*)