(*  Title:      HOL/Tools/sat_solver.ML

    ID:         $Id$

    Author:     Tjark Weber

    Copyright   2004-2006

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

*)

signature SAT_SOLVER =

sig

	exception NOT_CONFIGURED

	type assignment = int -> bool option

	type proof      = int list Inttab.table * int

	datatype result = SATISFIABLE of assignment

	                | UNSATISFIABLE of proof option

	                | UNKNOWN

	type solver     = PropLogic.prop_formula -> result

	(* auxiliary functions to create 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 read_std_result_file  : Path.T -> string * string * string -> result

	val make_external_solver  : string -> (PropLogic.prop_formula -> unit) -> (unit -> result) -> solver

	val read_dimacs_cnf_file : Path.T -> PropLogic.prop_formula

	(* generic solver 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;

(* ------------------------------------------------------------------------- *)

(* should be raised by an external SAT solver to indicate that the solver is *)

(* not configured properly                                                   *)

(* ------------------------------------------------------------------------- *)

	exception NOT_CONFIGURED;

(* ------------------------------------------------------------------------- *)

(* type of partial (satisfying) assignments: 'a i = NONE' means that 'a' is  *)

(*      a satisfying assignment regardless of the value of variable 'i'      *)

(* ------------------------------------------------------------------------- *)

	type assignment = int -> bool option;

(* ------------------------------------------------------------------------- *)

(* a proof of unsatisfiability, to be interpreted as follows: each integer   *)

(*      is a clause ID, each list 'xs' stored under the key 'x' in the table *)

(*      contains the IDs of clauses that must be resolved (in the given      *)

(*      order) to obtain the new clause 'x'.  Each list 'xs' must be         *)

(*      non-empty, and the literal to be resolved upon must always be unique *)

(*      (e.g. "A | ~B" must not be resolved with "~A | B").  Circular        *)

(*      dependencies of clauses are not allowed.  (At least) one of the      *)

(*      clauses in the table must be the empty clause (i.e. contain no       *)

(*      literals); its ID is given by the second component of the proof.     *)

(*      The clauses of the original problem passed to the SAT solver have    *)

(*      consecutive IDs starting with 0.  Clause IDs must be non-negative,   *)

(*      but do not need to be consecutive.                                   *)

(* ------------------------------------------------------------------------- *)

	type proof = int list Inttab.table * int;

(* ------------------------------------------------------------------------- *)

(* return type of SAT solvers: if the result is 'SATISFIABLE', a satisfying  *)

(*      assignment must be returned as well; if the result is                *)

(*      'UNSATISFIABLE', a proof of unsatisfiability may be returned         *)

(* ------------------------------------------------------------------------- *)

	datatype result = SATISFIABLE of assignment

	                | UNSATISFIABLE of proof option

	                | UNKNOWN;

(* ------------------------------------------------------------------------- *)

(* type of SAT solvers: given a propositional formula, a satisfying          *)

(*      assignment may be returned                                           *)

(* ------------------------------------------------------------------------- *)

	type solver = prop_formula -> result;

(* ------------------------------------------------------------------------- *)

(* 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' must be given in 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 list *)

		fun cnf_string fm =

		let

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

			fun cnf_string_acc True acc =

				error "formula is not in CNF"

			  | cnf_string_acc False acc =

				error "formula is not in CNF"

			  | cnf_string_acc (BoolVar i) acc =

				(assert (i>=1) "formula contains a variable index less than 1";

				string_of_int i :: acc)

			  | cnf_string_acc (Not (BoolVar i)) acc =

				"-" :: cnf_string_acc (BoolVar i) acc

			  | cnf_string_acc (Not _) acc =

				error "formula is not in CNF"

			  | cnf_string_acc (Or (fm1,fm2)) acc =

				cnf_string_acc fm1 (" " :: cnf_string_acc fm2 acc)

			  | cnf_string_acc (And (fm1,fm2)) acc =

				cnf_string_acc fm1 (" 0\n" :: cnf_string_acc fm2 acc)

		in

			cnf_string_acc fm []

		end

		val fm'               = cnf_True_False_elim fm

		val number_of_vars    = maxidx fm'

		val number_of_clauses = cnf_number_of_clauses fm'

	in

		File.write path

			("c This file was generated by SatSolver.write_dimacs_cnf_file\n" ^

			 "p cnf " ^ string_of_int number_of_vars ^ " " ^ string_of_int number_of_clauses ^ "\n");

		File.append_list path

			(cnf_string fm');

		File.append path

			" 0\n"

	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 list *)

		fun sat_string fm =

		let

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

			fun sat_string_acc True acc =

				"*()" :: acc

			  | sat_string_acc False acc =

				"+()" :: acc

			  | sat_string_acc (BoolVar i) acc =

				(assert (i>=1) "formula contains a variable index less than 1";

				string_of_int i :: acc)

			  | sat_string_acc (Not (BoolVar i)) acc =

				"-" :: sat_string_acc (BoolVar i) acc

			  | sat_string_acc (Not fm) acc =

				"-(" :: sat_string_acc fm (")" :: acc)

			  | sat_string_acc (Or (fm1,fm2)) acc =

				"+(" :: sat_string_acc_or fm1 (" " :: sat_string_acc_or fm2 (")" :: acc))

			  | sat_string_acc (And (fm1,fm2)) acc =

				"*(" :: sat_string_acc_and fm1 (" " :: sat_string_acc_and fm2 (")" :: acc))

			(* optimization to make use of n-ary disjunction/conjunction *)

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

			and sat_string_acc_or (Or (fm1,fm2)) acc =

				sat_string_acc_or fm1 (" " :: sat_string_acc_or fm2 acc)

			  | sat_string_acc_or fm acc =

				sat_string_acc fm acc

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

			and sat_string_acc_and (And (fm1,fm2)) acc =

				sat_string_acc_and fm1 (" " :: sat_string_acc_and fm2 acc)

			  | sat_string_acc_and fm acc =

				sat_string_acc fm acc

		in

			sat_string_acc fm []

		end

		val number_of_vars = Int.max (maxidx fm, 1)

	in

		File.write path

			("c This file was generated by SatSolver.write_dimacs_sat_file\n" ^

			 "p sat " ^ string_of_int number_of_vars ^ "\n" ^

			 "(");

		File.append_list path

			(sat_string fm);

		File.append path

			")\n"

	end;

(* ------------------------------------------------------------------------- *)

(* read_std_result_file: scans a SAT solver's output file for a satisfying   *)

(*      variable assignment.  Returns the assignment, or 'UNSATISFIABLE' if  *)

(*      the file contains 'unsatisfiable', or 'UNKNOWN' if the file contains *)

(*      neither 'satisfiable' nor 'unsatisfiable'.  Empty lines are ignored. *)

(*      The assignment must be given in one or more lines immediately after  *)

(*      the line that contains 'satisfiable'.  These lines must begin with   *)

(*      'assignment_prefix'.  Variables must be separated by " ".  Non-      *)

(*      integer strings are ignored.  If variable i is contained in the      *)

(*      assignment, then i is interpreted as 'true'.  If ~i is contained in  *)

(*      the assignment, then i is interpreted as 'false'.  Otherwise the     *)

(*      value of i is taken to be unspecified.                               *)

(* ------------------------------------------------------------------------- *)

	(* Path.T -> string * string * string -> result *)

	fun read_std_result_file path (satisfiable, assignment_prefix, unsatisfiable) =

	let

		(* string -> int list *)

		fun int_list_from_string s =

			List.mapPartial Int.fromString (space_explode " " s)

		(* int list -> assignment *)

		fun assignment_from_list [] i =

			NONE  (* the SAT solver didn't provide a value for this variable *)

		  | assignment_from_list (x::xs) i =

			if x=i then (SOME true)

			else if x=(~i) then (SOME false)

			else assignment_from_list xs i

		(* int list -> string list -> assignment *)

		fun parse_assignment xs [] =

			assignment_from_list xs

		  | parse_assignment xs (line::lines) =

			if String.isPrefix assignment_prefix line then

				parse_assignment (xs @ int_list_from_string line) lines

			else

				assignment_from_list xs

		(* 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 -> result *)

		fun parse_lines [] =

			UNKNOWN

		  | parse_lines (line::lines) =

			if is_substring unsatisfiable line then

				UNSATISFIABLE NONE

			else if is_substring satisfiable line then

				SATISFIABLE (parse_assignment [] lines)

			else

				parse_lines lines

	in

		(parse_lines o (List.filter (fn l => l <> "")) o split_lines o File.read) path

	end;

(* ------------------------------------------------------------------------- *)

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

(* ------------------------------------------------------------------------- *)

	(* string -> (PropLogic.prop_formula -> unit) -> (unit -> result) -> solver *)

	fun make_external_solver cmd writefn readfn fm =

		(writefn fm; system cmd; readfn ());

(* ------------------------------------------------------------------------- *)

(* read_dimacs_cnf_file: returns a propositional formula that corresponds to *)

(*      a SAT problem given in DIMACS CNF format                             *)

(* ------------------------------------------------------------------------- *)

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

	fun read_dimacs_cnf_file path =

	let

		(* string list -> string list *)

		fun filter_preamble [] =

			error "problem line not found in DIMACS CNF file"

		  | filter_preamble (line::lines) =

			if String.isPrefix "c " line orelse line = "c" then

				(* ignore comments *)

				filter_preamble lines

			else if String.isPrefix "p " line then

				(* ignore the problem line (which must be the last line of the preamble) *)

				(* Ignoring the problem line implies that if the file contains more clauses *)

				(* or variables than specified in its preamble, we will accept it anyway.   *)

				lines

			else

				error "preamble in DIMACS CNF file contains a line that does not begin with \"c \" or \"p \""

		(* string -> int *)

		fun int_from_string s =

			case Int.fromString s of

			  SOME i => i

			| NONE   => error ("token " ^ quote s ^ "in DIMACS CNF file is not a number")

		(* int list -> int list list *)

		fun clauses xs =

			let

				val (xs1, xs2) = take_prefix (fn i => i <> 0) xs

			in

				case xs2 of

				  []      => [xs1]

				| (0::[]) => [xs1]

				| (0::tl) => xs1 :: clauses tl

				| _       => sys_error "this cannot happen"

			end

		(* int -> PropLogic.prop_formula *)

		fun literal_from_int i =

			(assert (i<>0) "variable index in DIMACS CNF file is 0";

			if i>0 then

				PropLogic.BoolVar i

			else

				PropLogic.Not (PropLogic.BoolVar (~i)))

		(* PropLogic.prop_formula list -> PropLogic.prop_formula *)

		fun disjunction [] =

			error "empty clause in DIMACS CNF file"

		  | disjunction (x::xs) =

			(case xs of

			  [] => x

			| _  => PropLogic.Or (x, disjunction xs))

		(* PropLogic.prop_formula list -> PropLogic.prop_formula *)

		fun conjunction [] =

			error "no clause in DIMACS CNF file"

		  | conjunction (x::xs) =

			(case xs of

			  [] => x

			| _  => PropLogic.And (x, conjunction xs))

	in

		(conjunction

		o (map disjunction)

		o (map (map literal_from_int))

		o clauses

		o (map int_from_string)

		o List.concat

		o (map (String.fields (fn c => c mem [#" ", #"\t", #"\n"])))

		o filter_preamble

		o (List.filter (fn l => l <> ""))

		o split_lines

		o File.read)

			path

	end;

(* ------------------------------------------------------------------------- *)

(* 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) =

      let

        val the_solvers = !solvers;

        val _ = if AList.defined (op =) the_solvers name

          then warning ("SAT solver " ^ quote name ^ " was defined before")

          else ();

      in solvers := AList.update (op =) (name, new_solver) the_solvers end;

(* ------------------------------------------------------------------------- *)

(* 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 AList.lookup (op =) (!solvers)) name;

end;  (* SatSolver *)

(* ------------------------------------------------------------------------- *)

(* Predefined SAT solvers                                                    *)

(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)

(* Internal SAT solver, available as 'SatSolver.invoke_solver "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 -> SatSolver.result *)

		fun solver_loop xs =

			if PropLogic.eval (assignment_from_list xs) fm then

				SatSolver.SATISFIABLE (SOME o (assignment_from_list xs))

			else

				(case next_list xs indices of

				  SOME xs' => solver_loop xs'

				| NONE     => SatSolver.UNSATISFIABLE NONE)

	in

		(* start with the "first" assignment (all variables are interpreted as 'false') *)

		solver_loop []

	end

in

	SatSolver.add_solver ("enumerate", enum_solver)

end;

(* ------------------------------------------------------------------------- *)

(* Internal SAT solver, available as 'SatSolver.invoke_solver "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 'PropLogic.nnf fm' *)

			(* but that sometimes leads to worse performance due to the         *)

			(* introduction of additional variables.                            *)

			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)

			(* 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 _                 = 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 = valOf (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 -> assignment *)

			fun assignment_from_list [] i =

				NONE  (* the DPLL procedure didn't provide a value for this variable *)

			  | assignment_from_list (x::xs) i =

				if x=i then (SOME true)

				else if x=(~i) then (SOME false)

				else assignment_from_list xs i

		in

			(* initially, no variable is interpreted yet *)

			case dpll [] fm' of

			  SOME assignment => SatSolver.SATISFIABLE (assignment_from_list assignment)

			| NONE            => SatSolver.UNSATISFIABLE NONE

		end

	end  (* local *)

in

	SatSolver.add_solver ("dpll", dpll_solver)

end;

(* ------------------------------------------------------------------------- *)

(* Internal SAT solver, available as 'SatSolver.invoke_solver "auto"': uses  *)

(* the last installed solver (other than "auto" itself) that does not raise  *)

(* 'NOT_CONFIGURED'.  (However, the solver may return 'UNKNOWN'.)            *)

(* ------------------------------------------------------------------------- *)

let

	fun auto_solver fm =

	let

		fun loop [] =

			SatSolver.UNKNOWN

		  | loop ((name, solver)::solvers) =

			if name="auto" then

				(* do not call solver "auto" from within "auto" *)

				loop solvers

			else (

				(if name="dpll" orelse name="enumerate" then

					warning ("Using SAT solver " ^ quote name ^ "; for better performance, consider using an external solver.")

				else

					tracing ("Using SAT solver " ^ quote name ^ "."));

				(* apply 'solver' to 'fm' *)

				solver fm

					handle SatSolver.NOT_CONFIGURED => loop solvers

			)

	in

		loop (!SatSolver.solvers)

	end

in

	SatSolver.add_solver ("auto", auto_solver)

end;

(* ------------------------------------------------------------------------- *)

(* MiniSat 1.14                                                              *)

(* (http://www.cs.chalmers.se/Cs/Research/FormalMethods/MiniSat/)            *)

(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)

(* "minisat_with_proofs" requires a modified version of MiniSat 1.14 by John *)

(* Matthews, which can output ASCII proof traces.  Replaying binary proof    *)

(* traces generated by MiniSat-p_v1.14 has _not_ been implemented.           *)

(* ------------------------------------------------------------------------- *)

(* add "minisat_with_proofs" _before_ "minisat" to the available solvers, so *)

(* that the latter is preferred by the "auto" solver                         *)

(* There is a complication that is dealt with in the code below: MiniSat     *)

(* introduces IDs for original clauses in the proof trace.  It does not (in  *)

(* general) follow the convention that the original clauses are numbered     *)

(* from 0 to n-1 (where n is the number of clauses in the formula).          *)

let

	exception INVALID_PROOF of string

	fun minisat_with_proofs fm =

	let

		val _          = if (getenv "MINISAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

		val inpath     = File.tmp_path (Path.explode "isabelle.cnf")

		val outpath    = File.tmp_path (Path.explode "result")

		val proofpath  = File.tmp_path (Path.explode "result.prf")

		val cmd        = (getenv "MINISAT_HOME") ^ "/minisat " ^ (Path.implode inpath) ^ " -r " ^ (Path.implode outpath) ^ " -t " ^ (Path.implode proofpath) ^ "> /dev/null"

		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath fm

		fun readfn ()  = SatSolver.read_std_result_file outpath ("SAT", "", "UNSAT")

		val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

		val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

		val cnf        = PropLogic.defcnf fm

		val result     = SatSolver.make_external_solver cmd writefn readfn cnf

		val _          = try File.rm inpath

		val _          = try File.rm outpath

	in  case result of

	  SatSolver.UNSATISFIABLE NONE =>

		(let

			(* string list *)

			val proof_lines = (split_lines o File.read) proofpath

				handle IO.Io _ => raise INVALID_PROOF "Could not read file \"result.prf\""

			(* representation of clauses as ordered lists of literals (with duplicates removed) *)

			(* prop_formula -> int list *)

			fun clause_to_lit_list (PropLogic.Or (fm1, fm2)) =

				OrdList.union int_ord (clause_to_lit_list fm1) (clause_to_lit_list fm2)

			  | clause_to_lit_list (PropLogic.BoolVar i) =

				[i]

			  | clause_to_lit_list (PropLogic.Not (PropLogic.BoolVar i)) =

				[~i]

			  | clause_to_lit_list _ =

				raise INVALID_PROOF "Error: invalid clause in CNF formula."

			(* prop_formula -> int *)

			fun cnf_number_of_clauses (PropLogic.And (fm1, fm2)) =

				cnf_number_of_clauses fm1 + cnf_number_of_clauses fm2

			  | cnf_number_of_clauses _ =

				1

			val number_of_clauses = cnf_number_of_clauses cnf

			(* int list array *)

			val clauses = Array.array (number_of_clauses, [])

			(* initialize the 'clauses' array *)

			(* prop_formula * int -> int *)

			fun init_array (PropLogic.And (fm1, fm2), n) =

				init_array (fm2, init_array (fm1, n))

			  | init_array (fm, n) =

				(Array.update (clauses, n, clause_to_lit_list fm); n+1)

			val _ = init_array (cnf, 0)

			(* optimization for the common case where MiniSat "R"s clauses in their *)

			(* original order:                                                      *)

			val last_ref_clause = ref (number_of_clauses - 1)

			(* search the 'clauses' array for the given list of literals 'lits', *)

			(* starting at index '!last_ref_clause + 1'                          *)

			(* int list -> int option *)

			fun original_clause_id lits =

			let

				fun original_clause_id_from index =

					if index = number_of_clauses then

						(* search from beginning again *)

						original_clause_id_from 0

					(* both 'lits' and the list of literals used in 'clauses' are sorted, so *)

					(* testing for equality should suffice -- barring duplicate literals     *)

					else if Array.sub (clauses, index) = lits then (

						(* success *)

						last_ref_clause := index;

						SOME index

					) else if index = !last_ref_clause then

						(* failure *)

						NONE

					else

						(* continue search *)

						original_clause_id_from (index + 1)

			in

				original_clause_id_from (!last_ref_clause + 1)

			end

			(* string -> int *)

			fun int_from_string s = (

				case Int.fromString s of

				  SOME i => i

				| NONE   => raise INVALID_PROOF ("File format error: number expected (" ^ quote s ^ " encountered).")

			)

			(* parse the proof file *)

			val clause_table  = ref (Inttab.empty : int list Inttab.table)

			val empty_id      = ref ~1

			(* contains a mapping from clause IDs as used by MiniSat to clause IDs in *)

			(* our proof format, where original clauses are numbered starting from 0  *)

			val clause_id_map = ref (Inttab.empty : int Inttab.table)

			fun sat_to_proof id = (

				case Inttab.lookup (!clause_id_map) id of

				  SOME id' => id'

				| NONE     => raise INVALID_PROOF ("Clause ID " ^ Int.toString id ^ " used, but not defined.")

			)

			val next_id = ref (number_of_clauses - 1)

			(* string list -> unit *)

			fun process_tokens [] =

				()

			  | process_tokens (tok::toks) =

				if tok="R" then (

					case toks of

					  id::sep::lits =>

						let

							val _        = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"R\" disallowed after \"X\"."

							val cid      = int_from_string id

							val _        = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

							val ls       = sort int_ord (map int_from_string lits)

							val proof_id = case original_clause_id ls of

							                 SOME orig_id => orig_id

							               | NONE         => raise INVALID_PROOF ("Original clause (new ID is " ^ id ^ ") not found.")

						in

							(* extend the mapping of clause IDs with this newly defined ID *)

							clause_id_map := Inttab.update_new (cid, proof_id) (!clause_id_map)

								handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once (in \"R\").")

							(* the proof itself doesn't change *)

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"R\" followed by an insufficient number of tokens."

				) else if tok="C" then (

					case toks of

					  id::sep::ids =>

						let

							val _        = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"C\" disallowed after \"X\"."

							val cid      = int_from_string id

							val _        = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

							(* ignore the pivot literals in MiniSat's trace *)

							fun unevens []             = raise INVALID_PROOF "File format error: \"C\" followed by an even number of IDs."

							  | unevens (x :: [])      = x :: []

							  | unevens (x :: _ :: xs) = x :: unevens xs

							val rs       = (map sat_to_proof o unevens o map int_from_string) ids

							(* extend the mapping of clause IDs with this newly defined ID *)

							val proof_id = inc next_id

							val _        = clause_id_map := Inttab.update_new (cid, proof_id) (!clause_id_map)

							                 handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once (in \"C\").")

						in

							(* update clause table *)

							clause_table := Inttab.update_new (proof_id, rs) (!clause_table)

								handle Inttab.DUP _ => raise INVALID_PROOF ("Error: internal ID for clause " ^ id ^ " already used.")

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"C\" followed by an insufficient number of tokens."

				) else if tok="D" then (

					case toks of

					  [id] =>

						let

							val _ = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: \"D\" disallowed after \"X\"."

							val _ = sat_to_proof (int_from_string id)

						in

							(* simply ignore "D" *)

							()

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"D\" followed by an illegal number of tokens."

				) else if tok="X" then (

					case toks of

					  [id1, id2] =>

						let

							val _            = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: more than one end-of-proof statement."

							val _            = sat_to_proof (int_from_string id1)

							val new_empty_id = sat_to_proof (int_from_string id2)

						in

							(* update conflict id *)

							empty_id := new_empty_id

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"X\" followed by an illegal number of tokens."

				) else

					raise INVALID_PROOF ("File format error: unknown token " ^ quote tok ^ " encountered.")

			(* string list -> unit *)

			fun process_lines [] =

				()

			  | process_lines (l::ls) = (

					process_tokens (String.tokens (fn c => c = #" " orelse c = #"\t") l);

					process_lines ls

				)

			(* proof *)

			val _ = process_lines proof_lines

			val _ = if !empty_id <> ~1 then () else raise INVALID_PROOF "File format error: no conflicting clause specified."

		in

			SatSolver.UNSATISFIABLE (SOME (!clause_table, !empty_id))

		end handle INVALID_PROOF reason => (warning reason; SatSolver.UNSATISFIABLE NONE))

	| result =>

		result

	end

in

	SatSolver.add_solver ("minisat_with_proofs", minisat_with_proofs)

end;

let

	fun minisat fm =

	let

		val _          = if (getenv "MINISAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

		val inpath     = File.tmp_path (Path.explode "isabelle.cnf")

		val outpath    = File.tmp_path (Path.explode "result")

		val cmd        = (getenv "MINISAT_HOME") ^ "/minisat " ^ (Path.implode inpath) ^ " -r " ^ (Path.implode outpath) ^ " > /dev/null"

		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

		fun readfn ()  = SatSolver.read_std_result_file outpath ("SAT", "", "UNSAT")

		val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

		val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

		val result     = SatSolver.make_external_solver cmd writefn readfn fm

		val _          = try File.rm inpath

		val _          = try File.rm outpath

	in

		result

	end

in

	SatSolver.add_solver ("minisat", minisat)

end;

(* ------------------------------------------------------------------------- *)

(* zChaff (http://www.princeton.edu/~chaff/zchaff.html)                      *)

(* ------------------------------------------------------------------------- *)

(* ------------------------------------------------------------------------- *)

(* 'zchaff_with_proofs' applies the "zchaff" prover to a formula, and if     *)

(* zChaff finds that the formula is unsatisfiable, a proof of this is read   *)

(* from a file "resolve_trace" that was generated by zChaff.  See the code   *)

(* below for the expected format of the "resolve_trace" file.  Aside from    *)

(* some basic syntactic checks, no verification of the proof is performed.   *)

(* ------------------------------------------------------------------------- *)

(* add "zchaff_with_proofs" _before_ "zchaff" to the available solvers, so   *)

(* that the latter is preferred by the "auto" solver                         *)

let

	exception INVALID_PROOF of string

	fun zchaff_with_proofs fm =

	case SatSolver.invoke_solver "zchaff" fm of

	  SatSolver.UNSATISFIABLE NONE =>

		(let

			(* string list *)

			val proof_lines = ((split_lines o File.read) (Path.explode "resolve_trace"))

				handle IO.Io _ => raise INVALID_PROOF "Could not read file \"resolve_trace\""

			(* PropLogic.prop_formula -> int *)

			fun cnf_number_of_clauses (PropLogic.And (fm1, fm2)) = cnf_number_of_clauses fm1 + cnf_number_of_clauses fm2

			  | cnf_number_of_clauses _                          = 1

			(* string -> int *)

			fun int_from_string s = (

				case Int.fromString s of

				  SOME i => i

				| NONE   => raise INVALID_PROOF ("File format error: number expected (" ^ quote s ^ " encountered).")

			)

			(* parse the "resolve_trace" file *)

			val clause_offset = ref ~1

			val clause_table  = ref (Inttab.empty : int list Inttab.table)

			val empty_id      = ref ~1

			(* string list -> unit *)

			fun process_tokens [] =

				()

			  | process_tokens (tok::toks) =

				if tok="CL:" then (

					case toks of

					  id::sep::ids =>

						let

							val _   = if !clause_offset = ~1 then () else raise INVALID_PROOF ("File format error: \"CL:\" disallowed after \"VAR:\".")

							val _   = if !empty_id = ~1 then () else raise INVALID_PROOF ("File format error: \"CL:\" disallowed after \"CONF:\".")

							val cid = int_from_string id

							val _   = if sep = "<=" then () else raise INVALID_PROOF ("File format error: \"<=\" expected (" ^ quote sep ^ " encountered).")

							val rs  = map int_from_string ids

						in

							(* update clause table *)

							clause_table := Inttab.update_new (cid, rs) (!clause_table)

								handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ id ^ " defined more than once.")

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"CL:\" followed by an insufficient number of tokens."

				) else if tok="VAR:" then (

					case toks of

					  id::levsep::levid::valsep::valid::antesep::anteid::litsep::lits =>

						let

							val _   = if !empty_id = ~1 then () else raise INVALID_PROOF ("File format error: \"VAR:\" disallowed after \"CONF:\".")

							(* set 'clause_offset' to the largest used clause ID *)

							val _   = if !clause_offset = ~1 then clause_offset :=

								(case Inttab.max_key (!clause_table) of

								  SOME id => id

								| NONE    => cnf_number_of_clauses (PropLogic.defcnf fm) - 1  (* the first clause ID is 0, not 1 *))

								else

									()

							val vid = int_from_string id

							val _   = if levsep = "L:" then () else raise INVALID_PROOF ("File format error: \"L:\" expected (" ^ quote levsep ^ " encountered).")

							val _   = int_from_string levid

							val _   = if valsep = "V:" then () else raise INVALID_PROOF ("File format error: \"V:\" expected (" ^ quote valsep ^ " encountered).")

							val _   = int_from_string valid

							val _   = if antesep = "A:" then () else raise INVALID_PROOF ("File format error: \"A:\" expected (" ^ quote antesep ^ " encountered).")

							val aid = int_from_string anteid

							val _   = if litsep = "Lits:" then () else raise INVALID_PROOF ("File format error: \"Lits:\" expected (" ^ quote litsep ^ " encountered).")

							val ls  = map int_from_string lits

							(* convert the data provided by zChaff to our resolution-style proof format *)

							(* each "VAR:" line defines a unit clause, the resolvents are implicitly    *)

							(* given by the literals in the antecedent clause                           *)

							(* we use the sum of '!clause_offset' and the variable ID as clause ID for the unit clause *)

							val cid = !clause_offset + vid

							(* the low bit of each literal gives its sign (positive/negative), therefore  *)

							(* we have to divide each literal by 2 to obtain the proper variable ID; then *)

							(* we add '!clause_offset' to obtain the ID of the corresponding unit clause  *)

							val vids = filter (not_equal vid) (map (fn l => l div 2) ls)

							val rs   = aid :: map (fn v => !clause_offset + v) vids

						in

							(* update clause table *)

							clause_table := Inttab.update_new (cid, rs) (!clause_table)

								handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ string_of_int cid ^ " (derived from antecedent for variable " ^ id ^ ") already defined.")

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"VAR:\" followed by an insufficient number of tokens."

				) else if tok="CONF:" then (

					case toks of

					  id::sep::ids =>

						let

							val _   = if !empty_id = ~1 then () else raise INVALID_PROOF "File format error: more than one conflicting clause specified."

							val cid = int_from_string id

							val _   = if sep = "==" then () else raise INVALID_PROOF ("File format error: \"==\" expected (" ^ quote sep ^ " encountered).")

							val ls  = map int_from_string ids

							(* the conflict clause must be resolved with the unit clauses *)

							(* for its literals to obtain the empty clause                *)

							val vids         = map (fn l => l div 2) ls

							val rs           = cid :: map (fn v => !clause_offset + v) vids

							val new_empty_id = getOpt (Inttab.max_key (!clause_table), !clause_offset) + 1

						in

							(* update clause table and conflict id *)

							clause_table := Inttab.update_new (new_empty_id, rs) (!clause_table)

								handle Inttab.DUP _ => raise INVALID_PROOF ("File format error: clause " ^ string_of_int new_empty_id ^ " (empty clause derived from clause " ^ id ^ ") already defined.");

							empty_id     := new_empty_id

						end

					| _ =>

						raise INVALID_PROOF "File format error: \"CONF:\" followed by an insufficient number of tokens."

				) else

					raise INVALID_PROOF ("File format error: unknown token " ^ quote tok ^ " encountered.")

			(* string list -> unit *)

			fun process_lines [] =

				()

			  | process_lines (l::ls) = (

					process_tokens (String.tokens (fn c => c = #" " orelse c = #"\t") l);

					process_lines ls

				)

			(* proof *)

			val _ = process_lines proof_lines

			val _ = if !empty_id <> ~1 then () else raise INVALID_PROOF "File format error: no conflicting clause specified."

		in

			SatSolver.UNSATISFIABLE (SOME (!clause_table, !empty_id))

		end handle INVALID_PROOF reason => (warning reason; SatSolver.UNSATISFIABLE NONE))

	| result =>

		result

in

	SatSolver.add_solver ("zchaff_with_proofs", zchaff_with_proofs)

end;

let

	fun zchaff fm =

	let

		val _          = if (getenv "ZCHAFF_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

		val _          = if (getenv "ZCHAFF_VERSION") <> "2004.5.13" andalso

		                    (getenv "ZCHAFF_VERSION") <> "2004.11.15" then raise SatSolver.NOT_CONFIGURED else ()

			(* both versions of zChaff appear to have the same interface, so we do *)

			(* not actually need to distinguish between them in the following code *)

		val inpath     = File.tmp_path (Path.explode "isabelle.cnf")

		val outpath    = File.tmp_path (Path.explode "result")

		val cmd        = (getenv "ZCHAFF_HOME") ^ "/zchaff " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

		fun readfn ()  = SatSolver.read_std_result_file outpath ("Instance Satisfiable", "", "Instance Unsatisfiable")

		val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

		val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

		val result     = SatSolver.make_external_solver cmd writefn readfn fm

		val _          = try File.rm inpath

		val _          = try File.rm outpath

	in

		result

	end

in

	SatSolver.add_solver ("zchaff", zchaff)

end;

(* ------------------------------------------------------------------------- *)

(* BerkMin 561 (http://eigold.tripod.com/BerkMin.html)                       *)

(* ------------------------------------------------------------------------- *)

let

	fun berkmin fm =

	let

		val _          = if (getenv "BERKMIN_HOME") = "" orelse (getenv "BERKMIN_EXE") = "" then raise SatSolver.NOT_CONFIGURED else ()

		val inpath     = File.tmp_path (Path.explode "isabelle.cnf")

		val outpath    = File.tmp_path (Path.explode "result")

		val cmd        = (getenv "BERKMIN_HOME") ^ "/" ^ (getenv "BERKMIN_EXE") ^ " " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

		fun readfn ()  = SatSolver.read_std_result_file outpath ("Satisfiable          !!", "solution =", "UNSATISFIABLE          !!")

		val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

		val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

		val result     = SatSolver.make_external_solver cmd writefn readfn fm

		val _          = try File.rm inpath

		val _          = try File.rm outpath

	in

		result

	end

in

	SatSolver.add_solver ("berkmin", berkmin)

end;

(* ------------------------------------------------------------------------- *)

(* Jerusat 1.3 (http://www.cs.tau.ac.il/~ale1/)                              *)

(* ------------------------------------------------------------------------- *)

let

	fun jerusat fm =

	let

		val _          = if (getenv "JERUSAT_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()

		val inpath     = File.tmp_path (Path.explode "isabelle.cnf")

		val outpath    = File.tmp_path (Path.explode "result")

		val cmd        = (getenv "JERUSAT_HOME") ^ "/Jerusat1.3 " ^ (Path.implode inpath) ^ " > " ^ (Path.implode outpath)

		fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)

		fun readfn ()  = SatSolver.read_std_result_file outpath ("s SATISFIABLE", "v ", "s UNSATISFIABLE")

		val _          = if File.exists inpath then warning ("overwriting existing file " ^ quote (Path.implode inpath)) else ()

		val _          = if File.exists outpath then warning ("overwriting existing file " ^ quote (Path.implode outpath)) else ()

		val result     = SatSolver.make_external_solver cmd writefn readfn fm

		val _          = try File.rm inpath

		val _          = try File.rm outpath

	in

		result

	end

in

	SatSolver.add_solver ("jerusat", jerusat)

end;

(* ------------------------------------------------------------------------- *)

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

(* ------------------------------------------------------------------------- *)

(*

let

	fun mysolver fm = ...

in