(* 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
exception NOT_CONFIGURED
type assignment = int -> bool option
datatype result = SATISFIABLE of assignment
| UNSATISFIABLE
| 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 parse_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 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 assigment regardless of the value of variable 'i' *)
(* ------------------------------------------------------------------------- *)
type assignment = int -> bool option;
(* ------------------------------------------------------------------------- *)
(* return type of SAT solvers: if the result is 'SATISFIABLE', a satisfying *)
(* assignment must be returned as well *)
(* ------------------------------------------------------------------------- *)
datatype result = SATISFIABLE of assignment
| UNSATISFIABLE
| 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 *)
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
concat (cnf_string_acc fm [])
end
in
File.write path
(let
val fm' = cnf_True_False_elim fm
val number_of_vars = maxidx fm'
val number_of_clauses = cnf_number_of_clauses fm'
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 fm' ^ " 0\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 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 fm) acc =
"-(" :: sat_string_acc fm (")" :: acc)
| sat_string_acc (Or (fm1,fm2)) acc =
"+(" :: sat_string_acc fm1 (" " :: sat_string_acc fm2 (")" :: acc))
| sat_string_acc (And (fm1,fm2)) acc =
"*(" :: sat_string_acc fm1 (" " :: sat_string_acc fm2 (")" :: acc))
in
concat (sat_string_acc fm [])
end
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 '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 parse_std_result_file path (satisfiable, assignment_prefix, unsatisfiable) =
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 -> 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 satisfiable line then
SATISFIABLE (parse_assignment [] lines)
else if is_substring unsatisfiable line then
UNSATISFIABLE
else
parse_lines lines
in
(parse_lines o (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 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
xs1 :: clauses (tl xs2)
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 flat
o (map (String.fields (fn c => c mem [#" ", #"\t", #"\n"])))
o filter_preamble
o (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) =
(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.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)
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 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)
(* 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 -> 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
end
end (* local *)
in
SatSolver.add_solver ("dpll", dpll_solver)
end;
(* ------------------------------------------------------------------------- *)
(* Internal SAT solver, available as 'SatSolver.invoke_solver "auto"': uses *)
(* all installed solvers (other than "auto" itself) until one returns either *)
(* SATISFIABLE or UNSATISFIABLE *)
(* ------------------------------------------------------------------------- *)
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
());
(* apply 'solver' to 'fm' *)
(case solver fm of
SatSolver.SATISFIABLE a => SatSolver.SATISFIABLE a
| SatSolver.UNSATISFIABLE => SatSolver.UNSATISFIABLE
| SatSolver.UNKNOWN => loop solvers)
handle SatSolver.NOT_CONFIGURED => loop solvers
)
in
loop (rev (!SatSolver.solvers))
end
in
SatSolver.add_solver ("auto", auto_solver)
end;
(* ------------------------------------------------------------------------- *)
(* ZChaff, Version 2004.05.13 (http://www.princeton.edu/~chaff/zchaff.html) *)
(* ------------------------------------------------------------------------- *)
let
fun zchaff fm =
let
val _ = if (getenv "ZCHAFF_HOME") = "" then raise SatSolver.NOT_CONFIGURED else ()
val inpath = File.tmp_path (Path.unpack "isabelle.cnf")
val outpath = File.tmp_path (Path.unpack "result")
val cmd = (getenv "ZCHAFF_HOME") ^ "/zchaff " ^ (Path.pack inpath) ^ " > " ^ (Path.pack outpath)
fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)
fun readfn () = SatSolver.parse_std_result_file outpath ("Instance Satisfiable", "", "Instance Unsatisfiable")
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 result = SatSolver.make_external_solver cmd writefn readfn fm
val _ = (File.rm inpath handle _ => ())
val _ = (File.rm outpath handle _ => ())
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.unpack "isabelle.cnf")
val outpath = File.tmp_path (Path.unpack "result")
val cmd = (getenv "BERKMIN_HOME") ^ "/" ^ (getenv "BERKMIN_EXE") ^ " " ^ (Path.pack inpath) ^ " > " ^ (Path.pack outpath)
fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)
fun readfn () = SatSolver.parse_std_result_file outpath ("Satisfiable !!", "solution =", "UNSATISFIABLE !!")
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 result = SatSolver.make_external_solver cmd writefn readfn fm
val _ = (File.rm inpath handle _ => ())
val _ = (File.rm outpath handle _ => ())
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.unpack "isabelle.cnf")
val outpath = File.tmp_path (Path.unpack "result")
val cmd = (getenv "JERUSAT_HOME") ^ "/Jerusat1.3 " ^ (Path.pack inpath) ^ " > " ^ (Path.pack outpath)
fun writefn fm = SatSolver.write_dimacs_cnf_file inpath (PropLogic.defcnf fm)
fun readfn () = SatSolver.parse_std_result_file outpath ("s SATISFIABLE", "v ", "s UNSATISFIABLE")
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 result = SatSolver.make_external_solver cmd writefn readfn fm
val _ = (File.rm inpath handle _ => ())
val _ = (File.rm outpath handle _ => ())
in
result
end
in
SatSolver.add_solver ("jerusat", jerusat)
end;
(* ------------------------------------------------------------------------- *)
(* Add code for other SAT solvers below. *)
(* ------------------------------------------------------------------------- *)
(*
let
fun mysolver fm = ...
in
SatSolver.add_solver ("myname", (fn fm => if mysolver_is_configured then mysolver fm else raise SatSolver.NOT_CONFIGURED)); -- register the solver
end;
-- the solver is now available as SatSolver.invoke_solver "myname"
*)