1 ;;; -*- Mode: Lisp; Syntax: ANSI-Common-Lisp; Base: 10; -*-
3 ;;; satwrap.lisp --- SAT solvers wrapped for CL
5 ;; Copyright (C) 2010 Utz-Uwe Haus <lisp@uuhaus.de>
7 ;; This code is free software; you can redistribute it and/or modify
8 ;; it under the terms of the version 3 of the GNU General
9 ;; Public License as published by the Free Software Foundation, as
10 ;; clarified by the prequel found in LICENSE.Lisp-GPL-Preface.
12 ;; This code is distributed in the hope that it will be useful, but
13 ;; without any warranty; without even the implied warranty of
14 ;; merchantability or fitness for a particular purpose. See the GNU
15 ;; Lesser General Public License for more details.
17 ;; Version 3 of the GNU General Public License is in the file
18 ;; LICENSE.GPL that was distributed with this file. If it is not
19 ;; present, you can access it from
20 ;; http://www.gnu.org/copyleft/gpl.txt (until superseded by a
21 ;; newer version) or write to the Free Software Foundation, Inc., 59
22 ;; Temple Place, Suite 330, Boston, MA 02111-1307 USA
30 (eval-when (:compile-toplevel
:load-toplevel
)
31 (declaim (optimize (speed 3) (debug 1) (safety 1))))
34 (in-package #:satwrap
)
36 (defvar *default-sat-backend
* :minisat
37 "Name of the default backend to be used by make-sat-solver.")
39 (defparameter *satwrap-backends
*
40 `((:precosat .
,(find-class 'satwrap.precosat
:precosat-backend
))
41 (:minisat .
,(find-class 'satwrap.minisat
:minisat-backend
)))
42 "Alist of symbolic name to backend class name for all supported backends.")
44 (defun describe-supported-backends ()
45 "Return a list of (designator . description) pairs of the supported backends"
46 (loop :for
(key . class
) :in
*satwrap-backends
*
47 :as descr
:= (let ((instance (make-instance class
)))
48 (format nil
"~A version ~A"
49 (satwrap.backend
:sat-backend-name instance
)
50 (satwrap.backend
:sat-backend-version instance
)))
51 :collect
`(,key .
,descr
)))
55 (defclass sat-solver
()
56 ((backend :initarg
:backend
:accessor sat-solver-backend
)
57 (numvars :initform
0 :accessor sat-solver-numvars
)
58 ;; CNF; split into 'old' and 'new' to allow incremental solving
59 (new-clauses :initform
'() :accessor sat-solver-new-clauses
)
60 (old-clauses :initform
'() :accessor sat-solver-old-clauses
)
62 (:default-initargs
:backend
(make-instance (cdr (assoc *default-sat-backend
*
63 *satwrap-backends
*))))
64 (:documentation
"A Sat solver abstraction"))
66 (defmethod sat-solver-numclauses ((solver sat-solver
))
67 (+ (length (sat-solver-new-clauses solver
))
68 (length (sat-solver-old-clauses solver
))
71 (defmethod print-object ((solver sat-solver
) stream
)
72 (print-unreadable-object (solver stream
:type T
:identity T
)
73 (format stream
"~D vars, ~D clauses, backend ~A"
74 (sat-solver-numvars solver
)
75 (sat-solver-numclauses solver
)
76 (sat-solver-backend solver
))))
78 (defmethod clause-valid ((solver sat-solver
) (clause list
))
79 "Check that CLAUSE is a valid clause for SOLVER."
81 :with max
:= (sat-solver-numvars solver
)
84 :always
(and (not (zerop v
))
87 (defmacro iota
(n &optional
(start 1))
88 "Return a list of the N sequential integers starting at START (default: 1)."
89 (let ((i (gensym "i")))
94 (define-condition satwrap-condition
(error)
96 (:documentation
"Superclass of all conditions raised by satwrap code."))
98 (define-condition invalid-clause
(satwrap-condition)
99 ((solver :initarg
:solver
:reader invalid-clause-solver
)
100 (clause :initarg
:clause
:reader invalid-clause-clause
))
101 (:report
(lambda (condition stream
)
102 (format stream
"Invalid clause ~A for solver ~A"
103 (invalid-clause-clause condition
)
104 (invalid-clause-solver condition
)))))
106 (define-condition invalid-backend
(satwrap-condition)
107 ((name :initarg
:name
:reader invalid-backend-name
))
108 (:report
(lambda (condition stream
)
109 (declare (special *satwrap-backends
* *default-sat-backend
*))
110 (format stream
"Invalid backend ~S specified. Supported: ~{~S~^, ~}, default ~S."
111 (invalid-backend-name condition
)
112 (mapcar #'car
*satwrap-backends
*)
113 *default-sat-backend
*))))
115 (defmethod flush-to-backend ((solver sat-solver
))
116 "Populate backend, possibly flushing old backend contents."
117 (satwrap.backend
:synchronize-backend
(sat-solver-backend solver
)
118 (sat-solver-numvars solver
)
119 (sat-solver-new-clauses solver
)
120 '() ;; deleted clauses, not yet supported
121 (sat-solver-old-clauses solver
))
122 (loop :with deleted
:= '()
123 :for c
:in
(sat-solver-old-clauses solver
)
124 :unless
(find c deleted
:test
#'equal
)
125 :do
(push c
(sat-solver-new-clauses solver
)))
126 (setf (sat-solver-old-clauses solver
) (sat-solver-new-clauses solver
)
127 (sat-solver-new-clauses solver
) '()))
130 (defun make-sat-solver (&optional
(backend *default-sat-backend
*))
131 "Return a new sat solver instance. Optional argument BACKEND can be used to
132 specify which backend should be used. It defaults to *default-sat-backend*."
133 (let ((be (assoc backend
*satwrap-backends
* :test
#'eq
)))
135 (make-instance 'sat-solver
:backend
(make-instance (cdr be
)))
136 (error 'invalid-backend
:name backend
))))
138 (defgeneric add-variable
(solver)
139 (:documentation
"Add another variable to SOLVER. Returns new variable index.")
140 (:method
((solver sat-solver
))
141 (incf (sat-solver-numvars solver
))))
143 (defgeneric add-clause
(solver clause
)
144 (:documentation
"Add CLAUSE to SOLVER's cnf formula. Clause is consumed.")
145 (:method
((solver sat-solver
) (clause list
))
146 (if (clause-valid solver clause
)
147 (push clause
(sat-solver-new-clauses solver
))
148 (error 'invalid-clause
:clause clause
:solver solver
)))
149 (:method
((solver sat-solver
) (clause vector
))
150 (add-clause solver
(coerce clause
'list
))))
152 (defgeneric add-clauses
(solver clauses
)
153 (:documentation
"Add CLAUSES to SOLVER's cnf formula. Clauses are consumed.")
154 (:method
((solver sat-solver
) (clauses list
))
156 (add-clause solver c
))))
158 (defgeneric satisfiablep
(solver &key assume
)
159 (:documentation
"Check whether current CNF in SOLVER is satisfiable.
160 Keyword argument :ASSUME can provide a sequence of literals assumed TRUE or FALSE.
162 (:method
((solver sat-solver
) &key
(assume '()))
163 (if (clause-valid solver assume
) ;; misusing 'clause' concept here
165 (flush-to-backend solver
)
166 (satwrap.backend
:satisfiablep
(sat-solver-backend solver
) assume
))
167 (error 'invalid-clause
:clause assume
:solver solver
))))
169 (defgeneric solution
(solver &key interesting-vars
)
170 (:documentation
"Return solution for SOLVER. If unsat, return '(). If sat return
171 sequence of 0/1 values for variables [1...N]. Keyword argument INTERESTING-VARS can be used to restrict the variables whose values are reported. Solution components will be in the same order that INTERESTING-VARS lists the variables in.")
172 (:method
((solver sat-solver
)
173 &key
(interesting-vars (iota (sat-solver-numvars solver
))))
174 (satwrap.backend
:solution
(sat-solver-backend solver
) interesting-vars
)))
176 (defgeneric get-essential-variables
(solver)
177 (:documentation
"Compute the variables that are essential, i.e. fixed in all solutions of SOLVER.
178 Returns a list of literals fixed, sign according to phase.")
179 (:method
((solver sat-solver
))
180 ;; The backend may define a specialized method, but then it has to
181 ;; work on the one copy of data flushed to the solver. Precosat for example
182 ;; does not allow that, so we define the universal implementation here instead of
183 ;; a default implementation in the backend.
184 (let ((have-specialized-method
185 (find-method #'satwrap.backend
:get-essential-variables
187 `(,(class-of (sat-solver-backend solver
)))
189 (if have-specialized-method
191 (flush-to-backend solver
)
192 (satwrap.backend
:get-essential-variables
(sat-solver-backend solver
)))
194 (loop :for var
:of-type
(integer 1) :in
(iota (sat-solver-numvars solver
))
195 :as sat-for-
+ := (satisfiablep solver
:assume
`(,var
,@fixed
))
196 :as sat-for--
:= (satisfiablep solver
:assume
`(,(- var
) ,@fixed
))
198 ((and sat-for-
+ (not sat-for--
))
200 ((and (not sat-for-
+) sat-for--
)
201 (push (- var
) fixed
))))
205 (defmacro with-sat-solver
((name numatoms
&key
(backend *default-sat-backend
* backend-given
))
207 "Bind NAME to a fresh sat-solver with backend :BACKEND (default: *default-sat-backend*) and declare NUMATOMS variables numbered 1..NUMATOMS for the duration of BODY."
208 `(let ((,name
(make-sat-solver ,(if backend-given
210 '*default-sat-backend
*))))
211 (loop :repeat
,numatoms
:do
(add-variable ,name
))
215 (defmacro with-index-hash
((mapping &key
(test 'eq
)) objects
&body body
)
216 "Execute BODY while binding MAPPING to a hash-table (with predicate
217 TEST, defaults to #'cl:eq) mapping the elements of the sequence OBJECTS
218 to integer 1..[length objects].
219 Typically used to map objects to variables inside WITH-SAT-SOLVER."
220 (let ((o (gensym "objects")))
221 `(let* ((,o
,objects
)
222 (,mapping
(etypecase ,o
223 (list (loop :with ht
:= (make-hash-table :test
#',test
)
225 :for num
:of-type fixnum
:from
1
226 :do
(setf (gethash x ht
) num
)
227 :finally
(return ht
)))
228 (vector (loop :with ht
:= (make-hash-table :test
#',test
)
230 :for num
:of-type fixnum
:from
1
231 :do
(setf (gethash x ht
) num
)
232 :finally
(return ht
))))))
235 ;; convenience syntax for logical operations:
236 (defun standardize (tree)
237 "Convert tree of logical operations to elementary :NOT, :AND, :OR tree (preserving :ATOM).
238 Supports :IMPLY, :IFF, binary :XOR, and :NOR. "
241 (destructuring-bind (op . expr
)
245 (if (= 2 (length expr
))
246 `(:OR
(:NOT
,(standardize (car expr
)))
247 ,(standardize (cadr expr
)))
248 (error "non-binary :IMPLY operation: ~A" tree
)))
250 (if (= 2 (length expr
))
251 `(:AND
,(standardize `(:IMPLIES
,(car expr
) ,(cadr expr
)))
252 ,(standardize `(:IMPLIES
,(cadr expr
) ,(car expr
))))
253 (error "non-binary :IFF operation: ~A" tree
)))
255 (if (= 2 (length expr
))
256 (let ((e1 (standardize (first expr
)))
257 (e2 (standardize (second expr
))))
258 `(:OR
(:AND
,e1
(:NOT
,e2
))
259 (:AND
,e2
(:NOT
,e1
))))
260 (error "non-binary :XOR operation: ~A" tree
)))
262 `(:AND
,@(mapcar #'(lambda (p) `(:NOT
,(standardize p
))) expr
)))
264 (T `(,op
,@(mapcar #'standardize expr
))))))
267 (defun standard-tree->nnf
(tree)
268 "Move NOTs inward. Input must be AND/OR/NOT-only; returns NNF tree."
271 (destructuring-bind (op . expr
)
274 (:NOT
(if (= 1 (length expr
))
275 (let ((subexpr (first expr
)))
277 (case (first subexpr
)
278 ;; double-not: drop 2 levels
279 (:NOT
(standard-tree->nnf
(cadr subexpr
)))
281 (:AND
`(:OR
,@(mapcar #'(lambda (c)
282 (standard-tree->nnf
`(:NOT
,c
)))
285 (:OR
`(:AND
,@(mapcar #'(lambda (c)
286 (standard-tree->nnf
`(:NOT
,c
)))
288 ;; `quoted' atom: keep
289 (:ATOM
`(:NOT
,subexpr
))
290 (otherwise `(:NOT
,subexpr
)))
293 (error "non-unary :NOT expression: ~W" tree
)))
294 ;; other operators: AND, OR, ATOM: descend
296 `(,op
,@(mapcar #'standard-tree-
>nnf expr
))))))
300 (defun crossprod (list-of-lists)
301 "Return a list of all lists containing one element from each sublist of LIST-OF-LISTS."
302 (reduce #'(lambda (collected term
)
303 (loop :for x
:in term
:appending
304 (loop :for tail
:in collected
305 :collect
`(,x
,@tail
))))
307 :initial-value
'(())))
310 "Check that TREE is in CNF."
319 (and (not (eq :OR
(car f
)))
320 (not (eq :AND
(car f
))))
328 (every #'atomic
(cdr c
)))))
331 (defun distribute (tree)
332 "Pull out :ANDs to reduce standardized tree in NNF to CNF."
335 (:AND
(let ((sub-cnfs (mapcar #'distribute
(cdr tree
))))
336 (assert (every #'is-cnf sub-cnfs
)
338 "Not all in CNF: ~{~A~^~%~}" sub-cnfs
)
340 (loop :for sub-cnf
:in sub-cnfs
341 :if
(and (consp sub-cnf
)
342 (eq :AND
(car sub-cnf
)))
343 ;; AND(AND ..) = AND ..
344 :do
(dolist (clause (cdr sub-cnf
))
345 (push clause clauses
))
346 :else
:do
(push sub-cnf clauses
))
349 (dolist (term (mapcar #'distribute
(cdr tree
)))
350 (assert (is-cnf term
)
352 "~A not in CNF" term
)
355 (:AND
(push (cdr term
) ands
))
356 (:OR
(dolist (c (cdr term
))
358 (otherwise (push `(,term
) ands
)))
359 (push `(,term
) ands
)))
360 (let ((clauses (crossprod ands
)))
361 `(:AND
,@(mapcar #'(lambda (c)
364 `(:OR
,@ (loop :for subterm
:in c
365 :if
(and (consp subterm
)
366 (eq :OR
(car subterm
)))
367 ;; (OR(OR...)) == (OR ...)
368 :append
(cdr subterm
)
369 ;; if C is not an OR-clause
370 ;; it is considered atomic:
371 :else
:append
`(,subterm
)))))
378 (defun tree->cnf
(tree &optional
(mapping #'cl
:identity
) (atom-test #'eql
))
379 "Convert TREE to CNF, applying MAPPING to atoms and afterwards and removing duplicate literals in clauses using atom-test to test for equality. Also simplifies trivially fulfilled clauses to empty clause."
382 (and (not (eq :OR
(car f
)))
383 (not (eq :AND
(car f
))))
386 ;; destructively apply MAPPING
388 (setf cnf
(funcall mapping cnf
))
389 (setf (cdr cnf
) ;; we must be looking at an AND
390 (mapcar #'(lambda (clause)
392 (funcall mapping clause
)
393 `(:OR
,@(mapcar #'(lambda (lit)
394 (funcall mapping lit
))
399 (and (consp lit
) (eq :NOT
(car lit
))))
403 (:NOT
(atom-for (second lit
)))
407 (duplicate-literalsp (l1 l2
)
408 (funcall atom-test
(atom-for l1
) (atom-for l2
)))
410 (loop :for
(lit . haystack
) :on c
411 :as needle
:= (atom-for lit
)
412 :as negated
:= (and (consp lit
) (eq :NOT
(car lit
)))
413 :when
(find-if #'(lambda (x)
415 (not (negated-litp x
))
417 (funcall atom-test needle
(atom-for x
))))
420 :finally
(return NIL
)))
422 ;; destructively apply MAPPING
425 (setf (cdr cnf
) ;; we must be looking at an AND
426 (loop :for clause
:in
(cdr cnf
)
429 :else
:if
(not (trivial-clausep clause
))
430 :collect
`(:OR
,@(remove-duplicates
432 :test
#'duplicate-literalsp
)))))
438 (standardize tree
)))))))
441 (defgeneric add-formula
(solver formula
&key mapping atom-test
)
442 (:documentation
"Add logical FORMULA to SOLVER.
443 Formula is an expression tree that can contain high-level abbreviations :IMPLY, :IFF, :NOR and :XOR, as well as the basic :AND, :OR, :NOT. Everything else is considered an atom. Atoms can also be explicitly written as (:ATOM foo).
444 All atoms will be mapped through MAPPING before being added to the solver to convert your favorite atoms to phased variables. MAPPING is a function of 1 argument, the atom. It defaults to #'identity. Afterwards, clean clauses (removing duplicates and simplifying trivial clauses, comparing atoms using ATOM-TEST (default: #'eql).")
445 (:method
((solver sat-solver
) (formula list
) &key
(mapping #'identity
) (atom-test #'eql
))
446 (dolist (c (cdr ;; drop outer :AND
447 (tree->cnf formula mapping atom-test
)))
448 (add-clause solver
(cdr ;; drop :OR
451 (defgeneric add-and
(solver literals
&key mapping
)
452 (:documentation
"Add clauses for AND(literals) to SOLVER.")
453 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
454 (add-formula solver
`(:AND
,@literals
) mapping
)))
456 (defgeneric add-or
(solver literals
&key mapping
)
457 (:documentation
"Add clauses for OR(literals) to SOLVER.")
458 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
459 (add-formula solver
`(:OR
,@literals
) mapping
)))
461 (defgeneric add-xor
(solver literals
&key mapping
)
462 (:documentation
"Add clauses for binary XOR(literals) to SOLVER.")
463 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
464 (add-formula solver
`(:XOR
,@literals
) mapping
)))
466 (defgeneric add-if
(solver lhs rhs
&key mapping
)
467 (:documentation
"Add clauses for lhs -> rhs to SOLVER.")
468 (:method
((solver sat-solver
) (lhs list
) (rhs list
) &key
(mapping #'identity
))
469 (add-formula solver
`(:IMPLIES
,lhs
,rhs
) mapping
)))
471 (defgeneric add-iff
(solver lhs rhs
&key mapping
)
472 (:documentation
"Add clauses for lhs <-> rhs to SOLVER.")
473 (:method
((solver sat-solver
) (lhs list
) (rhs list
) &key
(mapping #'identity
))
474 (add-formula solver
`(:IFF
,lhs
,rhs
) mapping
)))