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20 ;; http://www.gnu.org/copyleft/gpl.txt (until superseded by a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 (defmethod clause-valid ((solver sat-solver
) (clause vector
))
88 "Check that CLAUSE is a valid clause for SOLVER."
90 :with max
:= (sat-solver-numvars solver
)
93 :always
(and (not (zerop v
))
96 (defmacro iota
(n &optional
(start 1))
97 "Return a list of the N sequential integers starting at START (default: 1)."
98 (let ((i (gensym "i")))
103 (define-condition satwrap-condition
(error)
105 (:documentation
"Superclass of all conditions raised by satwrap code."))
107 (define-condition invalid-clause
(satwrap-condition)
108 ((solver :initarg
:solver
:reader invalid-clause-solver
)
109 (clause :initarg
:clause
:reader invalid-clause-clause
))
110 (:report
(lambda (condition stream
)
111 (format stream
"Invalid clause ~A for solver ~A"
112 (invalid-clause-clause condition
)
113 (invalid-clause-solver condition
)))))
115 (define-condition invalid-backend
(satwrap-condition)
116 ((name :initarg
:name
:reader invalid-backend-name
))
117 (:report
(lambda (condition stream
)
118 (declare (special *satwrap-backends
* *default-sat-backend
*))
119 (format stream
"Invalid backend ~S specified. Supported: ~{~S~^, ~}, default ~S."
120 (invalid-backend-name condition
)
121 (mapcar #'car
*satwrap-backends
*)
122 *default-sat-backend
*))))
124 (defmethod flush-to-backend ((solver sat-solver
))
125 "Populate backend, possibly flushing old backend contents."
127 (satwrap.backend
:synchronize-backend
(sat-solver-backend solver
)
128 (sat-solver-numvars solver
)
129 (sat-solver-new-clauses solver
)
131 (sat-solver-old-clauses solver
))
132 (setf (sat-solver-old-clauses solver
)
133 (nconc (sat-solver-new-clauses solver
)
135 (delete-if #'(lambda (c)
136 (find c deleted
:test
#'equal
))
137 (sat-solver-old-clauses solver
))
138 (sat-solver-old-clauses solver
))))
139 (setf (sat-solver-new-clauses solver
) '())))
142 (defun make-sat-solver (&optional
(backend *default-sat-backend
*))
143 "Return a new sat solver instance. Optional argument BACKEND can be used to
144 specify which backend should be used. It defaults to *default-sat-backend*."
145 (let ((be (assoc backend
*satwrap-backends
* :test
#'eq
)))
147 (make-instance 'sat-solver
:backend
(make-instance (cdr be
)))
148 (error 'invalid-backend
:name backend
))))
150 (defgeneric add-variable
(solver)
151 (:documentation
"Add another variable to SOLVER. Returns new variable index.")
152 (:method
((solver sat-solver
))
153 (incf (sat-solver-numvars solver
))))
155 (defgeneric add-clause
(solver clause
)
156 (:documentation
"Add CLAUSE to SOLVER's cnf formula. Clause is consumed.")
157 (:method
((solver sat-solver
) (clause list
))
158 (if (clause-valid solver clause
)
159 (push clause
(sat-solver-new-clauses solver
))
160 (error 'invalid-clause
:clause clause
:solver solver
)))
161 (:method
((solver sat-solver
) (clause vector
))
162 (add-clause solver
(coerce clause
'list
))))
164 (defgeneric add-clauses
(solver clauses
)
165 (:documentation
"Add CLAUSES to SOLVER's cnf formula. Clauses are consumed.")
166 (:method
((solver sat-solver
) (clauses list
))
168 (add-clause solver c
))))
170 (defgeneric satisfiablep
(solver &key assume
)
171 (:documentation
"Check whether current CNF in SOLVER is satisfiable.
172 Keyword argument :ASSUME can provide a sequence of literals assumed TRUE or FALSE.
174 (:method
((solver sat-solver
) &key
(assume '()))
175 (if (clause-valid solver assume
) ;; misusing 'clause' concept here
177 (flush-to-backend solver
)
178 (satwrap.backend
:satisfiablep
(sat-solver-backend solver
) assume
))
179 (error 'invalid-clause
:clause assume
:solver solver
))))
181 (defgeneric solution
(solver &key interesting-vars
)
182 (:documentation
"Return solution for SOLVER. If unsat, return '(). If sat return
183 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.")
184 (:method
((solver sat-solver
)
185 &key
(interesting-vars (iota (sat-solver-numvars solver
))))
186 (satwrap.backend
:solution
(sat-solver-backend solver
) interesting-vars
)))
188 (defgeneric get-essential-variables
(solver)
189 (:documentation
"Compute the variables that are essential, i.e. fixed in all solutions of SOLVER.
190 Returns a list of literals fixed, sign according to phase.")
191 (:method
((solver sat-solver
))
192 ;; The backend may define a specialized method, but then it has to
193 ;; work on the one copy of data flushed to the solver. Precosat for example
194 ;; does not allow that, so we define the universal implementation here instead of
195 ;; a default implementation in the backend.
196 (let ((have-specialized-method
197 (find-method #'satwrap.backend
:get-essential-variables
199 `(,(class-of (sat-solver-backend solver
)))
201 (if have-specialized-method
203 (flush-to-backend solver
)
204 (satwrap.backend
:get-essential-variables
(sat-solver-backend solver
)))
206 (loop :for var
:of-type
(integer 1) :in
(iota (sat-solver-numvars solver
))
207 :as sat-for-
+ := (satisfiablep solver
:assume
`(,var
,@fixed
))
208 :as sat-for--
:= (satisfiablep solver
:assume
`(,(- var
) ,@fixed
))
210 ((and sat-for-
+ (not sat-for--
))
212 ((and (not sat-for-
+) sat-for--
)
213 (push (- var
) fixed
))))
217 (defmacro with-sat-solver
((name numatoms
&key
(backend *default-sat-backend
* backend-given
))
219 "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."
220 `(let ((,name
(make-sat-solver ,(if backend-given
222 '*default-sat-backend
*))))
223 (loop :repeat
,numatoms
:do
(add-variable ,name
))
227 (defmacro with-index-hash
((mapping &key
(test #'eq
)) objects
&body body
)
228 "Execute BODY while binding MAPPING to a hash-table (with predicate
229 TEST, defaults to #'cl:eq) mapping the elements of the sequence OBJECTS
230 to integer 1..[length objects].
231 Typically used to map objects to variables inside WITH-SAT-SOLVER."
232 (let ((o (gensym "objects")))
233 `(let* ((,o
,objects
)
234 (,mapping
(etypecase ,o
235 (list (loop :with ht
:= (make-hash-table :test
,test
)
237 :for num
:of-type fixnum
:from
1
238 :do
(setf (gethash x ht
) num
)
239 :finally
(return ht
)))
240 (vector (loop :with ht
:= (make-hash-table :test
,test
)
242 :for num
:of-type fixnum
:from
1
243 :do
(setf (gethash x ht
) num
)
244 :finally
(return ht
))))))
247 ;; convenience syntax for logical operations:
248 (defun standardize (tree)
249 "Convert tree of logical operations to elementary :NOT, :AND, :OR tree (preserving :ATOM).
250 Supports :IMPLY, :IFF, binary :XOR, and :NOR. "
253 (destructuring-bind (op . expr
)
257 (if (= 2 (length expr
))
258 `(:OR
(:NOT
,(standardize (car expr
)))
259 ,(standardize (cadr expr
)))
260 (error "non-binary :IMPLY operation: ~A" tree
)))
262 (if (= 2 (length expr
))
263 `(:AND
,(standardize `(:IMPLIES
,(car expr
) ,(cadr expr
)))
264 ,(standardize `(:IMPLIES
,(cadr expr
) ,(car expr
))))
265 (error "non-binary :IFF operation: ~A" tree
)))
267 (if (= 2 (length expr
))
268 (let ((e1 (standardize (first expr
)))
269 (e2 (standardize (second expr
))))
270 `(:OR
(:AND
,e1
(:NOT
,e2
))
271 (:AND
,e2
(:NOT
,e1
))))
272 (error "non-binary :XOR operation: ~A" tree
)))
274 `(:AND
,@(mapcar #'(lambda (p) `(:NOT
,(standardize p
))) expr
)))
276 (T `(,op
,@(mapcar #'standardize expr
))))))
279 (defun standard-tree->nnf
(tree)
280 "Move NOTs inward. Input must be AND/OR/NOT-only; returns NNF tree."
283 (destructuring-bind (op . expr
)
286 (:NOT
(if (= 1 (length expr
))
287 (let ((subexpr (first expr
)))
289 (case (first subexpr
)
290 ;; double-not: drop 2 levels
291 (:NOT
(standard-tree->nnf
(cadr subexpr
)))
293 (:AND
`(:OR
,@(mapcar #'(lambda (c)
294 (standard-tree->nnf
`(:NOT
,c
)))
297 (:OR
`(:AND
,@(mapcar #'(lambda (c)
298 (standard-tree->nnf
`(:NOT
,c
)))
300 ;; `quoted' atom: keep
301 (:ATOM
`(:NOT
,subexpr
))
302 (otherwise `(:NOT
,subexpr
)))
305 (error "non-unary :NOT expression: ~W" tree
)))
306 ;; other operators: AND, OR, ATOM: descend
308 `(,op
,@(mapcar #'standard-tree-
>nnf expr
))))))
312 (defun crossprod (list-of-lists)
313 "Return a list of all lists containing one element from each sublist of LIST-OF-LISTS."
314 (reduce #'(lambda (collected term
)
315 (loop :for x
:in term
:appending
316 (loop :for tail
:in collected
317 :collect
`(,x
,@tail
))))
319 :initial-value
'(())))
322 "Check that TREE is in CNF."
331 (and (not (eq :OR
(car f
)))
332 (not (eq :AND
(car f
))))
340 (every #'atomic
(cdr c
)))))
343 (defun distribute (tree)
344 "Pull out :ANDs to reduce standardized tree in NNF to CNF."
347 (:AND
(let ((sub-cnfs (mapcar #'distribute
(cdr tree
))))
348 (assert (every #'is-cnf sub-cnfs
)
350 "Not all in CNF: ~{~A~^~%~}" sub-cnfs
)
352 (loop :for sub-cnf
:in sub-cnfs
353 :if
(and (consp sub-cnf
)
354 (eq :AND
(car sub-cnf
)))
355 ;; AND(AND ..) = AND ..
356 :do
(dolist (clause (cdr sub-cnf
))
357 (push clause clauses
))
358 :else
:do
(push sub-cnf clauses
))
361 (dolist (term (mapcar #'distribute
(cdr tree
)))
362 (assert (is-cnf term
)
364 "~A not in CNF" term
)
367 (:AND
(push (cdr term
) ands
))
368 (:OR
(dolist (c (cdr term
))
370 (otherwise (push `(,term
) ands
)))
371 (push `(,term
) ands
)))
372 (let ((clauses (crossprod ands
)))
373 `(:AND
,@(mapcar #'(lambda (c)
376 `(:OR
,@ (loop :for subterm
:in c
377 :if
(and (consp subterm
)
378 (eq :OR
(car subterm
)))
379 ;; (OR(OR...)) == (OR ...)
380 :append
(cdr subterm
)
381 ;; if C is not an OR-clause
382 ;; it is considered atomic:
383 :else
:append
`(,subterm
)))))
390 (defun tree->cnf
(tree &optional
(mapping #'cl
:identity
) (atom-test #'eql
))
391 "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."
394 (and (not (eq :OR
(car f
)))
395 (not (eq :AND
(car f
))))
398 ;; destructively apply MAPPING
400 (setf cnf
(funcall mapping cnf
))
401 (setf (cdr cnf
) ;; we must be looking at an AND
402 (mapcar #'(lambda (clause)
404 (funcall mapping clause
)
405 `(:OR
,@(mapcar #'(lambda (lit)
406 (funcall mapping lit
))
411 (and (consp lit
) (eq :NOT
(car lit
))))
415 (:NOT
(atom-for (second lit
)))
419 (duplicate-literalsp (l1 l2
)
420 (funcall atom-test
(atom-for l1
) (atom-for l2
)))
422 (loop :for
(lit . haystack
) :on c
423 :as needle
:= (atom-for lit
)
424 :as negated
:= (and (consp lit
) (eq :NOT
(car lit
)))
425 :when
(find-if #'(lambda (x)
427 (not (negated-litp x
))
429 (funcall atom-test needle
(atom-for x
))))
432 :finally
(return NIL
)))
434 ;; destructively apply MAPPING
437 (setf (cdr cnf
) ;; we must be looking at an AND
438 (loop :for clause
:in
(cdr cnf
)
441 :else
:if
(not (trivial-clausep clause
))
442 :collect
`(:OR
,@(remove-duplicates
444 :test
#'duplicate-literalsp
)))))
450 (standardize tree
)))))))
453 (defgeneric add-formula
(solver formula
&key mapping atom-test
)
454 (:documentation
"Add logical FORMULA to SOLVER.
455 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).
456 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).")
457 (:method
((solver sat-solver
) (formula list
) &key
(mapping #'identity
) (atom-test #'eql
))
458 (dolist (c (cdr ;; drop outer :AND
459 (tree->cnf formula mapping atom-test
)))
460 (add-clause solver
(cdr ;; drop :OR
463 (defgeneric add-and
(solver literals
&key mapping
)
464 (:documentation
"Add clauses for AND(literals) to SOLVER.")
465 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
466 (add-formula solver
`(:AND
,@literals
) mapping
)))
468 (defgeneric add-or
(solver literals
&key mapping
)
469 (:documentation
"Add clauses for OR(literals) to SOLVER.")
470 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
471 (add-formula solver
`(:OR
,@literals
) mapping
)))
473 (defgeneric add-xor
(solver literals
&key mapping
)
474 (:documentation
"Add clauses for binary XOR(literals) to SOLVER.")
475 (:method
((solver sat-solver
) (literals list
) &key
(mapping #'identity
))
476 (add-formula solver
`(:XOR
,@literals
) mapping
)))
478 (defgeneric add-if
(solver lhs rhs
&key mapping
)
479 (:documentation
"Add clauses for lhs -> rhs to SOLVER.")
480 (:method
((solver sat-solver
) (lhs list
) (rhs list
) &key
(mapping #'identity
))
481 (add-formula solver
`(:IMPLIES
,lhs
,rhs
) mapping
)))
483 (defgeneric add-iff
(solver lhs rhs
&key mapping
)
484 (:documentation
"Add clauses for lhs <-> rhs to SOLVER.")
485 (:method
((solver sat-solver
) (lhs list
) (rhs list
) &key
(mapping #'identity
))
486 (add-formula solver
`(:IFF
,lhs
,rhs
) mapping
)))