satwrap.lisp

   1 ;;; -*- Mode: Lisp; Syntax: ANSI-Common-Lisp; Base: 10; -*-
   2 ;;;
   3 ;;; satwrap.lisp --- SAT solvers wrapped for CL
   4
   5 ;; Copyright (C) 2010 Utz-Uwe Haus <lisp@uuhaus.de>
   6 ;; $Id:$
   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.
  11 ;;
  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.
  16 ;;
  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  20 ;; http://www.gnu.org/copyleft/gpl.txt (until superseded by a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
  23 ;;
  24 ;; Commentary:
  25
  26 ;;
  27
  28 ;;; Code:
  29
  30 (eval-when (:compile-toplevel :load-toplevel)
  31   (declaim (optimize (speed 3) (debug 1) (safety 1))))
  32
  33
  34 (in-package #:satwrap)
  35 \f
  36 (defvar *default-sat-backend* :minisat
  37   "Name of the default backend to be used by make-sat-solver.")
  38
  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.")
  43
  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)))
  52
  53 \f
  54 ;;; Frontend:
  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)
  61    )
  62   (:default-initargs :backend (make-instance (cdr (assoc *default-sat-backend*
  63                                                          *satwrap-backends*))))
  64   (:documentation "A Sat solver abstraction"))
  65
  66 (defmethod sat-solver-numclauses ((solver sat-solver))
  67   (+ (length (sat-solver-new-clauses solver))
  68      (length (sat-solver-old-clauses solver))
  69      ))
  70
  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))))
  77
  78 (defmethod clause-valid ((solver sat-solver) (clause list))
  79   "Check that CLAUSE is a valid clause for SOLVER."
  80   (loop
  81      :with max := (sat-solver-numvars solver)
  82      :with min := (- max)
  83      :for v :in clause
  84      :always (and (not (zerop v))
  85                   (<= min v max))))
  86
  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")))
  90     `(loop :repeat ,n
  91         :for ,i :from ,start
  92         :collect ,i)))
  93
  94 (define-condition satwrap-condition (error)
  95   ()
  96   (:documentation "Superclass of all conditions raised by satwrap code."))
  97
  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)))))
 105
 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*))))
 114
 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) '()))
 128
 129 \f
 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)))
 134     (if be
 135         (make-instance 'sat-solver :backend (make-instance (cdr be)))
 136         (error 'invalid-backend :name backend))))
 137
 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))))
 142
 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))))
 151
 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))
 155     (dolist (c clauses)
 156       (add-clause solver c))))
 157
 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.
 161 Returns T or NIL.")
 162   (:method ((solver sat-solver) &key (assume '()))
 163     (if (clause-valid solver assume) ;; misusing 'clause' concept here
 164         (progn
 165           (flush-to-backend solver)
 166           (satwrap.backend:satisfiablep (sat-solver-backend solver) assume))
 167         (error 'invalid-clause :clause assume :solver solver))))
 168
 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)))
 175
 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
 186                         '()
 187                         `(,(class-of (sat-solver-backend solver)))
 188                         nil)))
 189       (if have-specialized-method
 190           (progn
 191             (flush-to-backend solver)
 192             (satwrap.backend:get-essential-variables (sat-solver-backend solver)))
 193           (let ((fixed '()))
 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))
 197                :do (cond
 198                      ((and sat-for-+ (not sat-for--))
 199                       (push var fixed))
 200                      ((and (not sat-for-+) sat-for--)
 201                       (push (- var) fixed))))
 202             fixed)))))
 203
 204 \f
 205 (defmacro with-sat-solver ((name numatoms &key (backend *default-sat-backend* backend-given))
 206                            &body body)
 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
 209                                       backend
 210                                       '*default-sat-backend*))))
 211      (loop :repeat ,numatoms :do (add-variable ,name))
 212      (locally
 213          ,@body)))
 214
 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)
 224                                 :for x :in ,o
 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)
 229                                   :for x :across ,o
 230                                   :for num :of-type fixnum :from 1
 231                                   :do (setf (gethash x ht) num)
 232                                   :finally (return ht))))))
 233        ,@body)))
 234 \f
 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. "
 239   (cond
 240     ((consp tree)
 241      (destructuring-bind (op . expr)
 242          tree
 243        (case op
 244          ((:IMPLY :IMPLIES)
 245           (if (= 2 (length expr))
 246               `(:OR (:NOT ,(standardize (car expr)))
 247                     ,(standardize (cadr expr)))
 248               (error "non-binary :IMPLY operation: ~A" tree)))
 249          (:IFF
 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)))
 254          (:XOR
 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)))
 261          (:NOR
 262           `(:AND ,@(mapcar #'(lambda (p) `(:NOT ,(standardize p))) expr)))
 263
 264          (T `(,op ,@(mapcar #'standardize expr))))))
 265     (T tree)))
 266
 267 (defun standard-tree->nnf (tree)
 268   "Move NOTs inward. Input must be AND/OR/NOT-only; returns NNF tree."
 269   (cond
 270     ((consp tree)
 271      (destructuring-bind (op . expr)
 272          tree
 273        (case op
 274          (:NOT (if (= 1 (length expr))
 275                    (let ((subexpr (first expr)))
 276                      (if (consp subexpr)
 277                          (case (first subexpr)
 278                            ;; double-not: drop 2 levels
 279                            (:NOT (standard-tree->nnf (cadr subexpr)))
 280                            ;; NOT AND:
 281                            (:AND `(:OR ,@(mapcar #'(lambda (c)
 282                                                      (standard-tree->nnf `(:NOT ,c)))
 283                                                  (cdr subexpr))))
 284                            ;; NOT OR:
 285                            (:OR  `(:AND ,@(mapcar #'(lambda (c)
 286                                                       (standard-tree->nnf `(:NOT ,c)))
 287                                                   (cdr subexpr))))
 288                            ;; `quoted' atom: keep
 289                            (:ATOM `(:NOT ,subexpr))
 290                            (otherwise `(:NOT ,subexpr)))
 291                          ;; non-quoted atom:
 292                          `(:NOT ,subexpr)))
 293                    (error "non-unary :NOT expression: ~W" tree)))
 294          ;; other operators: AND, OR, ATOM: descend
 295          (T
 296           `(,op ,@(mapcar #'standard-tree->nnf expr))))))
 297     ;; non-cons: stop
 298     (T tree)))
 299
 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))))
 306           list-of-lists
 307           :initial-value '(())))
 308
 309 (defun is-cnf (tree)
 310   "Check that TREE is in CNF."
 311   (flet ((and-form (f)
 312            (and (consp f)
 313                 (eq :AND (car f))))
 314          (or-form (f)
 315            (and (consp f)
 316                 (eq :OR (car f))))
 317          (atomic (f)
 318            (if (consp f)
 319                (and (not (eq :OR (car f)))
 320                     (not (eq :AND (car f))))
 321                T)))
 322     (if (atomic tree)
 323         T
 324         (and (and-form tree)
 325              (every #'(lambda (c)
 326                         (or (atomic c)
 327                             (and (or-form c)
 328                                  (every #'atomic (cdr c)))))
 329                     (cdr tree))))))
 330
 331 (defun distribute (tree)
 332   "Pull out :ANDs to reduce standardized tree in NNF to CNF."
 333   (if (consp tree)
 334       (case (car tree)
 335         (:AND (let ((sub-cnfs (mapcar #'distribute (cdr tree))))
 336                 (assert (every #'is-cnf sub-cnfs)
 337                         (sub-cnfs)
 338                         "Not all in CNF: ~{~A~^~%~}" sub-cnfs)
 339                 (let (clauses)
 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))
 347                   `(:AND ,@clauses))))
 348         (:OR (let (ands)
 349                (dolist (term (mapcar #'distribute (cdr tree)))
 350                  (assert (is-cnf term)
 351                          (term)
 352                          "~A not in CNF" term)
 353                  (if (consp term)
 354                      (case (car term)
 355                        (:AND (push (cdr term) ands))
 356                        (:OR (dolist (c (cdr term))
 357                               (push `(,c) ands)))
 358                        (otherwise (push `(,term) ands)))
 359                      (push `(,term) ands)))
 360                (let ((clauses (crossprod ands)))
 361                  `(:AND ,@(mapcar #'(lambda (c)
 362                                       (if (not (consp c))
 363                                           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)))))
 372                                   clauses)))))
 373         (otherwise
 374          tree))
 375       tree))
 376
 377
 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."
 380   (labels ((atomic (f)
 381              (if (consp f)
 382                  (and (not (eq :OR (car f)))
 383                       (not (eq :AND (car f))))
 384                  T))
 385            (map-literals (cnf)
 386              ;; destructively apply MAPPING
 387              (if (atomic cnf)
 388                  (setf cnf (funcall mapping cnf))
 389                  (setf (cdr cnf) ;; we must be looking at an AND
 390                        (mapcar #'(lambda (clause)
 391                                    (if (atomic clause)
 392                                        (funcall mapping clause)
 393                                        `(:OR ,@(mapcar #'(lambda (lit)
 394                                                            (funcall mapping lit))
 395                                                        (cdr clause)))))
 396                                (cdr cnf))))
 397              cnf)
 398            (negated-litp (lit)
 399              (and (consp lit) (eq :NOT (car lit))))
 400            (atom-for (lit)
 401              (if (consp lit)
 402                  (case (car lit)
 403                    (:NOT (atom-for (second lit)))
 404                    (:ATOM (second lit))
 405                    (otherwise lit))
 406                  lit))
 407            (duplicate-literalsp (l1 l2)
 408              (funcall atom-test (atom-for l1) (atom-for l2)))
 409            (trivial-clausep (c)
 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)
 414                                    (and (if negated
 415                                             (not (negated-litp x))
 416                                             (negated-litp x))
 417                                         (funcall atom-test needle (atom-for x))))
 418                                haystack)
 419                 :return T
 420                 :finally (return NIL)))
 421            (clean-clauses (cnf)
 422              ;; destructively apply MAPPING
 423              (if (atomic cnf)
 424                  (setf cnf cnf)
 425                  (setf (cdr cnf) ;; we must be looking at an AND
 426                        (loop :for clause :in (cdr cnf)
 427                           :if (atomic clause)
 428                           :collect clause
 429                           :else :if (not (trivial-clausep clause))
 430                           :collect `(:OR ,@(remove-duplicates
 431                                             (cdr clause)
 432                                             :test #'duplicate-literalsp)))))
 433              cnf))
 434     (clean-clauses
 435      (map-literals
 436       (distribute
 437        (standard-tree->nnf
 438         (standardize tree)))))))
 439
 440
 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
 449                           c)))))
 450
 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)))
 455
 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)))
 460
 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)))
 465
 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)))
 470
 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)))