zeroinstall/injector/sat.py

   1 """
   2 Internal implementation of a SAT solver, used by L{solver.SATSolver}.
   3 This is not part of the public API.
   4 """
   5
   6 # Copyright (C) 2010, Thomas Leonard
   7 # See the README file for details, or visit http://0install.net.
   8
   9 # The design of this solver is very heavily based on the one described in
  10 # the MiniSat paper "An Extensible SAT-solver [extended version 1.2]"
  11 # http://minisat.se/Papers.html
  12 #
  13 # The main differences are:
  14 #
  15 # - We care about which solution we find (not just "satisfiable" or "not").
  16 # - We take care to be deterministic (always select the same versions given
  17 #   the same input). We do not do random restarts, etc.
  18 # - We add an AtMostOneClause (the paper suggests this in the Excercises, and
  19 #   it's very useful for our purposes).
  20
  21 import tempfile, subprocess, os, sys
  22 from logging import warn
  23
  24 def debug(msg, *args):
  25         return
  26         print "SAT:", msg % args
  27
  28 # variables are numbered from 0
  29 # literals have the same number as the corresponding variable,
  30 # except they for negatives they are (-1-v):
  31 #
  32 # Variable     Literal     not(Literal)
  33 # 0            0           -1
  34 # 1            1           -2
  35 def neg(lit):
  36         return -1 - lit
  37
  38 def watch_index(lit):
  39         if lit >= 0:
  40                 return lit * 2
  41         return neg(lit) * 2 + 1
  42
  43 def makeAtMostOneClause(solver):
  44         class AtMostOneClause:
  45                 def __init__(self, lits):
  46                         """Preferred literals come first."""
  47                         self.lits = lits
  48
  49                         # The single literal from our set that is True.
  50                         # We store this explicitly because the decider needs to know quickly.
  51                         self.current = None
  52
  53                 def propagate(self, lit):
  54                         # Re-add ourselves to the watch list.
  55                         # (we we won't get any more notifications unless we backtrack,
  56                         # in which case we'd need to get back on the list anyway)
  57                         solver.watch_lit(lit, self)
  58
  59                         # value[lit] has just become True
  60                         assert solver.lit_value(lit) == True
  61                         assert lit >= 0
  62
  63                         #debug("%s: noticed %s has become True" % (self, solver.name_lit(lit)))
  64
  65                         # If we already propagated successfully when the first
  66                         # one was set then we set all the others to False and
  67                         # anyone trying to set one True will get rejected. And
  68                         # if we didn't propagate yet, current will still be
  69                         # None, even if we now have a conflict (which we'll
  70                         # detect below).
  71                         assert self.current is None
  72
  73                         self.current = lit
  74
  75                         # If we later backtrace, call our undo function to unset current
  76                         solver.get_varinfo_for_lit(lit).undo.append(self)
  77
  78                         count = 0
  79                         for l in self.lits:
  80                                 value = solver.lit_value(l)
  81                                 #debug("Value of %s is %s" % (solver.name_lit(l), value))
  82                                 if value is True and l is not lit:
  83                                         # Due to queuing, we might get called with current = None
  84                                         # and two versions already selected.
  85                                         debug("CONFLICT: already selected %s" % solver.name_lit(l))
  86                                         return False
  87                                 if value is None:
  88                                         # Since one of our lits is already true, all unknown ones
  89                                         # can be set to False.
  90                                         if not solver.enqueue(neg(l), self):
  91                                                 debug("CONFLICT: enqueue failed for %s", solver.name_lit(neg(l)))
  92                                                 return False    # Conflict; abort
  93
  94                         return True
  95
  96                 def undo(self, lit):
  97                         debug("(backtracking: no longer selected %s)" % solver.name_lit(lit))
  98                         assert lit == self.current
  99                         self.current = None
 100
 101                 # Why is lit True?
 102                 # Or, why are we causing a conflict (if lit is None)?
 103                 def cacl_reason(self, lit):
 104                         if lit is None:
 105                                 # Find two True literals
 106                                 trues = []
 107                                 for l in self.lits:
 108                                         if solver.lit_value(l) is True:
 109                                                 trues.append(l)
 110                                                 if len(trues) == 2: return trues
 111                         else:
 112                                 for l in self.lits:
 113                                         if l is not lit and solver.lit_value(l) is True:
 114                                                 return [l]
 115                                 # Find one True literal
 116                         assert 0        # don't know why!
 117
 118                 def best_undecided(self):
 119                         debug("best_undecided: %s" % (solver.name_lits(self.lits)))
 120                         for lit in self.lits:
 121                                 #debug("%s = %s" % (solver.name_lit(lit), solver.lit_value(lit)))
 122                                 if solver.lit_value(lit) is None:
 123                                         return lit
 124                         return None
 125
 126                 def __repr__(self):
 127                         return "<lone: %s>" % (', '.join(solver.name_lits(self.lits)))
 128
 129         return AtMostOneClause
 130
 131 def makeUnionClause(solver):
 132         class UnionClause:
 133                 def __init__(self, lits):
 134                         self.lits = lits
 135
 136                 # Try to infer new facts.
 137                 # We can do this only when all of our literals are False except one,
 138                 # which is undecided. That is,
 139                 #   False... or X or False... = True  =>  X = True
 140                 #
 141                 # To get notified when this happens, we tell the solver to
 142                 # watch two of our undecided literals. Watching two undecided
 143                 # literals is sufficient. When one changes we check the state
 144                 # again. If we still have two or more undecided then we switch
 145                 # to watching them, otherwise we propagate.
 146                 #
 147                 # Returns False on conflict.
 148                 def propagate(self, lit):
 149                         # value[get(lit)] has just become False
 150
 151                         #debug("%s: noticed %s has become False" % (self, solver.name_lit(neg(lit))))
 152
 153                         # For simplicity, only handle the case where self.lits[1]
 154                         # is the one that just got set to False, so that:
 155                         # - value[lits[0]] = None | True
 156                         # - value[lits[1]] = False
 157                         # If it's the other way around, just swap them before we start.
 158                         if self.lits[0] == neg(lit):
 159                                 self.lits[0], self.lits[1] = self.lits[1], self.lits[0]
 160
 161                         if solver.lit_value(self.lits[0]) == True:
 162                                 # We're already satisfied. Do nothing.
 163                                 solver.watch_lit(lit, self)
 164                                 return True
 165
 166                         assert solver.lit_value(self.lits[1]) == False
 167
 168                         # Find a new literal to watch now that lits[1] is resolved,
 169                         # swap it with lits[1], and start watching it.
 170                         for i in range(2, len(self.lits)):
 171                                 value = solver.lit_value(self.lits[i])
 172                                 if value != False:
 173                                         # Could be None or True. If it's True then we've already done our job,
 174                                         # so this means we don't get notified unless we backtrack, which is fine.
 175                                         self.lits[1], self.lits[i] = self.lits[i], self.lits[1]
 176                                         solver.watch_lit(neg(self.lits[1]), self)
 177                                         return True
 178
 179                         # Only lits[0], is now undefined.
 180                         solver.watch_lit(lit, self)
 181                         return solver.enqueue(self.lits[0], self)
 182
 183                 def undo(self, lit): pass
 184
 185                 # Why is lit True?
 186                 # Or, why are we causing a conflict (if lit is None)?
 187                 def cacl_reason(self, lit):
 188                         assert lit is None or lit is self.lits[0]
 189
 190                         # The cause is everything except lit.
 191                         return [neg(l) for l in self.lits if l is not lit]
 192
 193                 def __repr__(self):
 194                         return "<some: %s>" % (', '.join(solver.name_lits(self.lits)))
 195         return UnionClause
 196
 197 # Using an array of VarInfo objects is less efficient than using multiple arrays, but
 198 # easier for me to understand.
 199 class VarInfo(object):
 200         __slots__ = ['value', 'reason', 'level', 'undo', 'obj']
 201         def __init__(self, obj):
 202                 self.value = None               # True/False/None
 203                 self.reason = None              # The constraint that implied our value, if True or False
 204                 self.level = -1                 # The decision level at which we got a value (when not None)
 205                 self.undo = []                  # Constraints to update if we become unbound (by backtracking)
 206                 self.obj = obj                  # The object this corresponds to (for our caller and for debugging)
 207
 208         def __repr__(self):
 209                 return '%s=%s' % (self.name, self.value)
 210
 211         @property
 212         def name(self):
 213                 return str(self.obj)
 214
 215 class SATProblem(object):
 216         def __init__(self):
 217                 # Propagation
 218                 self.watches = []               # watches[2i,2i+1] = constraints to check when literal[i] becomes True/False
 219                 self.propQ = []                 # propagation queue
 220
 221                 # Assignments
 222                 self.assigns = []               # [VarInfo]
 223                 self.trail = []                 # order of assignments
 224                 self.trail_lim = []             # decision levels
 225
 226                 self.toplevel_conflict = False
 227
 228                 self.makeAtMostOneClause = makeAtMostOneClause(self)
 229                 self.makeUnionClause = makeUnionClause(self)
 230
 231         def get_decision_level(self):
 232                 return len(self.trail_lim)
 233
 234         def add_variable(self, obj):
 235                 debug("add_variable('%s')", obj)
 236                 index = len(self.assigns)
 237
 238                 self.watches += [[], []]        # Add watch lists for X and not(X)
 239                 self.assigns.append(VarInfo(obj))
 240                 return index
 241
 242         # lit is now True
 243         # reason is the clause that is asserting this
 244         # Returns False if this immediately causes a conflict.
 245         def enqueue(self, lit, reason):
 246                 debug("%s => %s" % (reason, self.name_lit(lit)))
 247                 old_value = self.lit_value(lit)
 248                 if old_value is not None:
 249                         if old_value is False:
 250                                 # Conflict
 251                                 return False
 252                         else:
 253                                 # Already set (shouldn't happen)
 254                                 return True
 255
 256                 if lit < 0:
 257                         var_info = self.assigns[neg(lit)]
 258                         var_info.value = False
 259                 else:
 260                         var_info = self.assigns[lit]
 261                         var_info.value = True
 262                 var_info.level = self.get_decision_level()
 263                 var_info.reason = reason
 264
 265                 self.trail.append(lit)
 266                 self.propQ.append(lit)
 267
 268                 return True
 269
 270         # Pop most recent assignment from self.trail
 271         def undo_one(self):
 272                 lit = self.trail[-1]
 273                 debug("(pop %s)", self.name_lit(lit))
 274                 var_info = self.get_varinfo_for_lit(lit)
 275                 var_info.value = None
 276                 var_info.reason = None
 277                 var_info.level = -1
 278                 self.trail.pop()
 279
 280                 while var_info.undo:
 281                         var_info.undo.pop().undo(lit)
 282
 283         def cancel(self):
 284                 n_this_level = len(self.trail) - self.trail_lim[-1]
 285                 debug("backtracking from level %d (%d assignments)" %
 286                                 (self.get_decision_level(), n_this_level))
 287                 while n_this_level != 0:
 288                         self.undo_one()
 289                         n_this_level -= 1
 290                 self.trail_lim.pop()
 291
 292         def cancel_until(self, level):
 293                 while self.get_decision_level() > level:
 294                         self.cancel()
 295
 296         # Process the propQ.
 297         # Returns None when done, or the clause that caused a conflict.
 298         def propagate(self):
 299                 #debug("propagate: queue length = %d", len(self.propQ))
 300                 while self.propQ:
 301                         lit = self.propQ[0]
 302                         del self.propQ[0]
 303                         var_info = self.get_varinfo_for_lit(lit)
 304                         wi = watch_index(lit)
 305                         watches = self.watches[wi]
 306                         self.watches[wi] = []
 307
 308                         debug("%s -> True : watches: %s" % (self.name_lit(lit), watches))
 309
 310                         # Notifiy all watchers
 311                         for i in range(len(watches)):
 312                                 clause = watches[i]
 313                                 if not clause.propagate(lit):
 314                                         # Conflict
 315
 316                                         # Re-add remaining watches
 317                                         self.watches[wi] += watches[i+1:]
 318
 319                                         # No point processing the rest of the queue as
 320                                         # we'll have to backtrack now.
 321                                         self.propQ = []
 322
 323                                         return clause
 324                 return None
 325
 326         def impossible(self):
 327                 self.toplevel_conflict = True
 328
 329         def get_varinfo_for_lit(self, lit):
 330                 if lit >= 0:
 331                         return self.assigns[lit]
 332                 else:
 333                         return self.assigns[neg(lit)]
 334
 335         def lit_value(self, lit):
 336                 if lit >= 0:
 337                         value = self.assigns[lit].value
 338                         return value
 339                 else:
 340                         v = -1 - lit
 341                         value = self.assigns[v].value
 342                         if value is None:
 343                                 return None
 344                         else:
 345                                 return not value
 346
 347         # Call cb when lit becomes True
 348         def watch_lit(self, lit, cb):
 349                 #debug("%s is watching for %s to become True" % (cb, self.name_lit(lit)))
 350                 self.watches[watch_index(lit)].append(cb)
 351
 352         # Returns the new clause if one was added, True if none was added
 353         # because this clause is trivially True, or False if the clause is
 354         # False.
 355         def _add_clause(self, lits, learnt):
 356                 if not lits:
 357                         assert not learnt
 358                         self.toplevel_conflict = True
 359                         return False
 360                 elif len(lits) == 1:
 361                         # A clause with only a single literal is represented
 362                         # as an assignment rather than as a clause.
 363                         if learnt:
 364                                 reason = "learnt"
 365                         else:
 366                                 reason = "top-level"
 367                         return self.enqueue(lits[0], reason)
 368
 369                 clause = self.makeUnionClause(lits)
 370                 clause.learnt = learnt
 371
 372                 if learnt:
 373                         # lits[0] is None because we just backtracked.
 374                         # Start watching the next literal that we will
 375                         # backtrack over.
 376                         best_level = -1
 377                         best_i = 1
 378                         for i in range(1, len(lits)):
 379                                 level = self.get_varinfo_for_lit(lits[i]).level
 380                                 if level > best_level:
 381                                         best_level = level
 382                                         best_i = i
 383                         lits[1], lits[best_i] = lits[best_i], lits[1]
 384
 385                 # Watch the first two literals in the clause (both must be
 386                 # undefined at this point).
 387                 for lit in lits[:2]:
 388                         self.watch_lit(neg(lit), clause)
 389
 390                 return clause
 391
 392         def name_lits(self, lst):
 393                 return [self.name_lit(l) for l in lst]
 394
 395         # For nicer debug messages
 396         def name_lit(self, lit):
 397                 if lit >= 0:
 398                         return self.assigns[lit].name
 399                 return "not(%s)" % self.assigns[neg(lit)].name
 400
 401         def add_clause(self, lits):
 402                 # Public interface. Only used before the solve starts.
 403                 assert lits
 404
 405                 debug("add_clause([%s])" % ', '.join(self.name_lits(lits)))
 406
 407                 if any(self.lit_value(l) == True for l in lits):
 408                         # Trivially true already.
 409                         return True
 410                 lit_set = set(lits)
 411                 for l in lits:
 412                         if neg(l) in lit_set:
 413                                 # X or not(X) is always True.
 414                                 return True
 415                 # Remove duplicates and values known to be False
 416                 lits = [l for l in lit_set if self.lit_value(l) != False]
 417
 418                 retval = self._add_clause(lits, learnt = False)
 419                 if not retval:
 420                         self.toplevel_conflict = True
 421                 return retval
 422
 423         def at_most_one(self, lits):
 424                 assert lits
 425
 426                 debug("at_most_one(%s)" % ', '.join(self.name_lits(lits)))
 427
 428                 # If we have zero or one literals then we're trivially true
 429                 # and not really needed for the solve. However, Zero Install
 430                 # monitors these objects to find out what was selected, so
 431                 # keep even trivial ones around for that.
 432                 #
 433                 #if len(lits) < 2:
 434                 #       return True     # Trivially true
 435
 436                 # Ensure no duplicates
 437                 assert len(set(lits)) == len(lits), lits
 438
 439                 # Ignore any literals already known to be False.
 440                 # If any are True then they're enqueued and we'll process them
 441                 # soon.
 442                 lits = [l for l in lits if self.lit_value(l) != False]
 443
 444                 clause = self.makeAtMostOneClause(lits)
 445
 446                 for lit in lits:
 447                         self.watch_lit(lit, clause)
 448
 449                 return clause
 450
 451         def analyse(self, cause):
 452                 # After trying some assignments, we've discovered a conflict.
 453                 # e.g.
 454                 # - we selected A then B then C
 455                 # - from A, B, C we got X, Y
 456                 # - we have a rule: not(A) or not(X) or not(Y)
 457                 #
 458                 # The simplest thing to do would be:
 459                 # 1. add the rule "not(A) or not(B) or not(C)"
 460                 # 2. unassign C
 461                 #
 462                 # Then we we'd deduce not(C) and we could try something else.
 463                 # However, that would be inefficient. We want to learn a more
 464                 # general rule that will help us with the rest of the problem.
 465                 #
 466                 # We take the clause that caused the conflict ("cause") and
 467                 # ask it for its cause. In this case:
 468                 #
 469                 #  A and X and Y => conflict
 470                 #
 471                 # Since X and Y followed logically from A, B, C there's no
 472                 # point learning this rule; we need to know to avoid A, B, C
 473                 # *before* choosing C. We ask the two variables deduced at the
 474                 # current level (X and Y) what caused them, and work backwards.
 475                 # e.g.
 476                 #
 477                 #  X: A and C => X
 478                 #  Y: C => Y
 479                 #
 480                 # Combining these, we get the cause of the conflict in terms of
 481                 # things we knew before the current decision level:
 482                 #
 483                 #  A and X and Y => conflict
 484                 #  A and (A and C) and (C) => conflict
 485                 #  A and C => conflict
 486                 #
 487                 # We can then learn (record) the more general rule:
 488                 #
 489                 #  not(A) or not(C)
 490                 #
 491                 # Then, in future, whenever A is selected we can remove C and
 492                 # everything that depends on it from consideration.
 493
 494
 495                 learnt = [None]         # The general rule we're learning
 496                 btlevel = 0             # The deepest decision in learnt
 497                 p = None                # The literal we want to expand now
 498                 seen = set()            # The variables involved in the conflict
 499
 500                 counter = 0
 501
 502                 while True:
 503                         # cause is the reason why p is True (i.e. it enqueued it).
 504                         # The first time, p is None, which requests the reason
 505                         # why it is conflicting.
 506                         if p is None:
 507                                 debug("Why did %s make us fail?" % cause)
 508                                 p_reason = cause.cacl_reason(p)
 509                                 debug("Because: %s => conflict" % (' and '.join(self.name_lits(p_reason))))
 510                         else:
 511                                 debug("Why did %s lead to %s?" % (cause, self.name_lit(p)))
 512                                 p_reason = cause.cacl_reason(p)
 513                                 debug("Because: %s => %s" % (' and '.join(self.name_lits(p_reason)), self.name_lit(p)))
 514
 515                         # p_reason is in the form (A and B and ...)
 516                         # p_reason => p
 517
 518                         # Check each of the variables in p_reason that we haven't
 519                         # already considered:
 520                         # - if the variable was assigned at the current level,
 521                         #   mark it for expansion
 522                         # - otherwise, add it to learnt
 523
 524                         for lit in p_reason:
 525                                 var_info = self.get_varinfo_for_lit(lit)
 526                                 if var_info not in seen:
 527                                         seen.add(var_info)
 528                                         if var_info.level == self.get_decision_level():
 529                                                 # We deduced this var since the last decision.
 530                                                 # It must be in self.trail, so we'll get to it
 531                                                 # soon. Remember not to stop until we've processed it.
 532                                                 counter += 1
 533                                         elif var_info.level > 0:
 534                                                 # We won't expand lit, just remember it.
 535                                                 # (we could expand it if it's not a decision, but
 536                                                 # apparently not doing so is useful)
 537                                                 learnt.append(neg(lit))
 538                                                 btlevel = max(btlevel, var_info.level)
 539                                 # else we already considered the cause of this assignment
 540
 541                         # At this point, counter is the number of assigned
 542                         # variables in self.trail at the current decision level that
 543                         # we've seen. That is, the number left to process. Pop
 544                         # the next one off self.trail (as well as any unrelated
 545                         # variables before it; everything up to the previous
 546                         # decision has to go anyway).
 547
 548                         # On the first time round the loop, we must find the
 549                         # conflict depends on at least one assignment at the
 550                         # current level. Otherwise, simply setting the decision
 551                         # variable caused a clause to conflict, in which case
 552                         # the clause should have asserted not(decision-variable)
 553                         # before we ever made the decision.
 554                         # On later times round the loop, counter was already >
 555                         # 0 before we started iterating over p_reason.
 556                         assert counter > 0
 557
 558                         while True:
 559                                 p = self.trail[-1]
 560                                 var_info = self.get_varinfo_for_lit(p)
 561                                 cause = var_info.reason
 562                                 self.undo_one()
 563                                 if var_info in seen:
 564                                         break
 565                                 debug("(irrelevant)")
 566                         counter -= 1
 567
 568                         if counter <= 0:
 569                                 assert counter == 0
 570                                 # If counter = 0 then we still have one more
 571                                 # literal (p) at the current level that we
 572                                 # could expand. However, apparently it's best
 573                                 # to leave this unprocessed (says the minisat
 574                                 # paper).
 575                                 break
 576
 577                 # p is the literal we decided to stop processing on. It's either
 578                 # a derived variable at the current level, or the decision that
 579                 # led to this level. Since we're not going to expand it, add it
 580                 # directly to the learnt clause.
 581                 learnt[0] = neg(p)
 582
 583                 debug("Learnt: %s" % (' or '.join(self.name_lits(learnt))))
 584
 585                 return learnt, btlevel
 586
 587         def run_solver(self, decide):
 588                 # Check whether we detected a trivial problem
 589                 # during setup.
 590                 if self.toplevel_conflict:
 591                         debug("FAIL: toplevel_conflict before starting solve!")
 592                         return False
 593
 594                 while True:
 595                         # Use logical deduction to simplify the clauses
 596                         # and assign literals where there is only one possibility.
 597                         conflicting_clause = self.propagate()
 598                         if not conflicting_clause:
 599                                 debug("new state: %s", self.assigns)
 600                                 if all(info.value != None for info in self.assigns):
 601                                         # Everything is assigned without conflicts
 602                                         debug("SUCCESS!")
 603                                         return True
 604                                 else:
 605                                         # Pick a variable and try assigning it one way.
 606                                         # If it leads to a conflict, we'll backtrack and
 607                                         # try it the other way.
 608                                         lit = decide()
 609                                         #print "TRYING:", self.name_lit(lit)
 610                                         assert lit is not None, "decide function returned None!"
 611                                         assert self.lit_value(lit) is None
 612                                         self.trail_lim.append(len(self.trail))
 613                                         r = self.enqueue(lit, reason = "considering")
 614                                         assert r is True
 615                         else:
 616                                 if self.get_decision_level() == 0:
 617                                         debug("FAIL: conflict found at top level")
 618                                         return False
 619                                 else:
 620                                         # Figure out the root cause of this failure.
 621                                         learnt, backtrack_level = self.analyse(conflicting_clause)
 622
 623                                         self.cancel_until(backtrack_level)
 624
 625                                         c = self._add_clause(learnt, learnt = True)
 626
 627                                         if c is not True:
 628                                                 # Everything except the first literal in learnt is known to
 629                                                 # be False, so the first must be True.
 630                                                 e = self.enqueue(learnt[0], c)
 631                                                 assert e is True