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