zeroinstall/injector/sat.py

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