2 Internal implementation of a SAT solver, used by L{solver.SATSolver}.
3 This is not part of the public API.
6 from __future__
import print_function
8 # Copyright (C) 2010, Thomas Leonard
9 # See the README file for details, or visit http://0install.net.
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
15 # The main differences are:
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).
23 def debug(msg
, *args
):
25 print("SAT:", msg
% args
)
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):
31 # Variable Literal not(Literal)
40 return neg(lit
) * 2 + 1
42 def makeAtMostOneClause(solver
):
43 class AtMostOneClause
:
44 def __init__(self
, lits
):
45 """Preferred literals come first."""
48 # The single literal from our set that is True.
49 # We store this explicitly because the decider needs to know quickly.
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
)
58 # value[lit] has just become True
59 assert solver
.lit_value(lit
) == True
62 #debug("%s: noticed %s has become True" % (self, solver.name_lit(lit)))
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
70 assert self
.current
is None
74 # If we later backtrace, call our undo function to unset current
75 solver
.get_varinfo_for_lit(lit
).undo
.append(self
)
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
))
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
95 debug("(backtracking: no longer selected %s)" % solver
.name_lit(lit
))
96 assert lit
== self
.current
100 # Or, why are we causing a conflict (if lit is None)?
101 def cacl_reason(self
, lit
):
103 # Find two True literals
106 if solver
.lit_value(l
) is True:
108 if len(trues
) == 2: return trues
111 if l
is not lit
and solver
.lit_value(l
) is True:
113 # Find one True literal
114 assert 0 # don't know why!
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:
125 return "<lone: %s>" % (', '.join(solver
.name_lits(self
.lits
)))
127 return AtMostOneClause
129 def makeUnionClause(solver
):
131 def __init__(self
, lits
):
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
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.
145 # Returns False on conflict.
146 def propagate(self
, lit
):
147 # value[get(lit)] has just become False
149 #debug("%s: noticed %s has become False" % (self, solver.name_lit(neg(lit))))
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]
159 if solver
.lit_value(self
.lits
[0]) == True:
160 # We're already satisfied. Do nothing.
161 solver
.watch_lit(lit
, self
)
164 assert solver
.lit_value(self
.lits
[1]) == False
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
])
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
)
177 # Only lits[0], is now undefined.
178 solver
.watch_lit(lit
, self
)
179 return solver
.enqueue(self
.lits
[0], self
)
181 def undo(self
, lit
): pass
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]
188 # The cause is everything except lit.
189 return [neg(l
) for l
in self
.lits
if l
is not lit
]
192 return "<some: %s>" % (', '.join(solver
.name_lits(self
.lits
)))
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)
207 return '%s=%s' % (self
.name
, self
.value
)
213 class SATProblem(object):
216 self
.watches
= [] # watches[2i,2i+1] = constraints to check when literal[i] becomes True/False
217 self
.propQ
= [] # propagation queue
220 self
.assigns
= [] # [VarInfo]
221 self
.trail
= [] # order of assignments
222 self
.trail_lim
= [] # decision levels
224 self
.toplevel_conflict
= False
226 self
.makeAtMostOneClause
= makeAtMostOneClause(self
)
227 self
.makeUnionClause
= makeUnionClause(self
)
229 def get_decision_level(self
):
230 return len(self
.trail_lim
)
232 def add_variable(self
, obj
):
233 debug("add_variable('%s')", obj
)
234 index
= len(self
.assigns
)
236 self
.watches
+= [[], []] # Add watch lists for X and not(X)
237 self
.assigns
.append(VarInfo(obj
))
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:
251 # Already set (shouldn't happen)
255 var_info
= self
.assigns
[neg(lit
)]
256 var_info
.value
= False
258 var_info
= self
.assigns
[lit
]
259 var_info
.value
= True
260 var_info
.level
= self
.get_decision_level()
261 var_info
.reason
= reason
263 self
.trail
.append(lit
)
264 self
.propQ
.append(lit
)
268 # Pop most recent assignment from self.trail
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
279 var_info
.undo
.pop().undo(lit
)
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:
290 def cancel_until(self
, level
):
291 while self
.get_decision_level() > level
:
295 # Returns None when done, or the clause that caused a conflict.
297 #debug("propagate: queue length = %d", len(self.propQ))
301 wi
= watch_index(lit
)
302 watches
= self
.watches
[wi
]
303 self
.watches
[wi
] = []
305 debug("%s -> True : watches: %s" % (self
.name_lit(lit
), watches
))
307 # Notifiy all watchers
308 for i
in range(len(watches
)):
310 if not clause
.propagate(lit
):
313 # Re-add remaining watches
314 self
.watches
[wi
] += watches
[i
+1:]
316 # No point processing the rest of the queue as
317 # we'll have to backtrack now.
323 def impossible(self
):
324 self
.toplevel_conflict
= True
326 def get_varinfo_for_lit(self
, lit
):
328 return self
.assigns
[lit
]
330 return self
.assigns
[neg(lit
)]
332 def lit_value(self
, lit
):
334 value
= self
.assigns
[lit
].value
338 value
= self
.assigns
[v
].value
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
)
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
352 def _add_clause(self
, lits
, learnt
):
355 self
.toplevel_conflict
= True
358 # A clause with only a single literal is represented
359 # as an assignment rather than as a clause.
364 return self
.enqueue(lits
[0], reason
)
366 clause
= self
.makeUnionClause(lits
)
369 # lits[0] is None because we just backtracked.
370 # Start watching the next literal that we will
374 for i
in range(1, len(lits
)):
375 level
= self
.get_varinfo_for_lit(lits
[i
]).level
376 if level
> best_level
:
379 lits
[1], lits
[best_i
] = lits
[best_i
], lits
[1]
381 # Watch the first two literals in the clause (both must be
382 # undefined at this point).
384 self
.watch_lit(neg(lit
), clause
)
388 def name_lits(self
, lst
):
389 return [self
.name_lit(l
) for l
in lst
]
391 # For nicer debug messages
392 def name_lit(self
, lit
):
394 return self
.assigns
[lit
].name
395 return "not(%s)" % self
.assigns
[neg(lit
)].name
397 def add_clause(self
, lits
):
398 # Public interface. Only used before the solve starts.
401 debug("add_clause([%s])" % ', '.join(self
.name_lits(lits
)))
403 if any(self
.lit_value(l
) == True for l
in lits
):
404 # Trivially true already.
408 if neg(l
) in lit_set
:
409 # X or not(X) is always True.
411 # Remove duplicates and values known to be False
412 lits
= [l
for l
in lit_set
if self
.lit_value(l
) != False]
414 retval
= self
._add
_clause
(lits
, learnt
= False)
416 self
.toplevel_conflict
= True
419 def at_most_one(self
, lits
):
422 debug("at_most_one(%s)" % ', '.join(self
.name_lits(lits
)))
424 # If we have zero or one literals then we're trivially true
425 # and not really needed for the solve. However, Zero Install
426 # monitors these objects to find out what was selected, so
427 # keep even trivial ones around for that.
430 # return True # Trivially true
432 # Ensure no duplicates
433 assert len(set(lits
)) == len(lits
), lits
435 # Ignore any literals already known to be False.
436 # If any are True then they're enqueued and we'll process them
438 lits
= [l
for l
in lits
if self
.lit_value(l
) != False]
440 clause
= self
.makeAtMostOneClause(lits
)
443 self
.watch_lit(lit
, clause
)
447 def analyse(self
, cause
):
448 # After trying some assignments, we've discovered a conflict.
450 # - we selected A then B then C
451 # - from A, B, C we got X, Y
452 # - we have a rule: not(A) or not(X) or not(Y)
454 # The simplest thing to do would be:
455 # 1. add the rule "not(A) or not(B) or not(C)"
458 # Then we we'd deduce not(C) and we could try something else.
459 # However, that would be inefficient. We want to learn a more
460 # general rule that will help us with the rest of the problem.
462 # We take the clause that caused the conflict ("cause") and
463 # ask it for its cause. In this case:
465 # A and X and Y => conflict
467 # Since X and Y followed logically from A, B, C there's no
468 # point learning this rule; we need to know to avoid A, B, C
469 # *before* choosing C. We ask the two variables deduced at the
470 # current level (X and Y) what caused them, and work backwards.
476 # Combining these, we get the cause of the conflict in terms of
477 # things we knew before the current decision level:
479 # A and X and Y => conflict
480 # A and (A and C) and (C) => conflict
481 # A and C => conflict
483 # We can then learn (record) the more general rule:
487 # Then, in future, whenever A is selected we can remove C and
488 # everything that depends on it from consideration.
491 learnt
= [None] # The general rule we're learning
492 btlevel
= 0 # The deepest decision in learnt
493 p
= None # The literal we want to expand now
494 seen
= set() # The variables involved in the conflict
499 # cause is the reason why p is True (i.e. it enqueued it).
500 # The first time, p is None, which requests the reason
501 # why it is conflicting.
503 debug("Why did %s make us fail?" % cause
)
504 p_reason
= cause
.cacl_reason(p
)
505 debug("Because: %s => conflict" % (' and '.join(self
.name_lits(p_reason
))))
507 debug("Why did %s lead to %s?" % (cause
, self
.name_lit(p
)))
508 p_reason
= cause
.cacl_reason(p
)
509 debug("Because: %s => %s" % (' and '.join(self
.name_lits(p_reason
)), self
.name_lit(p
)))
511 # p_reason is in the form (A and B and ...)
514 # Check each of the variables in p_reason that we haven't
515 # already considered:
516 # - if the variable was assigned at the current level,
517 # mark it for expansion
518 # - otherwise, add it to learnt
521 var_info
= self
.get_varinfo_for_lit(lit
)
522 if var_info
not in seen
:
524 if var_info
.level
== self
.get_decision_level():
525 # We deduced this var since the last decision.
526 # It must be in self.trail, so we'll get to it
527 # soon. Remember not to stop until we've processed it.
529 elif var_info
.level
> 0:
530 # We won't expand lit, just remember it.
531 # (we could expand it if it's not a decision, but
532 # apparently not doing so is useful)
533 learnt
.append(neg(lit
))
534 btlevel
= max(btlevel
, var_info
.level
)
535 # else we already considered the cause of this assignment
537 # At this point, counter is the number of assigned
538 # variables in self.trail at the current decision level that
539 # we've seen. That is, the number left to process. Pop
540 # the next one off self.trail (as well as any unrelated
541 # variables before it; everything up to the previous
542 # decision has to go anyway).
544 # On the first time round the loop, we must find the
545 # conflict depends on at least one assignment at the
546 # current level. Otherwise, simply setting the decision
547 # variable caused a clause to conflict, in which case
548 # the clause should have asserted not(decision-variable)
549 # before we ever made the decision.
550 # On later times round the loop, counter was already >
551 # 0 before we started iterating over p_reason.
556 var_info
= self
.get_varinfo_for_lit(p
)
557 cause
= var_info
.reason
561 debug("(irrelevant)")
566 # If counter = 0 then we still have one more
567 # literal (p) at the current level that we
568 # could expand. However, apparently it's best
569 # to leave this unprocessed (says the minisat
573 # p is the literal we decided to stop processing on. It's either
574 # a derived variable at the current level, or the decision that
575 # led to this level. Since we're not going to expand it, add it
576 # directly to the learnt clause.
579 debug("Learnt: %s" % (' or '.join(self
.name_lits(learnt
))))
581 return learnt
, btlevel
583 def run_solver(self
, decide
):
584 # Check whether we detected a trivial problem
586 if self
.toplevel_conflict
:
587 debug("FAIL: toplevel_conflict before starting solve!")
591 # Use logical deduction to simplify the clauses
592 # and assign literals where there is only one possibility.
593 conflicting_clause
= self
.propagate()
594 if not conflicting_clause
:
595 debug("new state: %s", self
.assigns
)
596 if all(info
.value
!= None for info
in self
.assigns
):
597 # Everything is assigned without conflicts
601 # Pick a variable and try assigning it one way.
602 # If it leads to a conflict, we'll backtrack and
603 # try it the other way.
605 #print "TRYING:", self.name_lit(lit)
606 assert lit
is not None, "decide function returned None!"
607 assert self
.lit_value(lit
) is None
608 self
.trail_lim
.append(len(self
.trail
))
609 r
= self
.enqueue(lit
, reason
= "considering")
612 if self
.get_decision_level() == 0:
613 debug("FAIL: conflict found at top level")
616 # Figure out the root cause of this failure.
617 learnt
, backtrack_level
= self
.analyse(conflicting_clause
)
619 self
.cancel_until(backtrack_level
)
621 c
= self
._add
_clause
(learnt
, learnt
= True)
624 # Everything except the first literal in learnt is known to
625 # be False, so the first must be True.
626 e
= self
.enqueue(learnt
[0], c
)