QBF Modeling: Exploiting Player Symmetry for Simplicity and Efficiency

1
QBF Modeling Exploiting Player Symmetry for
Simplicity and Efficiency

• Ashish Sabharwal, Carlos Ansotegui,Carla P.
  Gomes, Justin W. Hart, Bart Selman
• Cornell University
• SAT Conference, August 2006
• Seattle, WA

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The Goal of This Work

• To significantly extend the reach of QBF
  reasoning by
• Investigating and improving basic modeling
  framework
• Retaining the benefits of CNF for SAT/QBF solvers
• E.g., must avoid higher level representations
• Maintaining (or enhancing) simplicity of
  representation
• Our driving force
• Real-World Reasoning Program
• A set of challenging QBF benchmarks
• With many quantifier alternations
• Encoding a hard adversarial task chess-style end
  games

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Our Contribution

• We propose a simple but fundamental change in the
  way problems are modeled as QBF instances, and
  solved.
• A systematic modeling technique based on a game
  theoretic view and SAT-based planning ideas
• A split CNF-DNF dual encoding (existential player
  modeled as CNF, universal player as DNF)
• A new QBF solver Duaffle (dual-Quaffle)
• 2 orders of magnitude improvement through
• Better propagation across quantifiers
• Avoidance of illegal search space issue
• Simpler encoding w.r.t. previous approaches

4
Roadmap of the Talk
FourKey Challenges
The Basicsof QBF
Our ApproachFrom problem to ? games?
dual representation? dual solver
Summary
ExperimentalResults
5
Roadmap of the Talk
FourKey Challenges
The Basicsof QBF
Our ApproachFrom problem to ? games?
dual representation? dual solver
Summary
ExperimentalResults
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SAT, QBF, CNF, and DNF

• F a Boolean formula
• e.g. F (a or b) and (not (a and (b or
  c)))
• 3 satisfying assignments (a,b,c) (1,0,0),
  (0,1,0), (0,1,1)
• F in CNF FCNF (a or b) and (?a or ?b) and
  (b or ?c)
• F in DNF FDNF (?a and b) or (a and ?b and
  ?c)
• SAT Does F have any satisfying assignments?
• NP-complete for FCNF, trivial for FDNF
• QBF Is a given (totally) quantified Boolean
  formula True?
• e.g. G ?a,b ?c. (a or b) and (not (a and (b
  or c)))
• GCNF ?a,b ?c. FCNF, GDNF ?a,b ?c. FDNF
• In general, an unbounded number of quantifier
  layers
• PSPACE-complete for both CNF and DNF forms

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CNF Format and SAT

• Many good reasons to use the CNF format for SAT
• Fairly natural representation
• Many problems are a conjunction of several
  constraints
• Each constraint in itself is often simple and
  easy to satisfy
• Efficient pruning of unsat. parts of the search
  space
• Violation of any single constraint by a partial
  assignment can be detected immediately
• Simplicity
• Lends itself easily to clever techniques and data
  structures(e.g. watched literals, conflict
  graph, )
• Provides a clear uniform standard

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Is CNF Equally Good for QBF?

• Many advantages
• SAT techniques carry over to QBF(encoder
  format, clause learning, unit propagation,
  watched literals, restarts, )
• Can quickly extend existing SAT solvers to QBF
  solvers(search both assignments for universal
  variables)
• This approach led to the first QBF solvers based
  on DPLL, local search, Q-resolution, etc.

So far so good. The problem? Modern SAT solvers
scale very well (1M variables),but modern QBF
solvers dont! (10 K vars)
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The Message
Assuming CNF is a good modeling language for
SAT, a split CNF-DNF representation is the right
format for QBF

• Provides effective propagation
• Avoids QBF-specific search issues
• Results in a simpler encoding
• Improves state-of-the-art by orders of magnitude

10
Roadmap of the Talk
FourKey Challenges
The Basicsof QBF
Our ApproachFrom problem to ? games?
dual representation? dual solver
Summary
ExperimentalResults
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Challenge 1

• Most QBF benchmarks have only 2-3 quantifer
  levels
• Might as well translate into SAT (it often
  works!)
• Benchmarks with many levels are often the hardest
• Practical issues in both modeling and solving
  become much more apparent with many quantifier
  levels
• Our benchmarks encode chess-like problems with
  7-15 quantifier levels

Can QBF solvers be made to scale well with10
quantifier alternations?
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Challenge 2

• QBF solvers extremely sensitive to encoding!
• Especially with many quantifier levels, e.g.,
  evader-pursuer chess instances Madhusudan et
  al. 2003

Can we design generic QBF modeling
techniquesthat are simple and efficient for
solvers?
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Challenge 3

• For QBF, traditional encodings hinder unit
  propagation
• E.g. unsatisfiable reachability queries
• A SAT solver would have simply unit propagated
• QBF solvers need 1000s of backtracks and complex
  mechanisms like learning

Can we achieve unit propagation across
quantifiers?
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Lack of Effective Propagation
QuestionCan White reach thepink square
withoutbeing captured?
q-unsatWhite has one toofew available moves
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Challenge 4

• QBF solvers suffer from the illegal search space
  issue Ansotegui et al. 2005
• Auxiliary variables needed for conversion into
  CNF
• Can push solver into large irrelevant parts of
  search space
• Note negligible impact on SAT due to effective
  propagation
• Best fix for QBF condQuaffle (passes flags to
  the solver)

Can we somehow completely avoid the illegal
searchspace issue by using a better
representation?
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Aside Search Space for SAT
Effect of addingauxiliary variables
Search Space SAT Encoding 2NM
Original Search Space 2N
Space Searched by SAT Solvers 2N/C Nlog(N)
Poly(N)
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Aside Search Space for QBF
Search Space QBF Encoding 2NM
Can we reduce the search space With clever
encodings , streamlining, etc?
Original Search Space 2N
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Roadmap of the Talk
FourKey Challenges
The Basicsof QBF
Our ApproachFrom problem to ? games?
dual representation? dual solver
Summary
ExperimentalResults
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The Traditional Approach
CNF-basedQBF encoding
QBF Solver
Problemof interest
e.g. chess end-game, circuit minimization,advers
arial planning,
Solution!
Any discreteadversarial task
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Overview of Our Approach
Game G players E U,states, actions, rules,
goal
AdversarialTask
Planning as Satisfiability framework (standard)
e.g. chess end-game, circuit minimization,advers
arial planning,
Create CNF encodingseparately for E and
U initial state axioms, action implies
precondition,fact implies achieving
action, frame axioms, goal condition
Solution!
Dual (split)CNF-DNF encoding
QBF SolverDuaffle
NegateCNF part for U(creates DNF)
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From Adversarial Tasks To Games

• Example 1
• Circuit Minimization Given a circuit C, is
  there a smaller circuit computing the same
  function as C?
• Related QBF benchmarks adder circuits, sorting
  networks
• A game with 2 turns
• Moves First, E commits to a circuit CE second,
  U produces an input p and computations of CE, C
  on p.
• Rules CE must be a legal circuit smaller than C
  U must correctly compute CE(p) and C(p).
• Goal E wins if CE(p) C(p) no matter how U
  chooses p
• E wins iff there is a smaller circuit

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From Adversarial Tasks To Games

• Example 2
• The Chromatic Number Problem Given a graph G
  and a positive number k, does G have chromatic
  number k?
• Chromatic number minimum number of colors needed
  to color G so that every two adjacent vertices
  get different colors
• A game with 2 turns
• Moves First, E produces a coloring S of G
  second, U produces a coloring T of G
• Rules S must be a legal k-coloring of G T must
  be a legal (k-1)-coloring of G
• Goal E wins if S is valid and T is not
• E wins iff G has chromatic number k

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From Games to Formulas

• Use the planning as satisfiability framework
• I Initial conditions
• TrE Rules for legal transitions/moves of E
• TrU Rules for legal transitions/moves of U
• GE Goal of E (negation of goal of U)
• Two alternative formulations of the QBF Matrix

CNFclauses
Fits circuit minimization,chromatic number
problem, etc.
M1 I ? TrE ? (TrU ? GE)
M2 TrU ? (I ? TrE ? GE)
Fits games like chess, etc.
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The Dual Encoding
Two alternative formulations of the dual QBF
matrix
M1 (I ? TrE) ? (?TrU ? ?GU)
CNF
DNF
(negation of CNF clauses)
M2 (I ? TrE ? GE) ? ?TrU
In contrast withZhang, AAAI 06split,
non-redundant
Variables state vars S1, S2, , Sk1
action vars A1, A2, , Ak
?S1 ?A1?S2 ?A2?S3 ?A3?S4 ?Ak?Sk1 Mi
i ? 1,2
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The Dual Encoding Example

• Chess White as E, Black as U
• TrE Transition axioms for E CNF clauses
• e.g. ? Move(Wking, sqA, sqB, step 5) ?
  Loc(Wking, sqA, 5)
• TrU Transition axioms for U DNF terms(negated
  traditional axiom clauses)
• e.g. Move(Bking, sqA, sqB, step 5) ? ?
  Loc(Bking, sqA, 5)

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Our QBF Solver Duaffle
dual-Quaffle

• An extension of Quaffle Zhang-Malik 02
• Quaffle already supports DNF terms (cubes)
• However, its DNF terms are deduced from the CNF
  input
• For us, DNF and CNF parts are independent
• ? propagation mechanism changes
• Most features remain unchanged(e.g. parser, data
  structures, decision heuristic, clause and cube
  learning, fast backjumping, )

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Duaffle Input Format
c Dual QBF format c 100 variables c 25 CNF
clauses, 32 DNF terms c p cnfdnf and 100 25
32 c c Quantifiers e 1 2 5 9 23 56 0 a 6 7 21
22 0 0 c CNF clauses -4 -7 8 12 0 9 5 -55
0 0 c DNF terms 43 -61 -2 0 4 1 -100 0 0

• Straightforward extensionof QDIMACS format
• Specifies quantification,CNF clauses, DNF terms
• Additional flag for choosingbetween formulations
  M1 (connective ?) and M2 (connective ?)

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Duaffle Backtracking Policy

• E.g. what should we do when the CNF part is
  satisfied but the DNF part is not?
• Extension of Quaffles policy(Quaffle never
  encounters certain possibilities because its
  DNF part is logically deduced from the CNF part)

DNF part
DNF part
U
F
T
U
F
T
U
U
CNFpart
CNFpart
F
F
T
T
For formulation M2
For formulation M1
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Roadmap of the Talk
FourKey Challenges
The Basicsof QBF
Our ApproachFrom problem to ? games?
dual representation? dual solver
Summary
ExperimentalResults
30
Experimental Results
5-15 quantifier levels (reachability)
7-9 quantifier levels
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Experimental Results, contd.
7-9 quantifier levels
Duaffle (even without learning) on the dual
encoding dramatically outperforms all leading
CNF-based QBF solvers on these challenging
instances
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Summary

• A new QBF modeling approach
• Uses a split CNF-DNF representation
• Preserves benefits of CNF
• Leverages modern QBF solvers ability to handle
  DNF
• Based on a systematic view of problems as games,
  and the planning as satisfiability framework
• A dual format QBF solver, Duaffle
• Extends Quaffle
• Outperforms all existing QBF solvers (on xChess)
  by orders of magnitude, even without clause/cube
  learning