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

August 15, 2006 SAT 2006 2

The Goal of This Work

To significantly extend the reach of QBF reasoning by

1. Investigating and improving basic modeling framework

2. Retaining the benefits of CNF for SAT/QBF solvers E.g., must avoid “higher level” representations

3. 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

August 15, 2006 SAT 2006 3

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

August 15, 2006 SAT 2006 4

Roadmap of the Talk

The Basicsof QBF

FourKey Challenges

Our Approach:From problem to games dual representation dual solver

ExperimentalResults

Summary

August 15, 2006 SAT 2006 5

Roadmap of the Talk

The Basicsof QBF

FourKey Challenges

Our Approach:From problem to games dual representation dual solver

ExperimentalResults

Summary

August 15, 2006 SAT 2006 6

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

August 15, 2006 SAT 2006 7

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

August 15, 2006 SAT 2006 8

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 don’t! (~10 K vars)

August 15, 2006 SAT 2006 9

The Message

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

Assuming CNF is a good modeling language for SAT,

a split CNF-DNF representation is the right format for QBF

August 15, 2006 SAT 2006 10

Roadmap of the Talk

The Basicsof QBF

FourKey Challenges

Our Approach:From problem to games dual representation dual solver

ExperimentalResults

Summary

August 15, 2006 SAT 2006 11

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 with

10+ quantifier alternations?

August 15, 2006 SAT 2006 12

Challenge #2

QBF solvers extremely sensitive to encoding! Especially with many quantifier levels,

e.g., evader-pursuer chess instances [Madhusudan et al. 2003]

Instance (N, steps)

Model X [Madhusudan et al. 2003]

Model A [Ansotegui et al. 2005]

Model B [Ansotegui et al. 2005]

QuBEJ Semprop QuaffleBest other

solverCond-Quaffle

Best other solver

Cond-Quaffle

4 7 2030 >2030 >2030 7497 3 0.03 0.03

4 9 -- -- -- -- 28 0.06 0.04

8 7 -- -- -- -- 800 5 5

Can we design generic QBF modeling techniques

that are simple and efficient for solvers?

August 15, 2006 SAT 2006 13

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 1000’s of backtracks and complex

mechanisms like learning

Best solverwith only unit propagation

Best solver(conf-quaffle)with learning

conf-r1 2.5 0.2

conf-r5 8603 5.4

conf-r6 >21600 7.1

q-unsat: too few steps for White

?

Can we achieve unit propagation across quantifiers?

August 15, 2006 SAT 2006 14

Lack of Effective Propagation

QuickTime™ and aCinepak decompressor

are needed to see this picture.

q-unsat:White has one toofew available moves

Question:Can White reach thepink square withoutbeing captured?

August 15, 2006 SAT 2006 15

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 search

space issue by using a better representation?

August 15, 2006 SAT 2006 16

Aside: Search Space for SAT

OriginalSearch Space

2N

Search SpaceSAT Encoding

2N+M

Space Searchedby SAT Solvers

2N/C ; Nlog(N); Poly(N)

Original2N

Effect of addingauxiliary variables

August 15, 2006 SAT 2006 17

Aside: Search Space for QBF

OriginalSearch Space

2N

Search SpaceQBF Encoding

2N+M’

Can we reduce the search spaceWith clever encodings , streamlining, etc?

Search SpaceStandard QBF Encoding

2N+M’’

Original2N

August 15, 2006 SAT 2006 18

Roadmap of the Talk

The Basicsof QBF

FourKey Challenges

Our Approach:From problem to games dual representation dual solver

ExperimentalResults

Summary

August 15, 2006 SAT 2006 19

The Traditional Approach

Problemof interest

e.g. chess end-game, circuit minimization,adversarial planning,…

CNF-basedQBF encoding QBF Solver

Solution!

Any discreteadversarial task

August 15, 2006 SAT 2006 20

Overview of Our Approach

AdversarialTask

e.g. chess end-game, circuit minimization,adversarial planning,…

Game G:

players E & U,states, actions,

rules, goal

“Planning as Satisfiability”framework(standard)

Create CNF encodingseparately for E and U:

initial state axioms,action implies precondition,

fact implies achieving action,frame axioms,goal condition

Dual (split)CNF-DNF encoding

QBF SolverDuaffle

NegateCNF part for U(creates DNF)

Solution!

August 15, 2006 SAT 2006 21

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

August 15, 2006 SAT 2006 22

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

August 15, 2006 SAT 2006 23

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

M1 = I TrE (TrU GE)

M2 = TrU (I TrE GE)

Fits circuit minimization,chromatic number problem, etc.

Fits games like chess, etc.

CNFclauses

August 15, 2006 SAT 2006 24

The Dual Encoding

M’1 = (I TrE) (TrU GU)

M’2 = (I TrE GE) TrU

Two alternative formulations of the dual QBF matrix

CNF DNF

Variables : state vars S1, S2, …, Sk+1

action vars A1, A2, …, Ak

S1 A1S2 A2S3 A3S4 AkSk+1 M’i i {1,2}

(negation of CNF clauses)

In contrast with[Zhang, AAAI ’06]:split, non-redundant

August 15, 2006 SAT 2006 25

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)

August 15, 2006 SAT 2006 26

Our QBF Solver: Duaffle

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, …)

“dual-Quaffle”

August 15, 2006 SAT 2006 27

Duaffle: Input Formatc Dual QBF formatc 100 variablesc 25 CNF clauses, 32 DNF termsc p cnfdnf and 100 25 32cc Quantifierse 1 2 5 9 23 56 … 0a 6 7 21 22 … 0…0c CNF clauses-4 -7 8 12 09 5 -55 0…0c DNF terms43 -61 -2 04 1 -100 0…0

• Straightforward extensionof QDIMACS format

• Specifies quantification,CNF clauses, DNF terms

• Additional flag for choosingbetween formulations

M’1 (connective ) and

M’2 (connective )

August 15, 2006 SAT 2006 28

Duaffle: Backtracking Policy

E.g. what should we do when the CNF part is satisfied but the DNF part is not? Extension of Quaffle’s policy

(Quaffle never encounters certain possibilities because its DNF part is logically deduced from the CNF part)

BRN UNS BRN

UNS UNS UNS

BRN UNS SAT

U F T

U

F

T

CNFpart

DNF part

BRN BRN SAT

BRN UNS SAT

SAT SAT SAT

U F T

U

F

T

CNFpart

DNF part

For formulation M’1 For formulation M’2

August 15, 2006 SAT 2006 29

Roadmap of the Talk

The Basicsof QBF

FourKey Challenges

Our Approach:From problem to games dual representation dual solver

ExperimentalResults

Summary

August 15, 2006 SAT 2006 30

Experimental ResultsxChess instance Pure CNF Encoding Dual Encoding

name T/F Semprop sKizzo QuaffleCond-Quaffle

Duaffle(without learning!)

conf-r1 F 12 4.0 15 1.3 0.01

conf-r2 F 25 5.86 33 2.5 0.02

conf-r3 F 55 9.3 62 4.1 0.03

conf-r4 F 85 26 124 6.4 0.04

conf-r5 F 985 84 676 34 0.08

conf-r6 F 2042 73 713 49 0.10

conf01 F 1225 492 -- 539 6.4

conf02 F 93 30 6.0 1.0 0.0

conf03 T -- 1532 -- 83 1.4

conf04 T -- -- 2352 100 3.5

conf05 F 3290 448 510 196 0.1

conf06 F -- memout -- 633 30.6

conf07 F 261 42 78 3.5 0.0

conf08 T -- 1509 -- 1088 31.2

5-15quantifier

levels(reachability)

7-9quantifier

levels

August 15, 2006 SAT 2006 31

Experimental Results, contd.xChess instance Pure CNF Encoding Dual Encoding

name T/F Semprop sKizzo QuaffleCond-Quaffle

Duaffle(without learning!)

conf1a T 627 83 -- 161 1.8

conf1b F 682 176 2939 124 1.3

conf1c T 659 804 -- 156 2.1

conf1d F 706 1930 1473 148 2.2

conf2a T -- -- -- 438 65.9

conf2b F -- -- -- 275 56.9

conf3a T -- memout -- 653 5.2

conf3b F -- -- 2128 327 2.2

conf4 F -- -- -- 274 32.0

conf5 F 1018 427 142 11 0.1

7-9quantifier

levels

Duaffle (even without learning) on the dual encoding dramatically outperforms all leading CNF-based QBF solvers on these challenging instances

August 15, 2006 SAT 2006 32

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