Ten Challenges Ten Challenges ReduxRedux: : Recent Progress in Propositional Recent Progress in Propositional

Reasoning & SearchReasoning & SearchA Biased Random WalkA Biased Random Walk

Henry KautzHenry Kautz

University of WashingtonUniversity of Washington

Challenge 1: Prove that a hard 700 variable Challenge 1: Prove that a hard 700 variable random 3-SAT formula is unsatisfiablerandom 3-SAT formula is unsatisfiable

1997 1997 DPLL handles 400 variable random 3-SAT at DPLL handles 400 variable random 3-SAT at

4.25 clause/variable ratio 4.25 clause/variable ratio (Li & Anbulagan1996)(Li & Anbulagan1996)

Walksat handles 10,000 variable satisfiable Walksat handles 10,000 variable satisfiable (Selman, Cohen, & Kautz 1996)(Selman, Cohen, & Kautz 1996)

Limit of DPLL due to minimal proof tree size?Limit of DPLL due to minimal proof tree size?

20012001 ““Backbone based” variable selection heuristic Backbone based” variable selection heuristic

(Dubois & Dequen)(Dubois & Dequen) extends DPLL to 700 variables extends DPLL to 700 variables

Backbone Based HeuristicsBackbone Based Heuristics

BackboneBackbone Sat formulas: Set of variables that are fixed in Sat formulas: Set of variables that are fixed in

all satisfying assignmentsall satisfying assignments Unsat formulas: backbone of (some) max-sat Unsat formulas: backbone of (some) max-sat

subsetsubset

DPLL heuristic: branch on variables that DPLL heuristic: branch on variables that are likely to be in the backboneare likely to be in the backbone Identify using recursive version of MOM’sIdentify using recursive version of MOM’s

Survey PropagationSurvey Propagation

2002 – Survey propagation – identify 2002 – Survey propagation – identify backbone variables and values for backbone variables and values for satisfiable random k-SAT satisfiable random k-SAT (M(Méézard, Parisi, & zard, Parisi, & Zecchina)Zecchina) Linear scaling – 1,000,000+ variables at 4.25Linear scaling – 1,000,000+ variables at 4.25 Loopy belief propagationLoopy belief propagation

Challenge 1’: Develop survey propagation Challenge 1’: Develop survey propagation techniques for other interesting problem techniques for other interesting problem distributionsdistributions

Challenge 2: Solve the DIMACS 32-bit Challenge 2: Solve the DIMACS 32-bit parity problemparity problem

Extend DPLL by detecting chains of Extend DPLL by detecting chains of equivalent literalsequivalent literals Pre-processing Pre-processing (Warner & van Maaren 1999)(Warner & van Maaren 1999)

During execution of DPLL During execution of DPLL (Li 2000)(Li 2000)

Local search – clause re-weighting Local search – clause re-weighting promising promising (Wu & Wah 1999)(Wu & Wah 1999)

Challenge 2’: Solve the 32-bit problem Challenge 2’: Solve the 32-bit problem using local searchusing local search

Proof Complexity: Beyond DPLLProof Complexity: Beyond DPLL

DPLL < General Resolution < Frege SystemsDPLL < General Resolution < Frege Systems Challenge 3A: Demonstrate that a proof system Challenge 3A: Demonstrate that a proof system

more powerful than tree-like resolution can be more powerful than tree-like resolution can be practical for satisfiability testingpractical for satisfiability testing

Clause learning Clause learning (GRASP: Marques-Silva & Sakallah 1996; (GRASP: Marques-Silva & Sakallah 1996; Rel-Sat: Bayardo & Shrag 1997; SATO: Zhang 1997; Chaff: Rel-Sat: Bayardo & Shrag 1997; SATO: Zhang 1997; Chaff: Moskewicz, Madigan, Zhao, Zhang, & Malik 2001)Moskewicz, Madigan, Zhao, Zhang, & Malik 2001) Bounded model checking Bounded model checking (Velev & Bryant 2001)(Velev & Bryant 2001)

Alpha processor – 1M vars, 10M clauses Alpha processor – 1M vars, 10M clauses (Bjesse, (Bjesse, Leonard, & Mokkdem 2001)Leonard, & Mokkdem 2001)

What is formal power of clause learning?What is formal power of clause learning?

Conflict Clauses

[Beame, Kautz, Sabharwal ’03]FirstNewCut scheme

(x1 x2 x3)

Grasp’sDecision scheme

(p q b)

zChaff’s1-UIP scheme

t

p

q

b

a

t

x1

x2

x3

y

y

false

Pebbling Formulas

(a1 a2) (b1 b2) (c1 c2) (d1 d2)

(e1 e2)

(h1 h2)

(t1 t2)

(i1 i2)

(g1 g2)(f1 f2)

•Structure similar to precedence graphs, planning graphs

•No short proofs for DPLL (or even regular resolution)

•Short clause learning proofs in all common schemes

Branching Sequence

• B = (x1, x4, x3, x1, x8, x2, x4, x7, x1, x2)

• Analysis: can generate domain-dependent “pebbling” branching sequence

OLD: “Pick unassigned var x” NEW: “Pick next literal y from B; delete it from B; if y already assigned, repeat”

Results: Grid Pebbling

Unsatisfiable Satisfiable

Learning OFF Branching Seq OFF

45 vars 55 vars

Learning OFF Branching Seq ON

45 vars 55 vars

Learning ON Branching Seq OFF

2,000 vars 4,500 vars

Learning ON Branching Seq ON

2,500,000 vars 1,000,000 vars

zChaff settings

Max formula size solved24 hours; 512 MB memory

OriginalzChaff

ModifiedzChaff

NaiveDPLL

Challenge 3B: Demonstrate that a proof system Challenge 3B: Demonstrate that a proof system more powerful than general resolution can be more powerful than general resolution can be

made practical for satisfiability testingmade practical for satisfiability testing

Pigeon-hole problems – Pigeon-hole problems – E. ColiE. Coli of proof of proof complexitycomplexity

Detect & break symmetries Detect & break symmetries (Krishnamurphy 1985; (Krishnamurphy 1985; Crawford, Ginsberg, Luks & Roy 1996; Aloul, Markov, & Sakallah Crawford, Ginsberg, Luks & Roy 1996; Aloul, Markov, & Sakallah 2003)2003)

If {A, B} is a symmetry, adding A If {A, B} is a symmetry, adding A B preserves B preserves satisfiabilitysatisfiability

If (1, 0) is a model then so is (0, 1) – safe to kill (1,0)If (1, 0) is a model then so is (0, 1) – safe to kill (1,0) Significant speed-up on real-world problems, butSignificant speed-up on real-world problems, but

Can only find “obvious” symmetries – NP-hard in general!Can only find “obvious” symmetries – NP-hard in general! Additional clauses unwieldy – build into DPLL instead?Additional clauses unwieldy – build into DPLL instead?

Formula CachingFormula Caching

New idea: cache residual formulas instead of New idea: cache residual formulas instead of learned clauses learned clauses (Bacchus, Dalmao & Pitassi 2003; (Bacchus, Dalmao & Pitassi 2003; Beame, Impagliazzo, Pitassi, & Segerlind 2003)Beame, Impagliazzo, Pitassi, & Segerlind 2003)

Stronger than general resolution if check cache for Stronger than general resolution if check cache for subsumed formulassubsumed formulas

But not for pigeons…But not for pigeons…

Best approach for model counting?Best approach for model counting?

Challenge 4: Demonstrate that integer Challenge 4: Demonstrate that integer programming can be made practical for programming can be made practical for

satisfiability testingsatisfiability testing Cutting planes:Cutting planes:

Great in theory, but so far not in practiceGreat in theory, but so far not in practice Promising: extend DPLL to pseudo-Boolean Promising: extend DPLL to pseudo-Boolean

programming programming (Dixon & Ginsberg 2002)(Dixon & Ginsberg 2002)

1 2

1 3

2 3

1 2 3

1 2 3

1

1

1

1.5

2

x x

x x

x x

x x x

x x x

Challenge 5: Design a practical local search Challenge 5: Design a practical local search procedure for proving unsatisfiabilityprocedure for proving unsatisfiability

Need: small witnesses!Need: small witnesses! Backdoor sets? Backdoor sets? (Williams, Gomes, & Selman 2003)(Williams, Gomes, & Selman 2003)

Challenge 6: Handle variable dependencies Challenge 6: Handle variable dependencies more efficiently in local searchmore efficiently in local search

Random walk – unit propagation in nRandom walk – unit propagation in n22 time time (Papadimitriou 1995)(Papadimitriou 1995)

Walksat with unit-prop initialization Walksat with unit-prop initialization (UnitWalk: Hirsch & Kojevnikov 2001; Qingting: Li, (UnitWalk: Hirsch & Kojevnikov 2001; Qingting: Li, Stallman, & Brglez 2003)Stallman, & Brglez 2003)

Pre-process formulaPre-process formula Add clauses that capture long-range Add clauses that capture long-range

dependencies dependencies (Wei Wei & Selman 2002)(Wei Wei & Selman 2002)

Challenge 7: Successfully combine Challenge 7: Successfully combine stochastic & systematic searchstochastic & systematic search

Interleaved DPLL & local search Interleaved DPLL & local search (Maizure, (Maizure, Sais, & Gregoire 1996; Habet, Li, Devendeville, & Sais, & Gregoire 1996; Habet, Li, Devendeville, & Vasquez 2002)Vasquez 2002)

Randomized restart DPLL Randomized restart DPLL (Gomes, Selman, & (Gomes, Selman, & Kautz 1998)Kautz 1998) Heavy tailed run-time distributions Heavy tailed run-time distributions (Gomes, (Gomes,

Selman, Crater, & Kautz 2000)Selman, Crater, & Kautz 2000)

Issue: when to restart?Issue: when to restart?

Complete or no knowledge

P(t)

t

D

T*

Complete knowledge: calculate fixed cutoff to minimize E(Rt)

No knowledge: universal sequence 1, 1, 2, 1, 1, 2, 4, … (Luby 1993)

Run-Time Observations Can predict a particular run’s time to

solution (very roughly) based on features of a solver’s trace during an initial window

Can improve time to solution by immediately pruning runs that are predicted to be long (Horvitz, Gomes, Kautz, Ruan, Selman 2000-2003)

LongLongShortShortObservation horizonObservation horizon

Median run timeMedian run time

Partial Knowledge

Can incorporate partial knowledge about an ensemble RTD by updating beliefs after each run

Example: You know RTD of a SAT ensemble and an UNSAT ensemble, but you don’t know which ensemble current problem is from

Challenge 8: Characterize the computational Challenge 8: Characterize the computational properties of different encodings of real properties of different encodings of real

world domainsworld domains

CSP versus SAT encodings CSP versus SAT encodings (Walsh 1997; (Walsh 1997; Prestwich 2003; van Beek & Dechter 1997)Prestwich 2003; van Beek & Dechter 1997)

Effect of logically redundant clauses on power Effect of logically redundant clauses on power of unit propagation (local consistency)of unit propagation (local consistency)

Planning as satisfiability Planning as satisfiability (Kautz, McAllester, (Kautz, McAllester, Selman 1996; Kautz & Selman 1999)Selman 1996; Kautz & Selman 1999)

Much to be done!Much to be done!

Challenge 9: Find encodings of real-world Challenge 9: Find encodings of real-world domains so that “near models” are near domains so that “near models” are near

solutionssolutions

Challenge 10: Create random problem Challenge 10: Create random problem generator for instances similar to real-world generator for instances similar to real-world

problemsproblems Quasigroup completion problem Quasigroup completion problem (Gomes & (Gomes &

Selman 1997; Kautz, Ruan, Achlioptas, Gomes, Selman, Selman 1997; Kautz, Ruan, Achlioptas, Gomes, Selman, & Stickel 2001)& Stickel 2001)

Bounded-Model CheckingBounded-Model Checking

Growing libraries of real-world instancesGrowing libraries of real-world instances No one algorithm best for all – wide range No one algorithm best for all – wide range

of performance!of performance! Challenge 10’: Relate the specific kinds of Challenge 10’: Relate the specific kinds of

structures that appear in BMC problems to structures that appear in BMC problems to different solver techniques.different solver techniques.

Score CardScore Card

SolvedSolved 22

Partially solved:Partially solved: 66

Completely open:Completely open: 22

http://www.cs.washington.edu/homes/kautz/challenge