Ten Challenges Redux: Recent Progress in Propositional Reasoning & Search A Biased Random Walk Henry Kautz University of Washington

Challenge 1: Prove that a hard 700 variable random 3-SAT formula is unsatisfiable  1997  DPLL handles 400 variable random 3-SAT at 4.25 clause/variable ratio (Li & Anbulagan1996)  Walksat handles 10,000 variable satisfiable (Selman, Cohen, & Kautz 1996)  Limit of DPLL due to minimal proof tree size?  2001  “Backbone based” variable selection heuristic (Dubois & Dequen) extends DPLL to 700 variables

Backbone Based Heuristics  Backbone  Sat formulas: Set of variables that are fixed in all satisfying assignments  Unsat formulas: backbone of (some) max-sat subset  DPLL heuristic: branch on variables that are likely to be in the backbone  Identify using recursive version of MOM’s

Survey Propagation  2002 – Survey propagation – identify backbone variables and values for satisfiable random k-SAT (Mézard, Parisi, & Zecchina)  Linear scaling – 1,000,000+ variables at 4.25  Loopy belief propagation  Challenge 1’: Develop survey propagation techniques for other interesting problem distributions

Challenge 2: Solve the DIMACS 32-bit parity problem  Extend DPLL by detecting chains of equivalent literals  Pre-processing (Warner & van Maaren 1999)  During execution of DPLL (Li 2000)  Local search – clause re-weighting promising (Wu & Wah 1999)  Challenge 2’: Solve the 32-bit problem using local search

Proof Complexity: Beyond DPLL  DPLL < General Resolution < Frege Systems  Challenge 3A: Demonstrate that a proof system more powerful than tree-like resolution can be practical for satisfiability testing  Clause learning (GRASP: Marques-Silva & Sakallah 1996; Rel-Sat: Bayardo & Shrag 1997; SATO: Zhang 1997; Chaff: Moskewicz, Madigan, Zhao, Zhang, & Malik 2001)  Bounded model checking (Velev & Bryant 2001)  Alpha processor – 1M vars, 10M clauses (Bjesse, Leonard, & Mokkdem 2001)  What is formal power of clause learning?

Conflict Clauses [Beame, Kautz, Sabharwal ’03] FirstNewCut scheme (x 1  x 2  x 3 ) Grasp’s Decision scheme (p  q   b) zChaff’s 1-UIP scheme t  p p  q q b a  t t  x1 x1  x2 x2  x3 x3 y  y y false

Pebbling Formulas (a1  a2)(a1  a2)(b1  b2)(b1  b2)(c1  c2)(c1  c2)(d1  d2)(d1  d2) (e1  e2)(e1  e2) (h1  h2)(h1  h2) (t1  t2)(t1  t2) (i1  i2)(i1  i2) (g1  g2)(g1  g2)(f1  f2)(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 = (x 1, x 4,  x 3, x 1,  x 8,  x 2,  x 4, x 7,  x 1, x 2 ) 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 Original zChaff Modified zChaff Naive DPLL

Challenge 3B: Demonstrate that a proof system more powerful than general resolution can be made practical for satisfiability testing  Pigeon-hole problems – E. Coli of proof complexity  Detect & break symmetries (Krishnamurphy 1985; Crawford, Ginsberg, Luks & Roy 1996; Aloul, Markov, & Sakallah 2003)  If {A, B} is a symmetry, adding A  B preserves satisfiability  If (1, 0) is a model then so is (0, 1) – safe to kill (1,0)  Significant speed-up on real-world problems, but  Can only find “obvious” symmetries – NP-hard in general!  Additional clauses unwieldy – build into DPLL instead?

Formula Caching  New idea: cache residual formulas instead of learned clauses (Bacchus, Dalmao & Pitassi 2003; Beame, Impagliazzo, Pitassi, & Segerlind 2003)  Stronger than general resolution if check cache for subsumed formulas  But not for pigeons…  Best approach for model counting?

Challenge 4: Demonstrate that integer programming can be made practical for satisfiability testing  Cutting planes:  Great in theory, but so far not in practice  Promising: extend DPLL to pseudo-Boolean programming (Dixon & Ginsberg 2002)

Challenge 5: Design a practical local search procedure for proving unsatisfiability  Need: small witnesses!  Backdoor sets? (Williams, Gomes, & Selman 2003)

Challenge 6: Handle variable dependencies more efficiently in local search  Random walk – unit propagation in n 2 time (Papadimitriou 1995)  Walksat with unit-prop initialization (UnitWalk: Hirsch & Kojevnikov 2001; Qingting: Li, Stallman, & Brglez 2003)  Pre-process formula  Add clauses that capture long-range dependencies (Wei Wei & Selman 2002)

Challenge 7: Successfully combine stochastic & systematic search  Interleaved DPLL & local search (Maizure, Sais, & Gregoire 1996; Habet, Li, Devendeville, & Vasquez 2002)  Randomized restart DPLL (Gomes, Selman, & Kautz 1998)  Heavy tailed run-time distributions (Gomes, Selman, Crater, & Kautz 2000)  Issue: when to restart?

Complete or no knowledge P(t) t D T* Complete knowledge: calculate fixed cutoff to minimize E(R t ) 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 ) LongShort Observation horizon Median 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 properties of different encodings of real world domains  CSP versus SAT encodings (Walsh 1997; Prestwich 2003; van Beek & Dechter 1997)  Effect of logically redundant clauses on power of unit propagation (local consistency)  Planning as satisfiability (Kautz, McAllester, Selman 1996; Kautz & Selman 1999)  Much to be done!

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

Challenge 10: Create random problem generator for instances similar to real-world problems  Quasigroup completion problem (Gomes & Selman 1997; Kautz, Ruan, Achlioptas, Gomes, Selman, & Stickel 2001)

Bounded-Model Checking  Growing libraries of real-world instances  No one algorithm best for all – wide range of performance!  Challenge 10’: Relate the specific kinds of structures that appear in BMC problems to different solver techniques.

Score Card Solved2 Partially solved: 6 Completely open: 2