1 / 24

Ten Challenges Redux : Recent Progress in Propositional Reasoning & Search A Biased Random Walk

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

emmanuel-parsons
Download Presentation

Ten Challenges Redux : Recent Progress in Propositional Reasoning & Search A Biased Random Walk

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Ten Challenges Redux: Recent Progress in Propositional Reasoning & SearchA Biased Random Walk Henry Kautz University of Washington

  2. 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

  3. 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

  4. 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

  5. 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

  6. 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?

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

  8. Pebbling Formulas • Structure similar to precedence graphs, planning graphs • No short proofs for DPLL (or even regular resolution) • Short clause learning proofs in all common schemes (t1 t2) (h1 h2) (i1 i2) (e1 e2) (f1 f2) (g1 g2) (a1 a2) (b1 b2) (c1 c2) (d1 d2)

  9. 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 literalyfrom B; delete it from B; if y already assigned, repeat”

  10. Results: Grid Pebbling Naive DPLL Original zChaff Modified zChaff

  11. 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?

  12. 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?

  13. 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)

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

  15. Challenge 6: Handle variable dependencies more efficiently in local search • Random walk – unit propagation in n2 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)

  16. 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?

  17. Complete or no knowledge • Complete knowledge: calculate fixed cutoff to minimize E(Rt) • No knowledge: universal sequence 1, 1, 2, 1, 1, 2, 4, … (Luby 1993) P(t) D T* t

  18. Observation horizon Short Long Median run time 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)

  19. 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

  20. 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!

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

  22. 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)

  23. 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.

  24. Score Card http://www.cs.washington.edu/homes/kautz/challenge

More Related