1. 1 Understanding the Power of Clause Learning Ashish Sabharwal, Paul Beame, Henry Kautz University of Washington, Seattle IJCAI ConferenceAug 14, 2003

2. 2 SAT: The Satisfiability Problem Given a CNF formula F, [e.g. F = ( a  b)  (  a  c)   a ] determine if F has a satisfying assignment Your favorite problem NO SAT EncoderSAT Solver YES + a solution CNF formula

3. 3 SAT Solver: DPLL Algorithm Best complete CNF (un)satisfiability algorithms in practice are extensions of the DPLL algorithm [Davis-Putnam 60] [Davis-Logeman-Loveland 62] Recursive backtrack search Search space pruning based on falsified clauses

4. 4 DPLL( F ) // Perform unit propagation while exists unit clause (y)  F F = F | y if F is empty, report satisfiable and halt if F contains the empty clause  return else choose a literal x DPLL( F | x ) DPLL( F |  x ) DPLL Algorithm Remove all clauses containing y Shrink all clauses containing  y

5. 5 Extending DPLL: Clause Learning Added conflict clauses –Capture reasons of conflicts –Obtained via unit propagations from known ones –Reduce future search by producing conflicts sooner When backtracking in DPLL, add new clauses corresponding to causes of failure of the search EBL [Kleer-Williams 87, Stallman-Sussman 77, Genesereth 84, Davis 84] CL [Bayardo-Schrag 97, MarquesSilva-Sakallah 96, Zhang 97, Moskewicz et al. 01, Zhang et al. 01]

6. 6 Conflict Graphs Our New Scheme: FirstNewCut scheme (x 1  x 2  x 3 ) Decision scheme (p  q   b) 1-UIP scheme t pp qq b a x1x1 x2x2 x3x3 y yy false tt Known Clauses (p  q  a) (  a   b   t) (t   x 1 ) (t   x 2 ) (t   x 3 ) (x 1  x 2  x 3  y) (x 2   y) Current decisions p = false q = false b = true

7. 7 Restarts Clause Learning (CL) algorithms can be restarted at any point [Baptista-Silva 00] –Unset all variables and start over –But retain all clauses learned so far Avoids getting stuck in one part of the search space Evidently adds power to CL

8. 8 CL Critical to Performance Best current SAT algorithms rely heavily on CL for good behavior on real world problems GRASP [MarquesSilva-Sakallah 96], SATO [H.Zhang 97] zChaff [Moskewicz et al. 01], Berkmin [Goldberg-Novikov 02] However,  No good understanding of its strengths and weaknesses  Not much insight on why it works when it does

9. 9 Our Contribution Mathematical framework for analyzing clause learning Characterization of its power in relation to well-studied topics in proof complexity theory Ways to improve solver performance based on formal analysis

10. 10 Proofs of Unsatisfiability When F is unsatisfiable, DPLL refutation of F is a proof of its unsatisfiability Size lower bound on proofs of F gives time lower bound on executions of DPLL( F ) Size upper bound on proofs of F gives potential for quick executions of such algorithms (with or w/o learning depending on the class of proofs) (with the best possible branching heuristic, the best learning scheme, etc.)

11. 11 What about Satisfiable Formulas? [Achlioptas-Beame-Molloy 01] DPLL Tree Satisfying assignment Unsatisfiable sub-formula Bounds on unsatisfiable formulas imply bounds on satisfiable ones!

12. 12 Proof System: Resolution F = ( a  b)  (  a  c)   a  (  b  c)  (a   c ) Unsatisfiable CNF formula c cc Proof size = 9 empty clause  (a  b)(a  b)(  a  c)(  b  c)(a  c) aa (b  c)

13. 13 Special Cases of Resolution Tree-like resolution –Graph of inferences forms a tree  DPLL Regular resolution –Variable can be resolved on only once on any path from input to empty clause  Directed acyclic graph analog of DPLL tree –Natural to not branch on a variable once it has been eliminated –Used in original DP [Davis-Putnam 60]

14. 14 Proof System Hierarchy DPLL = Tree-like Space of polynomial time solvable formulas Regular RES [Bonet et al. 00] General RES [Alekhnovich et al. 02] Frege systems … … Pigeonhole principle [Haken 85]

15. 15 Our Results DPLL = Tree-like Regular RES General RES CL w/o restarts Trivial RES = Learned clauses DPLL = Tree-like Regular RES General RES

16. 16 Thm1. CL can beat Regular RES Regular RES General RES Formula f Poly-size RES proof  Exp-size Regular proof Such formulas exist! GT n Ordering principle Peb Pebbling formulas [Alekhnovich et al. 02] Formula PT(f,  ) Poly-size CL proof Exp-size Regular proof Regular RES CL w/o restarts

17. 17 PT(f,  ) : Proof Trace Extension Start with unsatisfiable formula f with poly-size RES proof  PT(f,  ) contains All clauses of f For each derived clause Q=(a  b  c) in , –Trace variable t Q –New clauses (t Q   a), (t Q   b), (t Q   c) CL proof of PT(f,  ) works by branching negatively on t Q ’s in bottom up order of clauses of 

18. 18 PT(f,  ) : Proof Trace Extension (a  b  x)(c   x) Q  (a  b  c)  ………… Formula f RES proof   t Q  a  b  c x  x false Trace variable t Q New clauses (t Q   a) (t Q   b) (t Q   c) PT(f,  ) FirstNewCut (a  b  c)

19. 19 How hard is PT(f,  ) ? Easy for CL: by construction CL branches exactly once on each trace variable  # branches = size(  ) Hard for Regular RES: reduction argument  Fact 1: PT(f,  ) | TraceVars = true  f  Fact 2: If  is a Regular RES proof of g, then  | x is a Regular RES proof of g | x  Fact 3: f does not have small Regular proofs!

20. 20 Implications? DPLL algorithms w/o clause learning are hopeless for some structured formulas CL algorithms have potential for small proofs Can we use such analysis to harness this potential?

21. 21 Branching Sequence B = (x 1, x 4,  x 3, x 1,  x 8,  x 2,  x 4, x 7,  x 1 ) DPLL picks branching literals from B Repetitions allowed Different from “branching order” How “good” is B ? –Depends on backtracking process, learning scheme, etc

22. 22 From Analysis to Practice Maximum size of grid pebbling formulas solvable by zChaff (in 1 day) can be substantially increased! Results extend to general randomized pebbling formulas [Sabharwal-Beame-Kautz SAT 2003]

23. 23 Trivial RES  Simple ladder structure  Distinct variables resolved upon  Properties: o Tree-like o Regular o Linear Clauses learned by CL are those that can be derived by trivial RES Known clauses: (a  b  c), (  a  x), (  b  c), (  c  y) Derived clause: ( x  y) (b  c  x) (c  x) (a  b  c)(  a  x) (  b  c) (  c  y) (x  y)

24. 24 Our Results DPLL = Tree-like Regular RES CL w/o restarts Trivial RES = Learned clauses

25. 25 CL-- A Variant of CL Allow branching on variables whose value is already implied by unit propagation –Equivalently, allow branching on variables not appearing at all in the residual formula How can this possibly help? Can learn a conflict clause that will reduce search later!

26. 26 Error in Paper Incorrect Theorem 1 in the paper: “CL + restarts is equivalent to RES” Correct version: “CL-- + restarts is equivalent to RES”

27. 27 Thm2. CL-- + restarts = General RES General RES can simulate CL + restarts  CL learns by unit propagation  Infer learned clauses using RES derivation CL-- + restarts can simulate General RES  If RES proof resolves (A  x) and (B   x) to obtain C, branch and set all literals of A and B to false, learn C, restart  Eventually learn empty clause

28. 28 Our Results DPLL = Tree-like Regular RES General RES = CL-- + restarts CL w/o restarts Trivial RES = Learned clauses

29. 29 Summary Formal framework CL can be stronger than Regular RES –FirstNewCut learning scheme CL-- + restarts is equivalent to RES

30. 30 Open Problems Fill in complexity hierarchy gaps 1.Can CL efficiently simulate Regular RES? 2.Can CL + restarts efficiently simulate RES? From Analysis to Practice Can we use this framework to improve SAT solvers for other classes of structured formulas? E.g. Planning as satisfiability Bounded model checking