Decision heuristics based on an Abstraction/Refinement model

1. Decision heuristics based on an Abstraction/Refinement model (HaifaSat) Ofer Strichman Roman Gershman Technion

2. SAT solving • “Naïve” point of view: • Searches in the decision tree, prunes subspaces. • Creates “blocking clauses” that restrain the solver from choosing the same bad path again. • This point of view fails to explain why • We can solve many formulas with 105 variables, • We cannot solve other formulas with 103 variables

3. A different point of view • Modern solvers act as proof engines based on resolution, rather than as search engines, with structured problems. • Evidence: adding the shortest conflict clauses is not the best strategy [R04]. • Furthermore: certain strategies resemble a proof by abstraction-refinement.

4. F! or simply: Abstraction of models and formulas • Model is an (over approximating) abstraction of M if: A degenerated case: • Formula is an (over-approximation) abstraction of F if:

5. over-approximates Abstraction of formulas • Now consider Binary Resolution:(AÇ x) Æ (BÇ:x) ! (AÇB) F!

6. Collapsed DAG with multi-degree nodes C-3 C-3 i4 c-2 O7 C-2 C-1 i3 c-1 O6 i2 O5 O7 i1 O1 O2 O5 O3 O4 O6 O4 O3 O1 O2 Resolution Graph Binary DAG with intermediate and conflict clauses. Each node in the graph is an abstraction of its descendants

7. F! , ! or simply: Refinement of models and formulas • An intermediate model is a refinement of if: • An intermediate formula is a refinement of if:

8. Why all this theory? … • Because Conflict Clauses are derived through a process of resolution. • Several modern Decision Heuristics are guided by the Conflict Clauses (e.g. Berkmin) • Hence, we can analyze them with the Abstraction/Refinement model.

9. Berkmin’s heuristic • Push conflict clauses to a stack. • Find the first unsatisfied clause and choose a variable from this clause. • If all conflict clauses are satisfied, choose a variable according to the VSIDS (Zchaff) heuristic.

10. Berkmin heuristic tail-first conflict clause A new conflict clause

11. Berkmin heuristic • Letφ denote the original formula • F abstracts φ (φ!F ) • is a refinement of F with respect toφ(φ! , ! F ) tail-first conflict clause F Check of abstract assignment fails

12. Berkmin heuristic • Does not focus on a specific Abstraction/Refinement path. • Generally: hundreds of clauses can be between a clause and its resolving clauses. C-3 C-2 C-1

13. A General Heuristic for choosing the next clause • Mark all roots. • Choose an unresolved marked clause V (If there are none - exit) • Decide a variable from V until it is satisfied. • Mark V’s children

14. The Clause-Move-To-Front (CMTF) heuristic • Is an instantiation of the general heuristic • Does not need to store the whole graph. • More focused than Berkmin.

15. Progressing on the resolve graph • Progress with “Best-First” according to some criterion. • Must store the whole resolve graph in memory – this is frequently infeasible. • HaifaSat’s strategy: • Do not store graph • Be more abstraction-focused than Berkmin

16. The CMTF heuristic • Position conflict clauses together with their resolving clauses in the end of a list. • Find the first unsatisfied clause and choose a variable from this clause. • If all conflict clauses are satisfied, choose a variable according to the VMTF (Siege) heuristic. Gives us the ‘first-layer approximation’ of the graph.

17. CMTF C-3 • When C-3 is created, C-0, C-1 are moved to the head of the list together with C-3. • C-2 is left in place. C-2 C-1 C-0

18. Given a clause: choose a variable. • The Activity of a variable v: • Activity score of a variable increases when it is a resolution variable, but… • only when the clause it helped resolving is currently relevant, and… • it happened recently • A recursive computation embedded in the First-UIP scheme.

19. Activity Score work invested in refutingx=1 (some of it seems wasted) C x=1 Refutation ofx=1 C5 C2 Decision Level C1 C4 C3 Decision Time Conflict

20. Activity Score C4 Weight is given to variables resolved-on in the process of resolving C C5 C C2 C0 C x=1 Refutation ofx=1 C5 C2 Decision Level C1 C4 C3 Decision Time Conflict

21. Results (sec., average)

22. (CMTF + RBS) Vs. Berkmin(both implemented inside HaifaSat)

23. HaifaSat Vs. zChaff 2004

24. Results –SAT05 (Industrial)

25. Results –SAT05 (Industrial)

26. Competition... • Independently, very similar principles were discovered by Dershowitz, Hana and Nadel [SAT’05] • Reached very similar conclusions • Their ‘black-box’ Eureka SAT solver took several first and second places in last year’s competition.

27. And now... • Two research directions • Better refinement strategies. • Hints.

28. Recall the general framework: • Mark all roots. • Choose an unresolved marked clause V (If there are none - exit) • Decide a variable from V until it is satisfied. • Mark V’s children

29. But, • HaifaSat does not really traverse the resolution graph. • The assumption is: the graph is too large to store in memory. • But, there are news: • A new technique developed in IBM-Haifa allows to shrink the graph stored in memory by two orders of magnitude. • The search for new refinement strategies is now open...

30. Refinement-driven Vs. Conflict-driven search. W(c) = ci 2 antecedents(c) W(ci) How should we balance between refinement-driven and conflict-driven strategies ? O-1 O-2 O-3 O-4 O-5 O-6 O-7 O-8

31. Hints • An (unpublished) idea by (Kroening, Yorav, Shacham) • Hints are constraints (clauses) that are conjectured to be true. • A separate BCP processes the set of conjectured clauses. • An implied literal becomes the next decision. • A conflict is ignored.

32. Hints (cont.) • The original use of hints: high-level knowledge. • We suggest: prune ‘seemingly hopeless branches (SHB)’ • Define a monotonically decreasing function f: decision-level  time-interval • If time at decision level dl > f(dl) prune the branch. This branch is seemingly hopeless.

33. Hints: Example l1 l2 l3 : l3 • A hint clause: (:l1:l2:l3) • Perhaps a better idea: • Keep track on which subset S of l1 ... l3 were used in the SHB. • The negation of literals in S is a better hint. SHB

34. Hints vs. restarts • Not entirely orthogonal techniques. • A restart is effective because of randomization and/or learning. • Hints are more directed: they push the solver away from seemingly hopeless branches. • Also: it is activated due to local consideration, and not a global clock. • Bart Selman: “You can not restart too much” • Perhaps now: “You can not hint too much”