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1 Decision heuristics based on an Abstraction/Refinement model Ofer Strichman Roman Gershman An earlier version was presented in IBMs verification conference (Haifa, Oct. 05). Technion (HaifaSat)

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Technion 2 SAT solving Naïve point of view: Searches in the decision tree, prunes subspaces. Creates blocking clauses that do not allow the solver to choose the same path again. This point of view fails to explain why We can solve many formulas with 10 5 variables, We cannot solve other formulas with 10 3 variables

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Technion 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.

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Technion 4 Abstraction of models and formulas Model is an (over approximating) abstraction of M if: A QF formula is an (over-approximating) abstraction of F if: F !F ! or simply:

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Technion 5 Model refines if: Formula refines if: Refinement of models and formulas F !, ! or simply:

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Technion 6 Abstraction of formulas Now consider Binary Resolution: ( A _ x ) ^ ( B _ : x ) ! ( A _ B ) Conflict Clauses are derived through a process of Binary Resolution. over-approximates F !F !

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Technion 7 Resolution Graph O1O1 O2O2 O3O3 O4O4 O6O6 O7O7 i1i1 C-1 i2i2 i3i3 O5O5 C-2 C-3 i4i4 Binary DAG with intermediate and conflict clauses. Each node in the graph is an abstraction of its antecedents Collapsed DAG with multi-degree nodes C-1 C- 3 C-2 O1O1 O2O2 O3O3 O4O4 O5O5 O6O6 O7O7

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Technion 8 Conflict clauses Several modern Decision Heuristics are guided by the Conflict Clauses (e.g. Berkmin) Hence, we can analyze them with the Abstraction/Refinement model.

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Technion 9 Berkmins heuristic Push conflict clauses to a stack. Find the first unsatisfied clause and choose a variable from this clause. Secondary heuristic: VSIDS (Zchaff).

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

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Technion 11 Let F denote the original formula abstracts F ( F ! ) refines ( F !, ! ) Berkmin heuristic tail- first conflict clause Check of abstract assignment fails

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

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Technion 13 Choosing the next clause by traversing the resolution graph: a general scheme 1. Mark all sinks. 2. Choose an unresolved marked clause C (If there are none - exit) 3. process C // attempt to satisfy C. 4. Mark C s antecedents. Berkmin chooses a clause only if all its descendents are already satisfied.

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Technion 14 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. HaifaSats strategy: Do not store graph Be more abstraction-focused than Berkmin

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Technion 15 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. secondary strategy: the VMTF (Siege) heuristic. Gives us the first-layer approximation of the graph.

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Technion 16 CMTF 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-1 C-3 C-2 C-0

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Technion 17 News A new technique developed in IBM-Haifa allows to shrink the graph stored in memory by ~ an order of magnitude. So, assuming the graph is in memory, the search for new strategies is now open… A contention between conflict-driven and abstraction-driven.

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Technion 18 Decision Conflict Decision Level Time Activity Score C x =1 C1C1 C5C5 C4C4 C3C3 C2C2 time dedicated for refuting x =1 time dedicated for creating C T1(C) Time C is asserting T2(C)

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Technion 19 Given a clause: choose a variable. The activity score. The Activity of a variable v : Let V be the set of variables that were resolved on in T1(C) on the resolution path of C. Add +1 for each v 2 V in the interval T2(C) Occasionally, divide all scores. A cheap recursive computation embedded in the First-UIP scheme.

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Technion 20 Results 165 industrial hard benchmarks from previous SAT competitions. All heuristics (Berkmin, VSIDS) implemented inside HaifaSat for fair comparison. Comparing…TimeFails CMTFBerkmin10%12-25% RBSVSIDS23%20-30% Berkmin a little better than zChaff 2004

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Technion 21 (CMTF + RBS) Vs. Berkmin (both implemented inside HaifaSat)

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Technion 22 HaifaSat Vs. zChaff 2004

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Technion 23 Results –SAT05 (Industrial)

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Technion 24 Results –SAT05 (Industrial)

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Technion 25 Competition... Independently, very similar principles were discovered by Dershowitz, Hanna and Nadel [SAT05] Reached very similar conclusions Their black-box Eureka SAT solver took several first and second places in last years competition.

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Technion 26 What we explore now We are currently investigating the possibility to use Hints for dealing with hard instances Hints: a yet-unpublished work by Kroening, Yorav, Shacham. The original use of hints: using presumed high- level knowledge to speed SAT.

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Technion 27 Hints 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 only used for deriving more hints.

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Technion 28 Hints (cont.) Use hints to prune seemingly hopeless branches (SHB) Define a monotonically decreasing function f : decision-level time-interval If time at decision level dl > f ( dl ) move to another branch. It is seemingly hopeless. Actively: force backtracking Passively: wait for a restart

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Technion 29 Hints: Example A hint clause: ( : l 1 : l 2 : l 3 ). Option #1: hint = negation of partial assignment Option #2: hint = projection of #1 to literals that were used in the SHB. l1l1 l2l2 l3l3 : l 3 SHB

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Technion 30 Combining Hints and restarts Some heuristics to be tried: Change restart policy, using f. Local restarts: only from a certain decision level …

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Technion 31 Hints and restarts A restart is effective because of randomization and/or learning. Hints will add new conjectures. Bart Selman: You can not restart too much Perhaps now: You can not hint too much

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Technion 32 Decision Conflict Decision Level Time work invested in refuting x =1 (some of it seems wasted) C x =1 Refutation of x =1 C1C1 C5C5 C4C4 C3C3 C2C2 Activity Score

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Technion 33 Conflict Decision C1C1 Level Time x =1 Refutation of x =1 C5C5 C4C4 C3C3 C2C2 C C5C5 C2C2 C4C4 Weight is given to variables resolved-on in the process of resolving C C Activity Score C0C0

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Technion 34 Results (sec., average) Benchmark(#)Berkmin+VSIDSCMTF+RBS Hanoi(5)530130 IP(4)395203 Hanoi03(4)1342426 Check-int(4)3323681 Bmc2(6)10301261 Fifo8(4)39441832 Fvp2(22)86381995 W08(3)53472680 Ibm02(9)97103875 01_rule(20)3364219171 11_rule_2(20)3400622974

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Technion 35 Refinement-driven Vs. Conflict- driven search. O-1O-2O-3O-4O-5O-6O-7O-8 W(c) = ci 2 antecedents(c) W(c i ) How should we balance between refinement-driven and conflict-driven strategies ?

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