1. Preprocessing in Incremental SAT Alexander Nadel 1, Vadim Ryvchin 1,2, and Ofer Strichman 2 1 – Intel, Haifa, Israel 2 – Technion, Haifa, Israel SAT’12, Trento, Italy

2. Introduction  Incremental SAT is at the core of a variety of applications  Preprocessing is essential for SAT’s performance … but is not compatible with incremental SAT  We propose a way to make them compatible thus increasing the capacity of SAT-based tools and algorithms

3. Incremental SAT

4. SatELite Preprocessing

5. The Problem

6. Previous Approaches 1. No preprocessing at all 2. Full preprocessing After each invocation return the eliminated clauses Redo preprocessing 3. Look Ahead: Do not eliminate “problematic” variables  Preprocessing is essential for performance  In many cases look ahead is not possible

7. Usages  Interactive communication with the user  Calculation of the next portion of the problem depends on previous invocation  In Intel, the conversion of BMC problems to CNF is done after applying some optimizations at the circuit level. Those optimizations cannot predict which variables might reappear in future calls

8. Definitions

9. Incremental Preprocessing Preprocess Solve // might run in-process

10. Incremental Preprocessing For each eliminated variable in elimination order: if (*) // decide heuristically Re-Eliminate else Re-Introduce Preprocess Solve // might run in-process

11. Incremental Preprocessing Remove subsumptions. For each eliminated variable in elimination order: if (*) // decide heuristically Re-Eliminate else Re-Introduce Preprocess Solve // might run in-process

12. Incremental Preprocess

13. Data structure for Eliminated Variables v1v1 v2v2 v3v3 v4v4 v5v5 … S1S1 S2S2 S3S3 S4S4 S5S5 Clauses containing v 3 but not v 1 nor v 2 may contain v 4, v 5, …

14. Start v1v1 v2v2 v3v3 v4v4 v5v5 … S1S1 S2S2 S3S3 S4S4 S5S5

15. Step 1 – Examine v 1 v1v1 v2v2 v3v3 v4v4 v5v5 … S1S1 S2S2 S3S3 S4S4 S5S5

16. Step 1 – Re-Eliminate v 1 v1v1 v2v2 v3v3 v4v4 v5v5 … S2S2 S3S3 S4S4 S5S5

17. Step 2 v1v1 v2v2 v3v3 v4v4 v5v5 … S2S2 S3S3 S4S4 S5S5

18. Step 2 – Examine v 2 v1v1 v2v2 v3v3 v4v4 v5v5 … S2S2 S3S3 S4S4 S5S5

19. Step 2 – Re-Eliminate v 2 v1v1 v2v2 v3v3 v4v4 v5v5 … S3S3 S4S4 S5S5

20. Step 3 v1v1 v2v2 v3v3 v4v4 v5v5 … S3S3 S4S4 S5S5

21. Step 3 – Examine v 3 v1v1 v2v2 v3v3 v4v4 v5v5 … S3S3 S4S4 S5S5

22. Step 3 – Re-Introduce v 3 v1v1 v2v2 v3v3 v4v4 v5v5 … S3S3 S4S4 S5S5 S3S3 S3S3

23. v1v1 v2v2 v4v4 v5v5 … S4S4 S5S5 S3S3 S3S3 May contain v 4 and/or v 5

24. Our Example

25. Our Example – Re-Elimination

29. Our Example

30. Our Example – Re-Introduction

31. c 1 c 2

32. Our Example – Re-Introduction

34. Additional Improvements

35. Removal of Resolvents

36. Example

37. Assumptions  How to deal with assumption variables?  Regular preprocessing: Freeze them (do not eliminate these variables).  Incremental preprocessing: Freeze them (the same). But what if variable was eliminated before? o Must Re-Introduce.  What if there are a lot of assumptions Our previous talk

38. Experimental Results  Benchmark Set: Implementation on top of Fiver (a new SAT solver under development at Intel). Large Bounded Model-checking instances from Intel divided into 4 different families. 206 different instances Timeout: 4000sec  Machines: Intel ® Xeon ® 4Ghz 32Gb of memory

39. Experimental Results

40. MethodTime-outsAvg. Total run-time Full-preprocessing682465.5 No-preprocessing421784.7 Incremental-preprocessing21221.3 Look-ahead01064.9

41. Incremental vs. No preprocessing

42. Incremental vs. Full preprocessing

43. Incremental vs. Look-ahead

44. Thank You!

45. Our Example ϕ 0 = c 1 c 2 : c 1 = (a v b) c 2 = (¬a v c) After preprocess ϕ 0 = c 3 = (b v c) Incremental call: ∆ 1 = c 4 c 5 c 4 = (¬b) c 5 =(¬a) ------------------------------------------------------------------------------- Incremental Preprocess: -Re-Eliminate: a Addition of: c 6 = (b) because of (c 1 x c 5 ) Solve  UNSAT

46. Our Example ϕ 0 = c 1 c 2 : c 1 = (a v b) c 2 = (¬a v c) After preprocess ϕ 0 = c 3 = (b v c) Incremental call: ∆ 1 = c 4 c 5 c 4 = (¬b) c 5 =(¬a) ------------------------------------------------------------------------------- Incremental Preprocess: -Re-Introduce: a Return c 1 and c 2 to formula Solve  UNSAT