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Faster Extraction of High-Level Minimal Unsatisfiable Cores

Faster Extraction of High-Level Minimal Unsatisfiable Cores. SAT’11 Conference Ann Arbor, USA June 21, 2011 . Ryvchin Vadim and Ofer Strichman Technion, Israel. Agenda. Introduction and motivation Optimizations A. Partial Resolution B. Selective clause minimization

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Faster Extraction of High-Level Minimal Unsatisfiable Cores

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  1. Faster Extraction of High-Level Minimal Unsatisfiable Cores SAT’11 ConferenceAnn Arbor, USA June 21, 2011 Ryvchin Vadim and Ofer Strichman Technion, Israel

  2. Agenda • Introduction and motivation • Optimizations • A. Partial Resolution • B. Selective clause minimization • C . Postponed IC-propagation • E. Selective learning of IC-clauses • G. Removal Strategy • Experimental results • Resolution vs. Selector variables

  3. High-Level UC • Given: • A set of interesting constraints (IC)  = { IC1, IC2 , …, ICm }, and • The remainder  • The set    is a high-level UC (HLUC) if    is unsatisfiable • HLUC is minimal (HLMUC) if removal of any IC makes    satisfiable

  4. Examples • Abstraction-refinement in model checking [MA’03] • Latches in core define the next abstraction. • Compositional Formal Equivalence Checking (FEC) [CGLNR’10] • Decompose the compared circuits to blocks. • Assume inputs to blocks are the same • Assumptions in the core need to be proved.

  5. Traditional UC Extraction:Stage 1: Translate to Clauses The remainder (the rest of the formula) An interesting constraint Each small square is a propositional clause, e.g. (a + b’)

  6. Traditional UC Extraction:Stage 2: Extract a Clause-Level UC The remainder (the rest of the formula) An interesting constraint Coloredsquares belong to the clause-level UC

  7. Traditional UC Extraction:Stage 3: Map UC back to ICs The remainder (the rest of the formula) An interesting constraint The UC contains three interesting constraints!

  8. A Mismatch between Mainstream Research and the Needs of Real-World Applications • Real-world applications: reduce # interesting constraints in the core • Latches/gates for abstraction refinement • Assumptions for compositional FEC • Vast majority ofexisting algorithms: reduce # of clauses in the core • 19/21 papers on UC extraction only consider clause-level UC extraction

  9. Small/Minimal Clause-Level UC  Small/Minimal High-Level UC A small clause-level UC, but the high-level UC is the largest possible: A large clause-level UC, but the high-level UC is empty:

  10. Resolution Refutation C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: Input clauses Derived clauses

  11. Resolution Refutation C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: Input clauses Derived clauses Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 } Unsat Core: { C4 C5 C6 C7 }

  12. Resolution Refutation C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: Empty Clause Cone Unsat Core Empty clause cone: { C4 C5 C6 C7 C13 C14 C19 C20 C23 } Unsat Core: { C4 C5 C6 C7 }

  13. Resolution with ICs C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: IC1 IC2 Remainder Input Clauses Derived clauses Derived Clauses

  14. Resolution with ICs C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: IC1 IC2 Remainder Input Clauses IC1 IC2 Remainder Derived Clauses

  15. HLUC C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: IC1 IC2 Remainder HLUC: { IC2 }

  16. HLMUC Algorithm [N’10] - remainder, - ICs Initialization: ’ =  =  Assumption:  Æis UNSAT Solve Æ UNSAT ’ = HLUC SAT • = ’ •  =  \ ICi  No unchecked ICs Remove one ICiϵ  that wasn’t already removed

  17. Contribution of this Work • Seven optimizations for single HLMUC. • improved run time and smaller HLMUC • Comparison between resolution and selector variables solvers.

  18. A. Partial Resolution • Observations: • IC-clauses usually between 5-15% of the problem clauses • We do not need the whole resolution table • Suggestion: • Keep only clauses relevant to IC resolutions • Result: • The size of the resolution graph reduced • Very effective on large CNFs

  19. A. Partial Resolution C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: IC1 IC2 Remainder

  20. A. Partial Resolution C22 C23=() C21 C17 C19 C20 C18 C16 C15 C11 C14 C13 C10 C12 C2 C4 C5 C8 C9 C1 C3 C6 C7 Legend: IC1 IC2 Not Needed

  21. A. Partial Resolution C22 C23=() C21 C17 C20 C16 C15 C10 C8 C1 C7 Legend: IC1 IC2

  22. A. Partial Resolution - Summary C22 C23=() C21 C17 C20 C16 C15 C10 C8 C1 C7 Legend: IC1 IC2 Keep only the needed resolutions

  23. B. Selective clause minimization • Technique for shrinking conflict clauses • The algorithm is based on traversing the resolution DAG backward from each literal in the learned clause • The problem: • May turn a non-IC-clause into a shorter IC-clause

  24. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6)

  25. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) c1 c2 v1 v2 v3

  26. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

  27. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

  28. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) 1-UIP based conflict analysis: c6= (¬v1Ǭv3 Ǭv4) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

  29. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) v1 v3 ¬v3 ¬v1

  30. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c6= (¬v1Ǭv3 Ǭv4) c6= (¬v1Ǭv4) v1 v3 ¬v3 ¬v1

  31. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) c6= (¬v1Ǭv4) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

  32. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3) (IC) c3= (¬v4Çv5) c4= (¬v5Çv6) c5= (¬v1Ǭv3 Ǭv4 Ǭv6) c1 c2 v1 v2 v3 c5 c5 ¬v6 c5 v4 c3 c4 v5 v6

  33. B. Selective clause minimization c1= (¬v1Çv2) c2= (¬v2Çv3)(IC) With minimization using c2: c6= (¬v1Ǭv4)(IC) Without minimization: c6= (¬v1Ǭv3 Ǭv4) (remainder)

  34. B. Selective clause minimization • Suggested solution: • Disable minimization if it adds dependency on IC-clause. • c6= (¬v1Ǭv3 Ǭv4) instead of c6= (¬v1Ǭv4) • Disabling minimization  reduces #derived IC-clauses  reduces #IC-clauses in UC and finds HLMUC faster

  35. C. Postpone IC-propagations • Change BCP order Run BCP Conflict Analyze Conflict no implications Next Operations

  36. C. Postpone IC-propagations • Change BCP order Run BCP over non IC-clause Conflict Analyze Conflict found implication no implications Propagate a single IC-clause Conflict no implications Next Operations

  37. C. Postpone IC-propagations • Increase chances to get conflicts in remainder • Decreases number of derived IC-clauses • Decreases number of IC-clauses in UC.

  38. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3

  39. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3

  40. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3 Learnt clause should be marked “IC-clause”

  41. E. Selective Learning • We refrain from learning IC-clauses • Instead, • do not learn it • learn a (non-asserting) remainder clause • make a decision • How ?

  42. E. Selective Learning • How ? • Treat the last IC-clause implication as decision • Perform new 1-UIP conflict analysis • The learnt clause is ‘remainder’

  43. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @5 @5 @5 @5 @5 X @5 @5 @5 @3 @3

  44. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @6 @5 @5 @5 @6 X @5 @5 @6 @3 @3

  45. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @6 @5 @5 @5 @6 X @5 @5 @6 @3 @3

  46. E. Selective Learning implication IC-clause implication @2 @2 @5 @5 @6 @5 @5 @5 @6 X @5 @5 @6 @3 @3

  47. G. Removal Strategy Recall: In each iteration one IC is chosen to be removed. Solve Æ UNSAT ’ = HLUC SAT • = ’ •  =  \ ICi  No unchecked ICs Remove one ICiϵ  that wasn’t already removed

  48. G. Removal Strategy • What is the effect of the removal order? • Which IC should we remove first?

  49. G. Removal Strategy • Criterion: #clauses in UC • Choose the one • that contains least clauses in UC • If UNSAT (not necessary), will converge faster • If UNSAT (not necessary), will likely allow further removals • that contains most clauses in UC • If SAT (necessary), clauses are added as ‘remainder’ fast

  50. Experimental Results • Benchmark Set: • Industrial set of problems from Intel • Average #clauses = 2,572,270 • Average #ICs = 3804 • Average #IC-clauses = 96568 (6% of #clauses) • Machines: • Intel® Xeon® 4Ghz 32Gb of memory

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