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An Efficient and Flexible Approach to Resolution Proof Reduction N. Sharygina Formal Verification and Security Group University of Lugano March 9, 2011 Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 1 / 60 Outline 1

  1. Related Work Features • Post-processing approach • SAT/SMT solving framework • Compression techniques • Clauses subsumption checking [Amjad07] • Proof reordering based on literals linking [Amjad07] • Proof reordering based on variable splitting [Cotton10] • Merging of shared substructures in subproofs [Sinz07] • Memoization of shared substructures [Amjad08,Cotton10] • Algebraic approach, resolution hypergraphs [Fontaine10] • Removal pivots redundancies along paths [Bar-Ilan08] Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 12 / 60

  2. Notation Resolution System • Literal p p Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

  3. Notation Resolution System • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . • Empty clause ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

  4. Notation Resolution System • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . • Empty clause ⊥ pC pD • Resolution rule p CD Antecedent Resolvent Pivot Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

  5. Notation Resolution System • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . • Empty clause ⊥ pC pD • Resolution rule p CD Antecedent Resolvent Pivot • Resolution proof of unsatisfiability of a set of clauses S Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

  6. Notation Resolution System • Literal p p • Clause p ∨ q ∨ r ∨ . . . → pqr . . . • Empty clause ⊥ pC pD • Resolution rule p CD Antecedent Resolvent Pivot • Resolution proof of unsatisfiability of a set of clauses S • Tree • Leaves as clauses of S • Intermediate nodes as resolvents • Root as unique empty clause Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 13 / 60

  7. Resolution Proofs Example • Set of clauses { pq , pq , qr , qr } • Proof of unsatisfiability pq pq qr qr p r q q q ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 14 / 60

  8. Pivots Redundancies [Bar-Ilan08] • No need to resolve more than once on a pivot in a path leaf-root Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

  9. Pivots Redundancies [Bar-Ilan08] • No need to resolve more than once on a pivot in a path leaf-root • O.Bar-Ilan, O.Fuhrmann, S.Hoory, O.Shacham and O.Strichman: RecyclePivots Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

  10. Pivots Redundancies [Bar-Ilan08] • No need to resolve more than once on a pivot in a path leaf-root • O.Bar-Ilan, O.Fuhrmann, S.Hoory, O.Shacham and O.Strichman: RecyclePivots • Perform DFS from root to leaves • Track pivots occurrences along paths • In case of multiple occurrences keep the closest one to root • Output regular proof Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 15 / 60

  11. RecyclePivots Example pq po p qr pq qo pq q q po pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 16 / 60

  12. RecyclePivots Example pq po p qr pq qo pq q q po pr p or o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 17 / 60

  13. RecyclePivots Example pq po p qr pq qo pq q q po pr p or { r , o } o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 18 / 60

  14. RecyclePivots Example pq po p qo pq qr pq q q po { r , o , p } pr p or { r , o } o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 19 / 60

  15. RecyclePivots Example pq po p qo { r , o , p , q } pq qr pq q q po { r , o , p } pr p or { r , o } o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 20 / 60

  16. RecyclePivots Example pq po p qo { r , o , p , q } pq qr pq q q po { r , o , p } pr p or { r , o } o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 21 / 60

  17. RecyclePivots Example pq po p qo { r , o , p , q } pq qr pq q q po { r , o , p } pr p or { r , o } o o r { r } r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 22 / 60

  18. RecyclePivots Example qr pq pq pq q q po pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 23 / 60

  19. RecyclePivots Example qr pq pq pq q q p pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 24 / 60

  20. RecyclePivots Example qr pq pq pq q q p pr p r o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 25 / 60

  21. RecyclePivots Example qr pq pq pq q q p pr p r o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 26 / 60

  22. RecyclePivots Example qr pq pq pq q q p pr p r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 27 / 60

  23. Outline 1 Background 2 Motivation and Related Work 3 Contribution Proof Reduction Framework Implementation and Evaluation 4 Summary and Future Work Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 28 / 60

  24. Outline 1 Background 2 Motivation and Related Work 3 Contribution Proof Reduction Framework Implementation and Evaluation 4 Summary and Future Work Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 29 / 60

  25. Transformation Framework Features • Local rewriting rules Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

  26. Transformation Framework Features • Local rewriting rules • B reduction • A perturbation Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

  27. Transformation Framework Features • Local rewriting rules • B reduction • A perturbation • Rule context pqC pD p qCD qE q CDE Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

  28. Transformation Framework Features • Local rewriting rules • B reduction • A perturbation • Rule context pqC pD p qCD qE q CDE • Exhaustiveness up to symmetry Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 30 / 60

  29. Transformation Framework Local rewriting rules • B rules pqC pqD p pqC pqE B 1 ⇒ q qCD pqE q pCE pCDE Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

  30. Transformation Framework Local rewriting rules • B rules pqC pqD p pqC pqE B 1 ⇒ q qCD pqE q pCE pCDE • Redundancy as reintroduction variable after elimination Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

  31. Transformation Framework Local rewriting rules • B rules pqC pqD p pqC pqE B 1 ⇒ q qCD pqE q pCE pCDE • Redundancy as reintroduction variable after elimination • Subproof simplification Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

  32. Transformation Framework Local rewriting rules • B rules pqC pqD p pqC pqE B 1 ⇒ q qCD pqE q pCE pCDE • Redundancy as reintroduction variable after elimination • Subproof simplification • Subproof root strengthening Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 31 / 60

  33. Transformation Framework Local rewriting rules • A rules pqC pD pqC qE p q A 2 ⇒ qCD qE q pCE pD p CDE CDE Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

  34. Transformation Framework Local rewriting rules • A rules pqC pD pqC qE p q A 2 ⇒ qCD qE q pCE pD p CDE CDE • Pivots swapping Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

  35. Transformation Framework Local rewriting rules • A rules pqC pD pqC qE p q A 2 ⇒ qCD qE q pCE pD p CDE CDE • Pivots swapping • Topology perturbation Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

  36. Transformation Framework Local rewriting rules • A rules pqC pD pqC qE p q A 2 ⇒ qCD qE q pCE pD p CDE CDE • Pivots swapping • Topology perturbation • Redundancies exposure Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 32 / 60

  37. Local rewriting rules pqC pqD pqC qE qE pqD p q A 1 ⇒ qCD qE pCE pDE q p CDE CDE pqC pD pqC qE p q A 2 ⇒ qCD qE q pCE pD p CDE CDE pqC pqD p pqC pqE B 1 ⇒ q qCD pqE q pCE pCDE pqC pD pqC pqE p q B 2 ⇒ qDC pqE pCE pD q p pCDE CDE pqC pD p pqC pqE B 2 ′ ⇒ q qDC pqE q pCE pCDE pqC pD p B 3 ⇒ qCD pqE pD q pCDE Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 33 / 60

  38. Rule-based Approach Example pq po p qr pq qo pq q q po pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 34 / 60

  39. Rule-based Approach Example pq po p qr pq qo pq q q po pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 35 / 60

  40. Rule-based Approach Example qr pq pq pq q q p pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 36 / 60

  41. Rule-based Approach Example qr pq pq pq q q p pr p or o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 37 / 60

  42. Rule-based Approach Example qr pq pq pq q q p pr p r o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 38 / 60

  43. Rule-based Approach Example qr pq pq pq q q p pr p r o o r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 39 / 60

  44. Rule-based Approach Example qr pq pq pq q q p pr p r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 40 / 60

  45. Rule-based Approach Example qr pq pq pq q q p pr p r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 41 / 60

  46. Rule-based Approach Example pq pq q p pq p qr q q r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 42 / 60

  47. Rule-based Approach Example pq pq q p pq p qr q q r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 43 / 60

  48. Rule-based Approach Example pq pq q p pq p qr q q r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 44 / 60

  49. Rule-based Approach Example pq pq p qr q q r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 45 / 60

  50. Rule-based Approach Example pq pq p qr q q r r r ⊥ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 46 / 60

  51. Comparison • RecyclePivots Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

  52. Comparison • RecyclePivots • Pros Global information Fast and effective • Cons Cannot expose redundancies Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

  53. Comparison • RecyclePivots • Pros Global information Fast and effective • Cons Cannot expose redundancies • Rule-based approach Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

  54. Comparison • RecyclePivots • Pros Global information Fast and effective • Cons Cannot expose redundancies • Rule-based approach • Pros Flexibility in rules application Flexibility in amount of transformation Can expose redundancies • Cons Local information Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 47 / 60

  55. Outline 1 Background 2 Motivation and Related Work 3 Contribution Proof Reduction Framework Implementation and Evaluation 4 Summary and Future Work Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 48 / 60

  56. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  57. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) • Parameterized in number of traversals and time limit Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  58. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) • Parameterized in number of traversals and time limit • Examination non-leaf clauses Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  59. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) • Parameterized in number of traversals and time limit • Examination non-leaf clauses • Pivot in both antecedents → update, match context, apply rule pqC ′ pD ′ qC ′ D ′ qE ′ p qC ′ D ′ qE ′ ⇒ ⇒ q q qC ′ D ′ qE ′ CDE C ′ D ′ E ′ q C ′ D ′ E ′ Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  60. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) • Parameterized in number of traversals and time limit • Examination non-leaf clauses • Pivot in both antecedents → update, match context, apply rule pqC ′ pD ′ qC ′ D ′ qE ′ p qC ′ D ′ qE ′ ⇒ ⇒ q q qC ′ D ′ qE ′ CDE C ′ D ′ E ′ q C ′ D ′ E ′ • Pivot not in both antecedents → remove resolution step C ′ D ′ qE ′ ⇒ q C ′ D ′ CDE Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  61. Implementation A Simple Algorithm • Based on a sequence of proof traversals (e.g. topological order) • Parameterized in number of traversals and time limit • Examination non-leaf clauses • Pivot in both antecedents → update, match context, apply rule pqC ′ pD ′ qC ′ D ′ qE ′ p qC ′ D ′ qE ′ ⇒ ⇒ q q qC ′ D ′ qE ′ CDE C ′ D ′ E ′ q C ′ D ′ E ′ • Pivot not in both antecedents → remove resolution step C ′ D ′ qE ′ ⇒ q C ′ D ′ CDE • Easy combination with RecyclePivots Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 49 / 60

  62. Evaluation Framework and Benchmarks • Implemented in C++ and integrated with OpenSMT • Available at www.inf.usi.ch/phd/rollini/hvc.html Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 50 / 60

  63. Evaluation Framework and Benchmarks • Implemented in C++ and integrated with OpenSMT • Available at www.inf.usi.ch/phd/rollini/hvc.html • Benchmarks Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 50 / 60

  64. Evaluation Framework and Benchmarks • Implemented in C++ and integrated with OpenSMT • Available at www.inf.usi.ch/phd/rollini/hvc.html • Benchmarks • SMT: SMT-LIB library • SAT: SAT competition • Academic and industrial problems Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 50 / 60

  65. Combined Approach Evaluation Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL # Avg nodes Avg edges Avg core T ( s ) Max nodes Max edges Max core RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 51 / 60

  66. Combined Approach Evaluation Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL # Avg nodes Avg edges Avg core T ( s ) Max nodes Max edges Max core RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 52 / 60

  67. Combined Approach Evaluation Experimental results over SMT: QF UF, QF IDL, QF LRA, QF RDL # Avg nodes Avg edges Avg core T ( s ) Max nodes Max edges Max core RP 1370 6.7% 7.5% 1.3% 1.7 65.1% 68.9% 39.1% Ratio 0.01 1366 8.9% 10.7% 1.4% 3.4 66.3% 70.2% 45.7% 0.025 1366 9.8% 11.9% 1.5% 3.6 77.2% 79.9% 45.7% 0.05 1366 10.7% 13.0% 1.6% 4.1 78.5% 81.2% 45.7% 0.075 1366 11.4% 13.8% 1.7% 4.5 78.5% 81.2% 45.7% 0.1 1364 11.8% 14.4% 1.7% 5.0 78.8% 83.6% 45.7% 0.25 1359 13.6% 16.6% 1.9% 7.6 79.6% 84.4% 45.7% 0.5 1348 15.0% 18.4% 2.0% 11.5 79.1% 85.2% 45.7% 0.75 1341 16.0% 19.5% 2.1% 15.1 79.9% 86.1% 45.7% 1 1337 16.7% 20.4% 2.2% 18.8 79.9% 86.1% 45.7% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 53 / 60

  68. Combined Approach Evaluation Experimental results over SAT # Avg nodes Avg edges Avg core T ( s ) Max nodes Max edges Max core RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 54 / 60

  69. Combined Approach Evaluation Experimental results over SAT # T ( s ) Avg nodes Avg edges Avg core Max nodes Max edges Max core RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 55 / 60

  70. Combined Approach Evaluation Experimental results over SAT # T ( s ) Avg nodes Avg edges Avg core Max nodes Max edges Max core RP 25 5.9% 6.5% 1.7% 10.8 33.1% 33.4% 30.3% Ratio 0.01 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.025 25 6.8% 7.9% 1.7% 32.3 34.0% 34.4% 30.5% 0.05 25 7.0% 8.2% 1.8% 40.0 34.0% 34.4% 30.5% 0.075 25 7.2% 8.4% 1.8% 49.3 34.7% 35.1% 30.5% 0.1 25 7.3% 8.4% 1.8% 60.2 34.7% 35.1% 30.5% 0.25 25 7.6% 8.8% 1.9% 125.3 39.8% 40.6% 31.7% 0.5 25 7.8% 9.1% 1.9% 243.5 41.0% 41.9% 32.1% 0.75 25 7.9% 9.3% 1.9% 360.0 41.6% 42.6% 32.1% 1 23 8.4% 9.9% 2.1% 175.6 33.1% 33.4% 30.6% • Ratio — time threshold as fraction of solving time • # — number of benchmarks solved • Avg nodes , Avg edges , Avg core — average reduction in proof size • T ( s ) — average transformation time in seconds • Max nodes , Max edges , Max core — max reduction in proof size Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 56 / 60

  71. Outline 1 Background 2 Motivation and Related Work 3 Contribution Proof Reduction Framework Implementation and Evaluation 4 Summary and Future Work Natasha Sharygina (USI) Flexible Proof Reduction March 9, 2011 57 / 60

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