Unsatisfiability Proofs for Parallel SAT Solver Portfolios with Clause Sharing and Inprocessing
- 1. Unsatisfiability Proofs for Parallel SAT Solver Portfolios with Clause Sharing and Inprocessing Tobias Philipp International Center for Computational Logic Technische Universität Dresden
- 3. The Portfolio Approach F0 Solver1 F0 Solver2 F0 Solver3 F0 Solver4 F0 F1 F2 F3 F4 SAT 1
- 4. The Cooperative Portfolio Approach F0 Solver1 F0 Solver2 F0 Solver3 F0 Solver4 F0 F1 F2 F3 F4 SAT 2
- 5. The Cooperative Portfolio Approach F0 Solver1 F0 Solver2 F0 Solver3 F0 Solver4 F0 F1 F2 F3 F4 SAT SAT 2
- 6. Inprocessing in Clause Sharing SAT Solvers ppfolio Roussel, 2012 ManySAT Hamadi et al, 2009 PLingeling Biere, 2010 PLingeling Biere, 2013 Inprocessing Clause Sharing 3
- 8. Asymmetric Tautologies Asymmetric literal addition alaF (C) = C ∪ {L | {L1, . . . , Ln, L} ∈ F and {L1, . . . , Ln} ⊆ C} Example F = {{p, q}, {p, ¬q, r}, {¬r, ¬q}} alaF ({p}) = {p, ¬q} C is an asymmetric tautology if there is n ∈ N st alan F (C) is a tautology Lemma 1. F ≡ F ∪ {C}, if C is an AT wrt F 2. {L1, . . . , Ln} is an AT wrt F iff F ∪ {L1} ∪ . . . ∪ {Ln} UP ⊥ 3. Linear resolvents are AT 4. CDCL-learned clauses are AT 4
- 9. Resolution Asymmetric Tautology Järvisalo et al.: Inprocessing Rules. In: IJCAR (2012). C is a RAT upon L wrt F, if C is an AT wrt F, or L ∈ C res(C, D, L) is an AT wrt F for every D ∈ F with ¬L ∈ D. {p}, {¬q} are RATS wrt F: 1. {p} p is pure 2. {¬q} one resolvent {p} that is an AT Lemma 1. F ≡sat F ∪ {C}, if C is RAT in F 2. All known formula simplifications can be characterized as RATs 5
- 10. Portfolios can be Described as State Transition Systems State transition system (∆, ;) ∆ is the set of states ; ⊆ ∆ × ∆ is the state transition relation. Local state for Solveri : working formula Fi melted literals Mi State ((M1, F1), . . . , (Mn, Fn)), SAT, UNSAT Initial state for init(F0, n) = ((∅, F0), . . . , (∅, F0)) Final states SAT, UNSAT Transition relation 6
- 11. SAT Termination Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) SAT some Fi is satisfiable 7
- 12. UNSAT Termination Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) UNSAT ∅ ∈ F0 8
- 13. AT Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi ∪ {C}) . . . (Mn, Fn) C is an AT wrt Fi 9
- 14. RAT Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi ∪ {L}, Fi ∪ {C}) . . . (Mn, Fn) C is RAT upon L wrt Fi 10
- 15. Clause Deletion Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi {C}) . . . (Mn, Fn) Fi ≡sat Fi {C} 11
- 16. Clause Sharing Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi ∪ {C} . . . (Mn, Fn) C ∈ Fj and C ∩ Mi = ∅, C ∩ Mj = ∅ 12
- 17. Soundness and Completeness Soundness: (i) if init(n, F0) ∗ ; SAT, then F0 is satisfiable, and (ii) if init(n, F0) ∗ ; UNSAT, then F0 is unsatisfiable Completeness: (i) if F0 is satisfiable, then init(n, F0) ∗ ; SAT (ii) if F0 is unsatisfiable, then init(n, F0) ∗ ; UNSAT Theorem: The formal model is sound and complete Proof idea: Semantical equivalences are preserved wrt a signature 13
- 18. Parallel DRAT A conservative extension of DRAT
- 19. Parallel DRAT Labeled clause ( , j, C), where ∈ {a, d}, j ∈ N PDRAT derivation D = (Di | 1 ≤ i ≤ n) in F if there is a run that is represented by D PDRAT refutation D = (Di | 1 ≤ i ≤ n) in F is a PDRAT derivation in F in which the empty clause occurs Theorem: 1. F is unsatisfiable iff there is a PDRAT refutation in F 2. We can efficiently check PDRAT refutations: AT > CS > RAT 14
- 21. AT Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi ∪ {C}) . . . (Mn, Fn) C is an AT wrt Fi Corresponding proof (a, i, C) 16
- 22. RAT Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi ∪ {L}, Fi ∪ {C}) . . . (Mn, Fn) C is RAT upon L wrt Fi Corresponding proof (a, i, C) 17
- 23. Clause Deletion Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi {C}) . . . (Mn, Fn) Fi ≡sat Fi {C} Corresponding proof (d, i, C) 18
- 24. Clause Sharing Rule (M0, F0) . . . (Mi , Fi ) . . . (Mn, Fn) (M0, F0) . . . (Mi , Fi ∪ {C} . . . (Mn, Fn) C ∈ Fj and C ∩ Mi = ∅, C ∩ Mj = ∅ Corresponding proof (a, i, C) 19
- 25. Example Formula F0 = {{p, q, r}, {p, ¬q, r}, {¬p, q, r}, {¬p, ¬q, r}} PDRAT derivation (a, 1, C1), (a, 1, C2), (a, 1, C3), (a, 2, C3) Corresponding run init(2, F0) ;AT ((∅, F0 ∪ {C1}), (∅, F0)) ;AT ((∅, F0 ∪ {C1, C2}), (∅, F0)) ;AT ((∅, F0 ∪ {C1, C2, C3}), (∅, F0)) ;CS ((∅, F0 ∪ {C1, C2, C3}), (∅, F0 ∪ {C3})) 20
- 26. Example Formula F0 = {{p, q, r}, {p, ¬q, r}, {¬p, q, r}, {¬p, ¬q, r}} PDRAT derivation (a, 1, C1), (a, 1, C2), (a, 1, C3), (a, 2, C3) Corresponding run init(2, F0) ;AT ((∅, F0 ∪ {C1}), (∅, F0)) ;AT ((∅, F0 ∪ {C1, C2}), (∅, F0)) ;AT ((∅, F0 ∪ {C1, C2, C3}), (∅, F0)) ;RAT ((∅, F0 ∪ {C1, C2, C3}), ({q}, F0 ∪ {C3})) 20
- 27. Open Issues Extend ManySAT,Plingeling, PRiss by PDRAT Develop a backward-checking procedure for PDRAT Develop a mechanically verified checker for PDRAT A mathematical, consistent and verified theory for RAT Is drat-trim sound and complete? A mechanically-verified and efficient checker for DRAT 21
- 28. Conclusion A formal model for parallel portfolios with restricted clause sharing arbritrary inprocessing Parallel DRAT can be used to certify answers from these solvers A prototypical checker implementation is available https://iccl.inf.tu-dresden.de/web/PDRAT 22
- 29. Unsatisfiability Proofs for Parallel SAT Solver Portfolios with Clause Sharing and Inprocessing Tobias Philipp Knowledge Representation and Reasoning Group Technische Universität Dresden Thank you for your attention.
- 30. UNSAT can be Incorrect: Applying Clause Addition Techniques in Two Solvers F = {{x, ¬y}, {¬x, y}, {x, z}, {y, ¬z}} C = {¬x, ¬z} is RAT in F upon ¬z D = {¬y, z} is RAT in F upon z F is satisfiable: I = {x, y, z} J = {x, y, ¬z} I |= C, J |= D F ∪ {C, D} is unsatisfiable F F F ∧ C F F ∧ C F ∧ D F ∧ C ∧ D F ∧ D UNSAT 23
- 31. Rule Preference Lemma (Monotonicity) If ((M1, F1), . . . (Sn, Fn)) ∗ ; ((M1, F1), . . . (Mn, Fn)) and Ti ⊆ Si , then: ((T1, F1), . . . (Tn, Fn)) ∗ ; ((T1, F1), . . . (Tn, Fn)) and Ti ⊆ Si Strategy Prefer AT-, and CS-rule over RAT-rule. 24