On Unit-Refutation Complete Formulae with Existentially Quantified Variables

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1 On Unit-Refutation Complete Formulae with Existentially Quantified Variables Lucas Bordeaux 1 Mikoláš Janota 2 Joao Marques-Silva 3 Pierre Marquis 4 1 Microsoft Research, Cambridge 2 INESC-ID, Lisboa 3 CASL, UCD & IST/INESC-ID 4 CRIL-CNRS, U. d Artois Bordeaux et al Unit-Refutation Complete Formulas 1 / 13

2 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

3 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Example of inference Propagation x x y ȳ z Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

4 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Example of inference Propagation x x y ȳ z u z Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

5 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Example of inference Propagation x x y ȳ z u z x z x z Refutation Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

6 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Example of inference Propagation x x y ȳ z u z x z x z z Refutation Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

7 CNF, Unit Propagation conjunctive normal form (CNF) is a popular language in solvers for its simple yet expressive structure unit propagation is an inference mechanism implemented virtually in all CNF-based solvers unit propagation can be computed polynomial time and moreover, efficient algorithms and data structures have been developed for it (watch literals) Example of inference Propagation x x y ȳ z u z x z x z z Refutation u Bordeaux et al Unit-Refutation Complete Formulas 2 / 13

8 Unit Refutation Completeness for general CNF, unit propagation is not complete, i.e. it does not let us infer all the facts Examples ū w u w x ū w x u w = x u x Bordeaux et al Unit-Refutation Complete Formulas 3 / 13

9 Unit Refutation Completeness for general CNF, unit propagation is not complete, i.e. it does not let us infer all the facts Examples ū w u w x ū w x u w = x u x ū w u w x ū w x u w x u Bordeaux et al Unit-Refutation Complete Formulas 3 / 13

10 Definition (URC-C) Languages a formula α CNF belongs to URC-C iff for every clause δ = l 1 l k if α = δ then α l 1... l k u Bordeaux et al Unit-Refutation Complete Formulas 4 / 13

11 Definition (URC-C) Languages a formula α CNF belongs to URC-C iff for every clause δ = l 1 l k if α = δ then α l 1... l k u Definition (Closures) if α 1 α n L then (α 1 α n ) L[ ] Bordeaux et al Unit-Refutation Complete Formulas 4 / 13

12 Definition (URC-C) Languages a formula α CNF belongs to URC-C iff for every clause δ = l 1 l k if α = δ then α l 1... l k u Definition (Closures) if α 1 α n L then (α 1 α n ) L[ ] if α L then ( X. α) L[ ] Bordeaux et al Unit-Refutation Complete Formulas 4 / 13

13 Definition (URC-C) Languages a formula α CNF belongs to URC-C iff for every clause δ = l 1 l k if α = δ then α l 1... l k u Definition (Closures) if α 1 α n L then (α 1 α n ) L[ ] if α L then ( X. α) L[ ] L[, ] enables both rules Bordeaux et al Unit-Refutation Complete Formulas 4 / 13

14 Motivation The Quest for The Perfect Knowledge Representation Structure inference should be fast (tractability) representation should not be too large (succinctness) Bordeaux et al Unit-Refutation Complete Formulas 5 / 13

15 Motivation The Quest for The Perfect Knowledge Representation Structure inference should be fast (tractability) representation should not be too large (succinctness) If a formula is unit refutation complete... which queries can be answered efficiently? how does the size of formulas correspond to other representations? Bordeaux et al Unit-Refutation Complete Formulas 5 / 13

16 Succinctness ( s, p ) Knowledge Compilation Map L 1 s L 2, if any formula in L 2 can be equivalently expressed in polynomially sized formula from L 1 Bordeaux et al Unit-Refutation Complete Formulas 6 / 13

17 Succinctness ( s, p ) Knowledge Compilation Map L 1 s L 2, if any formula in L 2 can be equivalently expressed in polynomially sized formula from L 1 L 1 p L 2, just as p but we also have an polynomial algorithm for it Bordeaux et al Unit-Refutation Complete Formulas 6 / 13

18 Succinctness ( s, p ) Knowledge Compilation Map L 1 s L 2, if any formula in L 2 can be equivalently expressed in polynomially sized formula from L 1 L 1 p L 2, just as p but we also have an polynomial algorithm for it Queries can we decide in polynomial time... consistency, clausal entailment, etc. Bordeaux et al Unit-Refutation Complete Formulas 6 / 13

19 Succinctness ( s, p ) Knowledge Compilation Map L 1 s L 2, if any formula in L 2 can be equivalently expressed in polynomially sized formula from L 1 L 1 p L 2, just as p but we also have an polynomial algorithm for it Queries can we decide in polynomial time... consistency, clausal entailment, etc. Transformations can we construct in polynomial time... Conditioning (α[x]), disjunction (α 1 α n ), etc. Bordeaux et al Unit-Refutation Complete Formulas 6 / 13

20 Trivia for URC-C for α, α 1, α 2 URC-C clausal entailment α = γ can be decided in polynomial time negate γ and run unit propagation Bordeaux et al Unit-Refutation Complete Formulas 7 / 13

21 Trivia for URC-C for α, α 1, α 2 URC-C clausal entailment α = γ can be decided in polynomial time negate γ and run unit propagation consistency of α can be decided in polynomial time check that the empty clause is an implicate of α Bordeaux et al Unit-Refutation Complete Formulas 7 / 13

22 Trivia for URC-C for α, α 1, α 2 URC-C clausal entailment α = γ can be decided in polynomial time negate γ and run unit propagation consistency of α can be decided in polynomial time check that the empty clause is an implicate of α equivalence of α 1, α 2 decidable in polynomial time check that each clause in α 1 is an implicate of α 2 check that each clause in α2 is an implicate of α 1 Bordeaux et al Unit-Refutation Complete Formulas 7 / 13

23 Trivia for URC-C for α, α 1, α 2 URC-C clausal entailment α = γ can be decided in polynomial time negate γ and run unit propagation consistency of α can be decided in polynomial time check that the empty clause is an implicate of α equivalence of α 1, α 2 decidable in polynomial time check that each clause in α 1 is an implicate of α 2 check that each clause in α2 is an implicate of α 1 if β CNF contains all of its prime implicates then β URC-C if β = γ, then there is a prime implicate γ γ and immediately β γ u Bordeaux et al Unit-Refutation Complete Formulas 7 / 13

24 Enabling Existential Variables Motivation: Using fresh variables in CNF enables... polynomial Boolean logic representation (Tseitin) cardinality encodings other constraints, e.g. XOR(x 1,..., x n ) Bordeaux et al Unit-Refutation Complete Formulas 8 / 13

25 Enabling Existential Variables Motivation: Using fresh variables in CNF enables... polynomial Boolean logic representation (Tseitin) cardinality encodings other constraints, e.g. XOR(x 1,..., x n ) Definition ( URC-C) a formula X. α CNF[ ] belongs to URC-C iff for every clause δ = l 1 l k if X. α = δ then α l 1... l k u Bordeaux et al Unit-Refutation Complete Formulas 8 / 13

26 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

27 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

28 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n [Tseitin] α = X τ 1... τ n. τ 1 α 1... τ n α n τ 1 τ n Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

29 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n [Tseitin] α = X τ 1... τ n. τ 1 α 1... τ n α n τ 1 τ n α is not necessarily unit refutation complete! Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

30 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n [Tseitin] α = X τ 1... τ n. τ 1 α 1... τ n α n τ 1 τ n α is not necessarily unit refutation complete! if α = γ, then ( X i. α i ) = γ, for i 1..n Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

31 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n [Tseitin] α = X τ 1... τ n. τ 1 α 1... τ n α n τ 1 τ n α is not necessarily unit refutation complete! if α = γ, then ( X i. α i ) = γ, for i 1..n since α i URC-C, then α τ i γ u, for i 1..n Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

32 URC-C p URC-C[ ] α = ( X 1. α 1 ) ( X n. α n ) [prenex] α = X. α 1 α n [Tseitin] α = X τ 1... τ n. τ 1 α 1... τ n α n τ 1 τ n α is not necessarily unit refutation complete! if α = γ, then ( X i. α i ) = γ, for i 1..n since α i URC-C, then α τ i γ u, for i 1..n add new variables that simulate unit propagation on the disjuncts and derive if all derive Bordeaux et al Unit-Refutation Complete Formulas 9 / 13

33 Queries Results L CO VA CE IM EQ SE CT ME MC URC-C URC-C[, ] URC-C means satisfies means does not satisfy unless P=NP Bordeaux et al Unit-Refutation Complete Formulas 10 / 13

34 Succinctness Results 1. URC-C s URC-C[, ] < s URC-C < s PI 2. URC-C s CNF and CNF s URC-C 3. URC-C s CNF and CNF s URC-C 4. URC-C s DNF, URC-C s SDNNF, and URC-C s d-dnnf, 5. DNF s URC-C, SDNNF s URC-C, and FBDD s URC-C 6. URC-C s DNNF 7. URC-C < s DNF 8. URC-C < s SDNNF 9. URC-C < s d-dnnf L 1 s L 2 means that L 1 is not at least as succinct as L 2 unless PH collapses Bordeaux et al Unit-Refutation Complete Formulas 11 / 13

35 Transformations Results L CD FO SFO C BC C BC C URC-C URC-C[, ] URC-C?? means satisfies means does not satisfy means does not satisfy unless P=NP Bordeaux et al Unit-Refutation Complete Formulas 12 / 13

36 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

37 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C the languages have number of favorable KR properties Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

38 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C the languages have number of favorable KR properties URC-C is powerful in answering queries, e.g. clausal entailment, equivalence Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

39 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C the languages have number of favorable KR properties URC-C is powerful in answering queries, e.g. clausal entailment, equivalence URC-C loses some ability to answer queries but is (strictly) more succinct than number of interesting languages Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

40 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C the languages have number of favorable KR properties URC-C is powerful in answering queries, e.g. clausal entailment, equivalence URC-C loses some ability to answer queries but is (strictly) more succinct than number of interesting languages in the future we are interested in practical algorithms for compilation into URC-C Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

41 Conclusions and Future Work we studied the unit refutation complete language URC-C and its existential extension URC-C the languages have number of favorable KR properties URC-C is powerful in answering queries, e.g. clausal entailment, equivalence URC-C loses some ability to answer queries but is (strictly) more succinct than number of interesting languages in the future we are interested in practical algorithms for compilation into URC-C how can existential variables be employed ( URC-C) Bordeaux et al Unit-Refutation Complete Formulas 13 / 13

This is an author-deposited version published in : Eprints ID : 16897

This is an author-deposited version published in : Eprints ID : 16897 Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited