Effective interpolation for guarded logics

Transcription

1 Effective interpolation for guarded logics Michael Benedikt 1, Balder ten Cate 2, Michael Vanden Boom 1 1 University of Oxford 2 LogicBlox and UC Santa Cruz LogIC Seminar at Imperial College London December / 20

2 Some decidable fragments of first-order logic ML FO finite model property tree-like model property Craig interpolation ML 2 / 20

3 Some decidable fragments of first-order logic FO 2 constrain number of variables ML FO ML FO 2 finite model property tree-like model property Craig interpolation 2 / 20

4 Some decidable fragments of first-order logic constrain number of variables ML FO 2 GF FO constrain quantification [Andréka, van Benthem, Németi 95-98] x (G(xy) ψ(xy)) x (G(xy) ψ(xy)) ML FO 2 GF finite model property tree-like model property Craig interpolation 2 / 20

5 Some decidable fragments of first-order logic constrain number of variables ML FO 2 GF FO constrain quantification [Andréka, van Benthem, Németi 95-98] x (G(xy) ψ(xy)) x (G(xy) ψ(xy)) UNF constrain negation [ten Cate, Segoufin 11] ML FO 2 GF UNF finite model property tree-like model property Craig interpolation x (ψ(xy)) ψ(x) 2 / 20

6 Some decidable fragments of first-order logic constrain number of variables ML FO 2 UNF GF GNF FO constrain quantification [Andréka, van Benthem, Németi 95-98] x (G(xy) ψ(xy)) x (G(xy) ψ(xy)) constrain negation [ten Cate, Segoufin 11] [Bárány, ten Cate, Segoufin 11] ML FO 2 GF UNF GNF finite model property tree-like model property Craig interpolation x (ψ(xy)) G(xy) ψ(xy) 2 / 20

7 Interpolation φ ψ 3 / 20

8 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ 3 / 20

9 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R 4 / 20

10 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R b a c 4 / 20

11 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R b a c 4 / 20

12 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R b a c interpolant χ = xyz(rxy Ryz Rzx) there is a 3-cycle using R 4 / 20

13 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R b a c GNF interpolant χ = xyz(rxy Ryz Rzx) there is a 3-cycle using R 4 / 20

14 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (Bárány+Benedikt+ten Cate 13) Given GNF formulas φ and ψ such that φ ψ, there is a GNF interpolant χ (but model theoretic proof implies no bound on size of χ). 5 / 20

15 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (Bárány+Benedikt+ten Cate 13) Given GNF formulas φ and ψ such that φ ψ, there is a GNF interpolant χ (but model theoretic proof implies no bound on size of χ). Even when input is in GF, no idea how to compute interpolants (or other rewritings related to interpolation). 5 / 20

16 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (Bárány+Benedikt+ten Cate 13) Given GNF formulas φ and ψ such that φ ψ, there is a GNF interpolant χ (but model theoretic proof implies no bound on size of χ). Theorem (Benedikt+ten Cate+VB. 14) Given GNF formulas φ and ψ such that φ ψ, we can construct a GNF interpolant χ of doubly exponential DAG-size (in size of input). 5 / 20

17 Mosaics A mosaic τ(a) for φ is a collection of subformulas of φ over some guarded set a of parameters. τ 1 (ab) Raa Sa z(rbz Sz) Sb Rba τ 2 (bc) Sb Rbb Rbc Sc Rcb Sc τ 3 (d) Sd Sd yz(ryz Sz) z(rdz) Rdd Sd 6 / 20

18 Mosaics A mosaic τ(a) for φ is a collection of subformulas of φ over some guarded set a of parameters. τ 1 (ab) Raa Sa z(rbz Sz) Sb Rba τ 2 (bc) Sb Rbb Rbc Sc Rcb Sc τ 3 (d) Internally inconsistent (e.g. S d & S d) 6 / 20

19 Mosaics A mosaic τ(a) for φ is a collection of subformulas of φ over some guarded set a of parameters. τ 1 (ab) τ 2 (bc) τ 3 (d) a b b c Internally inconsistent (e.g. S d & S d) Internally consistent mosaics are windows into a (guarded) piece of a structure. 6 / 20

20 Linking mosaics Mosaics can be linked together to fulfill an existential requirement if they agree on all formulas that use only shared parameters. τ 1 a b z(r bz Sz) 7 / 20

21 Linking mosaics Mosaics can be linked together to fulfill an existential requirement if they agree on all formulas that use only shared parameters. τ 1 τ 2 a b c z(r bz Sz) 7 / 20

22 Linking mosaics Mosaics can be linked together to fulfill an existential requirement if they agree on all formulas that use only shared parameters. τ 1 τ 2 a b c z(r bz Sz) We say a set S of mosaics is saturated if every existential requirement in a mosaic τ S is fulfilled in τ or in some linked τ S. 7 / 20

23 Mosaics Fix some set P of size 2 width(φ) and let M φ be the set of mosaics for φ over parameters P. The size of M φ is doubly exponential in the size of φ. Theorem φ is satisfiable iff there is a saturated set S of internally consistent mosaics from M φ that contains some τ with φ τ. 8 / 20

24 Mosaics Fix some set P of size 2 width(φ) and let M φ be the set of mosaics for φ over parameters P. The size of M φ is doubly exponential in the size of φ. Theorem φ is satisfiable iff there is a saturated set S of internally consistent mosaics from M φ that contains some τ with φ τ. τ 3 τ τ 2 τ 3 τ S = { 1,,, 4 } 8 / 20

25 Mosaics Fix some set P of size 2 width(φ) and let M φ be the set of mosaics for φ over parameters P. The size of M φ is doubly exponential in the size of φ. Theorem φ is satisfiable iff there is a saturated set S of internally consistent mosaics from M φ that contains some τ with φ τ. τ 3 τ τ 2 τ 3 τ S = { 1,,, 4 } τ 4 8 / 20

26 Mosaics Fix some set P of size 2 width(φ) and let M φ be the set of mosaics for φ over parameters P. The size of M φ is doubly exponential in the size of φ. Theorem φ is satisfiable iff there is a saturated set S of internally consistent mosaics from M φ that contains some τ with φ τ. τ τ 2 τ 3 τ S = { 1,,, 4 } τ 3 τ 4 τ 1 8 / 20

27 Mosaics Fix some set P of size 2 width(φ) and let M φ be the set of mosaics for φ over parameters P. The size of M φ is doubly exponential in the size of φ. Theorem φ is satisfiable iff there is a saturated set S of internally consistent mosaics from M φ that contains some τ with φ τ. τ τ 2 τ 3 τ S = { 1,,, 4 } τ 2 τ 3 τ 4 τ 1 8 / 20

28 Mosaic elimination algorithm for satisfiability testing M φ τ 4 τ 5 τ 6 τ 2 τ 3 τ 1 τ 7 9 / 20

29 Mosaic elimination algorithm for satisfiability testing Stage 1. Eliminate mosaics with internal inconsistencies. M φ τ 4 τ 5 τ 6 τ 2 τ 3 τ 1 τ 7 9 / 20

30 Mosaic elimination algorithm for satisfiability testing Stage 1. Eliminate mosaics with internal inconsistencies. Stage i + 1. Eliminate mosaics with existential requirements that can only be fulfilled using mosaics eliminated in earlier stages. M φ τ 3 τ 4 τ 6 τ 5 τ 2 τ 1 τ 7 9 / 20

31 Mosaic elimination algorithm for satisfiability testing Stage 1. Eliminate mosaics with internal inconsistencies. Stage i + 1. Eliminate mosaics with existential requirements that can only be fulfilled using mosaics eliminated in earlier stages. M φ τ 3 τ 4 τ 6 τ 5 τ 2 Continue until fixpoint M reached. The set M is a saturated set of internally consistent mosaics. τ 1 τ 7 Theorem φ is satisfiable iff there is some mosaic τ M with φ τ. 9 / 20

32 Mosaics for interpolation Assume φ L φ R. Idea: Construct interpolant from proof that φ L φ R is unsatisfiable. 10 / 20

33 Mosaics for interpolation Assume φ L φ R. Idea: Construct interpolant from proof that φ L φ R is unsatisfiable. Consider mosaics for φ L φ R. Annotate each mosaic and each formula with a provenance L or R. L τ 1 (ab) L Raa R Sa R z(rbz Sz) R Rbb L Sb R Rba... R τ 2 (bc) L Sb R Rbb R Rbc Sc R Rbc L Rcb R z(rbz Sz) R Sc... L τ 3 (d) L Sd R Sd R Rdd Sd R yz(ryz Sz) L z(rdz) L Rdd Sd... Linking must respect the provenance annotations. 10 / 20

34 Mosaics for interpolation Assume φ L φ R. Test satisfiability of φ L φ R using mosaic elimination. M φl φ R τ 4 τ 5 τ 6 τ 2 τ 3 τ 1 τ 7 11 / 20

35 Mosaics for interpolation Assume φ L φ R. Test satisfiability of φ L φ R using mosaic elimination. M φl φ R θ 5 τ 4 τ 5 Assign a mosaic interpolant θ τ to each eliminated mosaic τ such that τ L θ τ and θ τ τ R. Mosaic interpolants θ τ describe why the mosaic τ was eliminated. τ 3 τ 6 τ 1 τ 7 τ 2 θ 7 11 / 20

36 Mosaics for interpolation Assume φ L φ R. Test satisfiability of φ L φ R using mosaic elimination. M φl φ R θ 5 τ 4 τ 5 Assign a mosaic interpolant θ τ to each eliminated mosaic τ such that τ L θ τ and θ τ τ R. Mosaic interpolants θ τ describe why the mosaic τ was eliminated. τ 3 τ 6 τ 1 τ 7 τ 2 θ 7 11 / 20

37 Mosaics for interpolation Assume φ L φ R. Test satisfiability of φ L φ R using mosaic elimination. M φl φ R θ 5 τ 4 τ 5 Assign a mosaic interpolant θ τ to each eliminated mosaic τ such that τ L θ τ and θ τ τ R. Mosaic interpolants θ τ describe why the mosaic τ was eliminated. τ 3 τ 6 τ 1 θ 6 τ 7 τ 2 θ 7 11 / 20

38 Mosaics for interpolation Assume φ L φ R. Test satisfiability of φ L φ R using mosaic elimination. M φl φ R θ 5 τ 4 τ 5 Assign a mosaic interpolant θ τ to each eliminated mosaic τ such that τ L θ τ and θ τ τ R. Mosaic interpolants θ τ describe why the mosaic τ was eliminated. τ 3 τ 6 τ 1 θ 6 τ 7 τ 2 θ 7 Theorem An interpolant χ for φ L φ R of at most doubly exponential DAG-size can be constructed from the mosaic interpolants. 11 / 20

39 Shape of interpolants Mosaic interpolants θ τ satisfy τ L θ τ and θ τ τ R. They describe why the mosaic τ was eliminated. Stage 1: L Rab L Rab θ τ = Internal R Rab R Rab θ τ = inconsistency L Rab R Rab θ τ = Rab R Rab L Rab θ τ = Rab 12 / 20

40 Shape of interpolants Mosaic interpolants θ τ satisfy τ L θ τ and θ τ τ R. They describe why the mosaic τ was eliminated. Stage 1: L Rab L Rab θ τ = Internal R Rab R Rab θ τ = inconsistency L Rab R Rab θ τ = Rab R Rab L Rab θ τ = Rab Stage i + 1: Unfulfilled L z [G(bz) ψ(bz)] θ τ = τ (bc) z [ τ τ θ τ (bz)] there is a mosaic τ that can be linked to τ to fulfil the requirement, but no matter what R-formulas are added, the resulting mosaic τ has already been eliminated 12 / 20

41 Mosaics for interpolation Challenge: ensure interpolant χ is in GNF 13 / 20

42 Mosaics for interpolation Challenge: ensure interpolant χ is in GNF Solution: place further restrictions on the formulas in the mosaics 13 / 20

43 Mosaics for interpolation Challenge: ensure interpolant χ is in GNF Solution: place further restrictions on the formulas in the mosaics Idea: in an L-mosaic, only allow R-formulas that are guarded by some L-atom in the common signature. L τ full info about L-formulas partial info about R-formulas R τ full info about R-formulas partial info about L-formulas This makes it harder to prove completeness of mosaic method, but makes it easier to prove properties about the mosaic interpolants. 13 / 20

44 Stronger interpolation theorems for GNF Lyndon interpolation: χ respects polarity of relations A relation R occurs positively (respectively, negatively) in χ iff R occurs positively (respectively, negatively) in both φ L and φ R. Relativized interpolation: χ respects quantification pattern If the quantification in φ L and φ R is relativized to a distinguished set of unary predicates U, then χ is U-relativized. i.e. quantification is of the form x (Ux ψ(xy)) for U U 14 / 20

45 Bonus: effective preservation theorems φ is preserved under extensions if A φ and A B implies B φ. φ is in existential GNF if no quantifier is in the scope of a negation. Corollary (Analog of Loś-Tarski) If φ is preserved under extensions and in GNF, then we can construct an equivalent existential GNF formula φ of doubly exponential DAG-size. 15 / 20

46 Some decidable fragments of FO FO 2 ML GF FO UNF GNF ML FO 2 GF UNF GNF finite model property tree-like model property Craig interpolation 16 / 20

47 Some decidable fragments of FO+LFP L µ GFP UNFP GNFP FO + LFP L µ GFP UNFP GNFP finite model property tree-like model property Craig interpolation??? 17 / 20

48 Some decidable fragments of FO+LFP L µ GFP UNFP GNFP FO + LFP L µ GFP UNFP GNFP finite model property tree-like model property Craig interpolation? 17 / 20

49 Some decidable fragments of FO+LFP L µ GFP UNFP GNFP FO + LFP L µ GFP UNFP GNFP finite model property tree-like model property Craig interpolation 17 / 20

50 Uniform interpolation The modal mu-calculus (L µ ) has uniform interpolation [D Agostino+Hollenberg 00]... Uniform interpolation: χ depends only on antecedent and the signature of the consequent Given φ L and a sub-signature σ, there is an interpolant χ over σ such that for all φ R with φ L φ R and common signature σ, φ L χ and χ φ R 18 / 20

51 Uniform interpolation for UNFP k Let UNFP k denote the k-variable fragment of UNFP (when written in a normal form...) Theorem (Benedikt+ten Cate+VB. unpublished) UNFP k has effective uniform interpolation. UNFP has Craig interpolation. 19 / 20

52 Uniform interpolation for UNFP k Let UNFP k denote the k-variable fragment of UNFP (when written in a normal form...) Theorem (Benedikt+ten Cate+VB. unpublished) UNFP k has effective uniform interpolation. UNFP has Craig interpolation. Relational structures UNFP k φ Coded structures (tree decompositions of width k) L µ φ 19 / 20

53 Uniform interpolation for UNFP k Let UNFP k denote the k-variable fragment of UNFP (when written in a normal form...) Theorem (Benedikt+ten Cate+VB. unpublished) UNFP k has effective uniform interpolation. UNFP has Craig interpolation. Relational structures UNFP k φ Coded structures (tree decompositions of width k) L µ φ [D Agostino+Hollenberg 00] L µ χ over subsignature encoding 19 / 20

54 Uniform interpolation for UNFP k Let UNFP k denote the k-variable fragment of UNFP (when written in a normal form...) Theorem (Benedikt+ten Cate+VB. unpublished) UNFP k has effective uniform interpolation. UNFP has Craig interpolation. Relational structures UNFP k φ Coded structures (tree decompositions of width k) L µ φ UNFP k χ over subsignature L µ χ over subsignature encoding [D Agostino+Hollenberg 00] 19 / 20

55 Summary Guarded logics have attractive computational and model-theoretic properties, including interpolation. Craig interpolation ML GF UNF GNF adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] 20 / 20

56 Summary Guarded logics have attractive computational and model-theoretic properties, including interpolation. ML GF UNF GNF L µ GFP UNFP GNFP Craig interpolation adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] 20 / 20

57 Summary Guarded logics have attractive computational and model-theoretic properties, including interpolation. ML GF UNF GNF L µ GFP UNFP GNFP Craig interpolation adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] used uniform interpolation for L µ [Benedikt,ten Cate,VB. unpublished] 20 / 20

58 Effective preservation theorems φ is monotone if A φ implies that A φ for any A obtained from A by adding tuples to the interpretation of some relation. φ is positive if every relation appears within the scope of an even number of negations. Corollary (Monotone = Positive) If φ is monotone and in GNF, then we can construct an equivalent positive GNF formula φ of doubly exponential DAG-size.

59 Effective preservation theorems φ is monotone if A φ implies that A φ for any A obtained from A by adding tuples to the interpretation of some relation. φ is positive if every relation appears within the scope of an even number of negations. Corollary (Monotone = Positive) If φ is monotone and in GNF, then we can construct an equivalent positive GNF formula φ of doubly exponential DAG-size. Let φ i be the result of replacing every relation R with a copy R i. The Lyndon interpolant χ for y (R 1 y R 2 y) φ 1 φ 2 R can only use relations of the form R 2, and these must all be positive. Replacing every R 2 with R in χ yields positive φ equivalent to φ.

60 Loś-Tarski preservation theorem φ is preserved under extensions if A φ and A B implies B φ. φ is in existential GNF if no quantifier is in the scope of a negation. Corollary (Analog of Loś-Tarski) If φ is preserved under extensions and in GNF, then we can construct an equivalent existential GNF formula φ of doubly exponential DAG-size.

61 Loś-Tarski preservation theorem φ is preserved under extensions if A φ and A B implies B φ. φ is in existential GNF if no quantifier is in the scope of a negation. Corollary (Analog of Loś-Tarski) If φ is preserved under extensions and in GNF, then we can construct an equivalent existential GNF formula φ of doubly exponential DAG-size. Let U = {U 1, U 2 } be a set of fresh unary predicates. Let φ i be the result of relativizing every quantification to U i. The relativized Lyndon interpolant χ for y (U 1 y U 2 y) φ 1 φ 2 is an existential GNF formula. Replacing every U 2 z in χ with yields the desired existential GNF φ.