Interpolation for guarded logics
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1 Interpolation for guarded logics Michael Vanden Boom University of Oxford Highlights 2014 Paris, France Joint work with Michael Benedikt and Balder ten Cate 1 / 7
2 Some guarded logics ML GF FO constrain quantification x(g(xy) ψ(xy)) x(g(xy) ψ(xy)) [Andréka, van Benthem, Németi 95-98] 2 / 7
3 Some guarded logics ML GF UNF FO constrain quantification x(g(xy) ψ(xy)) x(g(xy) ψ(xy)) [Andréka, van Benthem, Németi 95-98] constrain negation x(ψ(xy)) ψ(x) [ten Cate, Segoufin 11] 2 / 7
4 Some guarded logics constrain quantification GF x(g(xy) ψ(xy)) x(g(xy) ψ(xy)) ML GNF FO [Andréka, van Benthem, Németi 95-98] UNF constrain negation x(ψ(xy)) G(xy) ψ(xy) [ten Cate, Segoufin 11] [Bárány, ten Cate, Segoufin 11] 2 / 7
5 Some guarded logics constrain quantification GF x(g(xy) ψ(xy)) x(g(xy) ψ(xy)) ML GNF FO [Andréka, van Benthem, Németi 95-98] UNF constrain negation x(ψ(xy)) G(xy) ψ(xy) [ten Cate, Segoufin 11] [Bárány, ten Cate, Segoufin 11] These guarded logics are decidable, and expressive enough to capture many query languages and integrity constraints of interest in databases and knowledge representation. 2 / 7
6 Interpolation φ ψ 3 / 7
7 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ 3 / 7
8 Interpolation example xyz(txyz Rxy Ryz Rzx) xy(rxy ((Sx Sy) ( Sx Sy))) there is a T-guarded 3-cycle using R 4 / 7
9 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 / 7
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 / 7
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 interpolant χ = xyz(rxy Ryz Rzx) there is a 3-cycle using R 4 / 7
12 Why do we care? 5 / 7
13 Why do we care? Why might someone care? 5 / 7
14 Why do we care? Why might someone care? interpolation is a benchmark property of modal logic 5 / 7
15 Why do we care? Why might someone care? interpolation is a benchmark property of modal logic interpolation implies the Beth definability property (implicit definability = explicit definability) which indicates a good balance between syntax and semantics 5 / 7
16 Why do we care? Why might someone care? interpolation is a benchmark property of modal logic interpolation implies the Beth definability property (implicit definability = explicit definability) which indicates a good balance between syntax and semantics for these guarded logics with connections to databases, interpolation is related to query rewriting over views 5 / 7
17 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (ten Cate, Segoufin 11; Barany, Benedikt, ten Cate 13) Given GNF (respectively, UNF) formulas φ and ψ such that φ ψ, there is a GNF (respectively, UNF) interpolant χ. 6 / 7
18 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (ten Cate, Segoufin 11; Barany, Benedikt, ten Cate 13) Given GNF (respectively, UNF) formulas φ and ψ such that φ ψ, there is a GNF (respectively, UNF) interpolant χ. No idea how to compute interpolants (or other rewritings related to interpolation). 6 / 7
19 Interpolation interpolant φ χ ψ only uses relations in both φ and ψ Theorem (ten Cate, Segoufin 11; Barany, Benedikt, ten Cate 13) Given GNF (respectively, UNF) formulas φ and ψ such that φ ψ, there is a GNF (respectively, UNF) interpolant χ. Theorem (Benedikt, ten Cate, VB. 14) Given GNF (respectively, UNF) formulas φ and ψ s.t. φ ψ, we can construct a GNF (respectively, UNF) interpolant χ of doubly exponential DAG-size. 6 / 7
20 Conclusion Interpolation? ML GF UNF GNF adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] 7 / 7
21 Conclusion ML GF UNF GNF L µ GFP UNFP GNFP Interpolation? adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] 7 / 7
22 Conclusion ML GF UNF GNF L µ GFP UNFP GNFP Interpolation? adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] used automata for L µ [Benedikt,ten Cate,VB. unpublished] 7 / 7
23 Conclusion ML GF UNF GNF L µ GFP UNFP GNFP Interpolation? adapted mosaic method from ML [Benedikt,ten Cate,VB. 14] used automata for L µ [Benedikt,ten Cate,VB. unpublished] Open question Is there a decidable extension of GNFP that has interpolation? 7 / 7
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