Interpol a tion for g ua rded logi c s Micha el Va nden B oom U ni v - - PowerPoint PPT Presentation
Interpol a tion for g ua rded logi c s Micha el Va nden B oom U ni v ersit y of Ox ford H ighlights 20 14 Pa ris, F r a n c e J oint w ork w ith M i c h a el B enedikt a nd Ba lder ten Ca te 1 / 7 Some guarded logi c s constrain quantification
Michael Vanden Boom
University of Oxford
Highlights 2014 Paris, France
Joint work with Michael Benedikt and Balder ten Cate
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Some guarded logics
ML GF constrain quantification
∃x(G(xy) ∧ ψ(xy)) ∀x(G(xy) → ψ(xy))
[Andr´ eka, van Benthem, N´ emeti ’95-’98]
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Some guarded logics
ML GF UNF constrain quantification
∃x(G(xy) ∧ ψ(xy)) ∀x(G(xy) → ψ(xy))
[Andr´ eka, van Benthem, N´ emeti ’95-’98]
constrain negation
∃x(ψ(xy)) ¬ψ(x)
[ten Cate, Segoufin ’11]
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Some guarded logics
ML GF GNF UNF constrain quantification
∃x(G(xy) ∧ ψ(xy)) ∀x(G(xy) → ψ(xy))
[Andr´ eka, van Benthem, N´ emeti ’95-’98]
constrain negation
∃x(ψ(xy)) G(xy) ∧ ¬ψ(xy)
[ten Cate, Segoufin ’11] [B´ ar´ any, ten Cate, Segoufin ’11]
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Some guarded logics
ML GF GNF UNF constrain quantification
∃x(G(xy) ∧ ψ(xy)) ∀x(G(xy) → ψ(xy))
[Andr´ eka, van Benthem, N´ emeti ’95-’98]
constrain negation
∃x(ψ(xy)) G(xy) ∧ ¬ψ(xy)
[ten Cate, Segoufin ’11] [B´ ar´ any, 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.
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Interpolation
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Interpolation
relations in both φ and ψ
interpolant
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Interpolation example
∃xyz(Txyz ∧ Rxy ∧ Ryz ∧ Rzx) ⊧ ∃xy(Rxy ∧ ((Sx ∧ Sy) ∨ (¬Sx ∧ ¬Sy))) “there is a T-guarded 3-cycle using R”
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Interpolation example
∃xyz(Txyz ∧ Rxy ∧ Ryz ∧ Rzx) ⊧ ∃xy(Rxy ∧ ((Sx ∧ Sy) ∨ (¬Sx ∧ ¬Sy))) “there is a T-guarded 3-cycle using R”
a b c
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Interpolation example
∃xyz(Txyz ∧ Rxy ∧ Ryz ∧ Rzx) ⊧ ∃xy(Rxy ∧ ((Sx ∧ Sy) ∨ (¬Sx ∧ ¬Sy))) “there is a T-guarded 3-cycle using R”
a b c
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Interpolation example
∃xyz(Txyz ∧ Rxy ∧ Ryz ∧ Rzx) ⊧ ∃xy(Rxy ∧ ((Sx ∧ Sy) ∨ (¬Sx ∧ ¬Sy))) “there is a T-guarded 3-cycle using R”
a b c interpolant χ ∶= ∃xyz(Rxy ∧ Ryz ∧ Rzx)
“there is a 3-cycle using R”
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Why do we care?
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Why do we care?
Why might someone care?
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Why do we care?
Why might someone care? interpolation is a benchmark property of modal logic
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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
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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
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Interpolation
relations in both φ and ψ
interpolant
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.
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Interpolation
relations in both φ and ψ
interpolant
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).
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Interpolation
relations in both φ and ψ
interpolant
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.
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Conclusion
ML GF UNF GNF Interpolation? ✓ ✗ ✓ ✓ adapted mosaic method from ML
[Benedikt,ten Cate,VB.’14]
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Conclusion
ML GF UNF GNF Lµ GFP UNFP GNFP Interpolation? ✓ ✗ ✓ ✓ ✓ ✗
adapted mosaic method from ML
[Benedikt,ten Cate,VB.’14]
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Conclusion
ML GF UNF GNF Lµ GFP UNFP GNFP Interpolation? ✓ ✗ ✓ ✓ ✓ ✗
adapted mosaic method from ML
[Benedikt,ten Cate,VB.’14]
used automata for Lµ
[Benedikt,ten Cate,VB. unpublished]
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Conclusion
ML GF UNF GNF Lµ GFP UNFP GNFP Interpolation? ✓ ✗ ✓ ✓ ✓ ✗
adapted mosaic method from ML
[Benedikt,ten Cate,VB.’14]
used automata for Lµ
[Benedikt,ten Cate,VB. unpublished]
Open question Is there a decidable extension of GNFP that has interpolation?
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