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 ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [Andr´ eka, van Benthem, FO N´ emeti ’95-’98 ] ML 2 / 7

Some guarded logi c s constrain quantification ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [Andr´ eka, van Benthem, FO N´ emeti ’95-’98 ] ML c onstr a in UNF neg a tion ∃ x ( ψ ( xy )) ¬ ψ ( x ) [ ten Ca te, S egoufin ’11 ] 2 / 7

Some guarded logi c s constrain quantification ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [Andr´ eka, van Benthem, FO N´ emeti ’95-’98 ] ML GNF c onstr a in UNF neg a tion ∃ x ( ψ ( xy )) G ( xy ) ∧ ¬ ψ ( xy ) [ ten Ca te, S egoufin ’11 ] [B´ a r ´ a ny, ten Ca te, S egoufin ’11 ] 2 / 7

Some guarded logi c s constrain quantification ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [Andr´ eka, van Benthem, FO N´ emeti ’95-’98 ] ML GNF c onstr a in UNF neg a tion ∃ x ( ψ ( xy )) G ( xy ) ∧ ¬ ψ ( xy ) [ ten Ca te, S egoufin ’11 ] [B´ a r ´ a ny, ten Ca te, S egoufin ’11 ] T hese gu a rded logi c s a re de c id ab le, a nd e x pressi v e eno u gh to ca pt u re m a n y q u er y l a ng u a ges a nd integrit y c onstr a ints of interest in d a t aba ses a nd kno w ledge represent a tion. 2 / 7

Int erpol a tion φ ψ ⊧ 3 / 7

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ onl y u ses rel a tions in b oth φ a nd ψ 3 / 7

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -guarded 3-cyc le u sing R ” 4 / 7

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -guarded 3-cyc le u sing R ” b a c 4 / 7

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -guarded 3-cyc le u sing R ” b a c 4 / 7

Int erpol a tion e xa mple ∃ xyz ( Txyz ∧ Rxy ∧ Ryz ∧ Rzx ) ∃ xy ( Rxy ∧ (( Sx ∧ Sy ) ∨ ( ¬ Sx ∧ ¬ Sy ))) ⊧ “there is a T -guarded 3-cyc le u sing R ” b a c interpol a nt χ ∶ = ∃ xyz ( Rxy ∧ Ryz ∧ Rzx ) “there is a 3 - cyc le u sing R ” 4 / 7

Why do we care? 5 / 7

Why do we care? Why might someone care? 5 / 7

Why do we care? Why might someone care? inter pol a tion is a b en c hm a rk propert y of mod a l logi c 5 / 7

Why do we care? Why might someone care? inter pol a tion is a b en c hm a rk propert y of mod a l logi c interpol a tion implies the B eth defin ab ilit y propert y (impli c it defin ab ilit y = e x pli c it defin ab ilit y ) w hi c h indi ca tes a good ba l a n c e b et w een s y nt ax a nd sem a nti c s 5 / 7

Why do we care? Why might someone care? inter pol a tion is a b en c hm a rk propert y of mod a l logi c interpol a tion implies the B eth defin ab ilit y propert y (impli c it defin ab ilit y = e x pli c it defin ab ilit y ) w hi c h indi ca tes a good ba l a n c e b et w een s y nt ax a nd sem a nti c s for these g ua rded logi c s w ith c onne c tions to d a t aba ses, interpol a tion is rel a ted to q u er y re w riting o v er v ie w s 5 / 7

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ onl y u ses rel a tions in b oth φ a nd ψ T heorem (ten Ca te, S ego u fin ’11; Ba r a n y , B enedikt, ten Ca te ’1 3 ) G i v en GNF (respe c ti v el y , UNF ) form u l a s φ a nd ψ s uc h th a t φ ⊧ ψ , there is a GNF (respe c ti v el y , UNF ) interpol a nt χ . 6 / 7

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ onl y u ses rel a tions in b oth φ a nd ψ T heorem (ten Ca te, S ego u fin ’11; Ba r a n y , B enedikt, ten Ca te ’1 3 ) G i v en GNF (respe c ti v el y , UNF ) form u l a s φ a nd ψ s uc h th a t φ ⊧ ψ , there is a GNF (respe c ti v el y , UNF ) interpol a nt χ . N o ide a ho w to c omp u te interpol a nts (or other re w ritings rel a ted to interpol a tion). 6 / 7

Int erpol a tion interpol a nt φ χ ψ ⊧ ⊧ onl y u ses rel a tions in b oth φ a nd ψ T heorem (ten Ca te, S ego u fin ’11; Ba r a n y , B enedikt, ten Ca te ’1 3 ) G i v en GNF (respe c ti v el y , UNF ) form u l a s φ a nd ψ s uc h th a t φ ⊧ ψ , there is a GNF (respe c ti v el y , UNF ) interpol a nt χ . T heorem ( B enedikt, ten Ca te, VB . ’14) G i v en GNF (respe c ti v el y , UNF ) form u l a s φ a nd ψ s.t. φ ⊧ ψ , w e ca n c onstr uc t a GNF (respe c ti v el y , UNF ) interpol a nt χ of do ub l y e x ponenti a l DAG -si z e. 6 / 7

Conc l u sion ML GF UNF GNF Inter pol a tion? ✓ ✗ ✓ ✓ a d a pted mos a i c method from ML [B enedikt,ten Ca te, VB .’14] 7 / 7

Conc l u sion ML GF UNF GNF L µ GFP UNFP GNFP Inter pol a tion? ✓ ✗ ✓ ✓ ✓ ✗ ✓ ✗ a d a pted mos a i c method from ML [B enedikt,ten Ca te, VB .’14] 7 / 7

Conc l u sion ML GF UNF GNF L µ GFP UNFP GNFP Inter pol a tion? ✓ ✗ ✓ ✓ ✓ ✗ ✓ ✗ a d a pted u sed mos a i c method au tom a t a from ML for L µ [B enedikt,ten Ca te, VB .’14] [Benedikt,ten C a te, VB . u np ub lished] 7 / 7

Conc l u sion ML GF UNF GNF L µ GFP UNFP GNFP Inter pol a tion? ✓ ✗ ✓ ✓ ✓ ✗ ✓ ✗ a d a pted u sed mos a i c method au tom a t a from ML for L µ [B enedikt,ten Ca te, VB .’14] [Benedikt,ten C a te, VB . u np ub lished] O pen q u estion Is there a de c id ab le e x tension of GNFP th a t h a s interpol a tion? 7 / 7