Interpol a tion w ith de c id ab le fi x point logi c s Micha el Va - PowerPoint PPT Presentation

Interpol a tion w ith de c id ab le fi x point logi c s Micha el Va nden B oom U ni v ersit y of Ox ford H ighlights 20 15 P r a g u e J oint w ork w ith M i c h a el B enedikt a nd Ba lder ten Ca te 1 / 8

Fixpoint logi c s Fixpoint logi c s gi v e me c h a nism to e x press d y n a mi c , re cu rsi v e properties. Exa mple b in a r y rel a tion R , u n a r y rel a tion P “from y , it is possi b le to R -re ac h some P -element” [ lfp Y , y . Py ∨ ∃ z ( Ryz ∧ Yz )]( y ) 2 / 8

Some decidab le fi x point logi c s M od a l m u - ca l cu l u s (L µ ) [Kozen ’83 ] extension of moda l logi c w ith fi x points des c ri b es tr a nsition s y stems (rel a tions of a rit y a t most 2 ) de c id ab le s a tisfi ab ilit y ( EXPTIME - c omplete) tree model propert y 3 / 8

Some decidab le fi x point logi c s M od a l m u - ca l cu l u s (L µ ) U n a r y neg a tion fi x point logi c ( UNFP ) [Kozen ’83] [S egoufin, ten Ca te ’11] extension of mod a l logi c fr a gment of LFP with mon a di c fixpoints with fixpoints a nd neg a tion of formul a s with a t most one free v a ri ab le des c ri b es tr a nsition systems des c ri b es rel a tion a l stru c tures (rel a tions of a rity a t most 2) (rel a tions of a r b itr a ry a rity) de c id ab le s a tisfi ab ility de c id ab le s a tisfi ab ility ( EXPTIME - c omplete) (2 EXPTIME - c omplete) tree model property tree-like model property (models of b ounded tree-width) 3 / 8

UNFP UNFP is expressive: mod a l logi c a nd L µ , even with bac kw a rds mod a lities; positi v e e x istenti a l FO (i.e. u nions of c onj u n c ti v e q u eries); des c ription logi c s in c l u ding ALC , ALCHIO , ELI ; mon a di c Da t a log. 4 / 8

UNFP UNFP is expressive: mod a l logi c a nd L µ , even with bac kw a rds mod a lities; positi v e e x istenti a l FO (i.e. u nions of c onj u n c ti v e q u eries); des c ription logi c s in c l u ding ALC , ALCHIO , ELI ; mon a di c Da t a log. UNFP sh a res some properties w ith L µ ... 4 / 8

UNFP UNFP is expressive: mod a l logi c a nd L µ , even with bac kw a rds mod a lities; positi v e e x istenti a l FO (i.e. u nions of c onj u n c ti v e q u eries); des c ription logi c s in c l u ding ALC , ALCHIO , ELI ; mon a di c Da t a log. UNFP sh a res some properties w ith L µ ... ... w h a t ab o u t interpol a tion? 4 / 8

Int erpol a tion φ ψ ⊧ 5 / 8

Int erpol a tion interpol a nt φ θ ψ ⊧ ⊧ only uses rel a tions c ommon to φ a nd ψ 5 / 8

Int erpol a tion interpol a nt φ θ ψ ⊧ ⊧ only uses rel a tions c ommon to φ a nd ψ C r a ig interpol a tion: θ depends on φ a nd ψ 5 / 8

Int erpol a tion interpol a nt φ θ ψ ⊧ ⊧ only uses rel a tions c ommon to φ a nd ψ C r a ig interpol a tion: θ depends on φ a nd ψ U niform interpol a tion: θ depends only on φ a nd c ommon sign a ture (not on a p a rti c ul a r ψ ) 5 / 8

Int erpol a tion for L µ a nd UNFP T heorem (D’Agostino, Hollenberg ’ 00 ) L µ h a s effe c ti v e u niform interpol a tion. 6 / 8

Int erpol a tion for L µ a nd UNFP T heorem (D’Agostino, Hollenberg ’ 00 ) L µ h a s effe c ti v e u niform interpol a tion. L et UNFP k denote the k - va ri ab le fr a gment of UNFP (in norm a l form...). T heorem ( B enedikt, ten Ca te, VB . ’15) UNFP k h a s effe c ti v e u niform interpol a tion. UNFP h a s effe c ti v e C r a ig interpol a tion. 6 / 8

Int erpol a tion for L µ a nd UNFP T heorem (D’Agostino, Hollenberg ’ 00 ) L µ h a s effe c ti v e u niform interpol a tion. L et UNFP k denote the k - va ri ab le fr a gment of UNFP (in norm a l form...). T heorem ( B enedikt, ten Ca te, VB . ’15) UNFP k h a s effe c ti v e u niform interpol a tion. UNFP h a s effe c ti v e C r a ig interpol a tion. P roof str a teg y : B ootstr a p from mod a l w orld, m a king u se of res u lts/ide a s of [G r¨ a del, Wa l u kie w i cz ’99], [Gr¨ a del, H irs c h, O tto ’ 00 ], [D’Agostino, Hollenberg ’ 00 ] . 6 / 8

Uniform int erpol a tion for UNFP k T heorem (Benedikt, ten C a te, VB . ’15) UNFP k h a s effe c tive uniform interpol a tion. 7 / 8

Uniform int erpol a tion for UNFP k T heorem (Benedikt, ten C a te, VB . ’15) UNFP k h a s effe c tive uniform interpol a tion. P roof str uc t u re: C oded stru c tures R el a tion a l (tree de c ompositions of stru c tures width k ) UNFP k φ L µ ̂ φ 7 / 8

Uniform int erpol a tion for UNFP k T heorem (Benedikt, ten C a te, VB . ’15) UNFP k h a s effe c tive uniform interpol a tion. P roof str uc t u re: C oded stru c tures R el a tion a l (tree de c ompositions of stru c tures width k ) UNFP k φ L µ ̂ φ [D ’ A gostino, H ollen b erg’ 00 ] L µ ̂ θ over su b sign a ture en c oding 7 / 8

Uniform int erpol a tion for UNFP k T heorem (Benedikt, ten C a te, VB . ’15) UNFP k h a s effe c tive uniform interpol a tion. P roof str uc t u re: C oded stru c tures R el a tion a l (tree de c ompositions of stru c tures width k ) UNFP k φ L µ ̂ φ [D ’ A gostino, H ollen b erg’ 00 ] UNFP k θ L µ ̂ θ over su b sign a ture over su b sign a ture en c oding 7 / 8

Conc l u sion UNFP is a n expressive, de c id ab le fixpoint logi c with effe c tive interpol a tion. 8 / 8

Conc l u sion UNFP is a n expressive, de c id ab le fixpoint logi c with effe c tive interpol a tion. I s there some de c id ab le extension of UNFP th a t h a s interpol a tion? ( W e a lre a dy know th a t the guarded negation fixpoint logi c ( GNFP ) f a ils to h a ve interpol a tion.) 8 / 8

Conc l u sion UNFP is a n expressive, de c id ab le fixpoint logi c with effe c tive interpol a tion. I s there some de c id ab le extension of UNFP th a t h a s interpol a tion? ( W e a lre a dy know th a t the guarded negation fixpoint logi c ( GNFP ) f a ils to h a ve interpol a tion.) Ca n this result ab out UNFP help us a nswer a ny interesting query rewriting pro b lems? 8 / 8

Uniform int erpol a tion e xa mple “ S holds a t x , a nd from every position y where S holds, there is a n R -neigh b or z where S holds” φ ( x ) ∶ = Sx ∧ ∀ y ( Sy → ∃ z ( Ryz ∧ Sz )) ≡ Sx ∧ ¬∃ y ( Sy ∧ ¬∃ z ( Ryz ∧ Sz ))

Uniform int erpol a tion e xa mple “ S holds a t x , a nd from every position y where S holds, there is a n R -neigh b or z where S holds” φ ( x ) ∶ = Sx ∧ ∀ y ( Sy → ∃ z ( Ryz ∧ Sz )) ≡ Sx ∧ ¬∃ y ( Sy ∧ ¬∃ z ( Ryz ∧ Sz )) U niform interpol a nt of φ o v er s ub sign a t u re { R } “there is a n infinite R -p a th from x ” [ gfp Y , y . ∃ z ( Ryz ∧ Yz )]( x ) ≡ ¬ [ lfp Y , y . ¬∃ z ( Ryz ∧ ¬ Yz )]( x )