The Complexity of Boundedness for Gu a rded L ogi c s Micha el B enedikt 1 , Ba lder ten Ca te 2 , T hom a s C ol c om b et 3 , M i c h a el Va nden B oom 1 1 U ni v ersit y of Ox ford 2 L ogi cB lo x a nd UC Sa nt a C r uz 3 U ni v ersit ´ e Pa ris D iderot LICS 20 15 Ky oto, Ja p a n 1 / 12
Least fixpoint Consider ψ ( y , Y ) positive in Y (of arity m = ∣ y ∣ ). For a ll str uc t u res A , the form u l a ψ ind uc es a monotone oper a tion P ( A m ) ⟶ P ( A m ) V ⟼ ψ A ( V ) ∶ = { a ∈ A m ∶ A , a , V ⊧ ψ } ⇒ there is a u niq u e le a st fi x point [ lfp Y , y . ψ ( y , Y )] A ∶ = ⋃ α ψ α A ψ 0 A ∶ = ∅ ψ α + 1 ∶ = ψ A ( ψ α A ) A ψ λ A ∶ = ⋃ ψ α A α < λ 2 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll str uc t u res A , ψ n A = ψ n + 1 A ? (i.e. the le a st fi x point is a l way s re ac hed w ithin n iter a tions) 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) un b ounded 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) un b ounded [ lfp Y , xy . ψ 1 ] ≡ tr a nsitive c losure of R 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) un b ounded [ lfp Y , xy . ψ 1 ] ≡ tr a nsitive c losure of R ψ 2 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Yzy ) 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) un b ounded [ lfp Y , xy . ψ 1 ] ≡ tr a nsitive c losure of R ψ 2 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Yzy ) [ lfp Y , xy . ψ 2 ]( xy ) ≡ Rxy ∨ ∃ z ( Rzy ) 3 / 1 2
Boundedness prob lem B o u ndedness pro b lem for L I np u t: ψ ( y , Y ) ∈ L positive in Y Qu estion: is there n ∈ N s.t. for a ll stru c tures A , ψ n A = ψ n + 1 A ? (i.e. the le a st fixpoint is a lw a ys re ac hed within n iter a tions) ψ 1 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Rxz ∧ Yzy ) un b ounded [ lfp Y , xy . ψ 1 ] ≡ tr a nsitive c losure of R ψ 2 ( xy , Y ) ∶ = Rxy ∨ ∃ z ( Yzy ) b ounded [ lfp Y , xy . ψ 2 ]( xy ) ≡ Rxy ∨ ∃ z ( Rzy ) 3 / 1 2
Some prior resu lts Boundedness is u nde c id ab le for bin a ry predi ca te in positive existenti a l FO (i.e. Da t a log) [H ille b r a nd, Ka nell a kis, Ma irson, Va rdi ’95] mon a di c predi ca te in existenti a l FO with inequ a lities [Ga ifm a n, Ma irson, Sa giv, Va rdi ’87 ] mon a di c predi ca te in FO 2 [K ol a itis, O tto ’98 ] 4 / 1 2
Some prior resu lts Boundedness is u nde c id ab le for B o u ndedness is de c id ab le for bin a ry predi ca te in positive mon a di c predi ca te in positi v e existenti a l FO (i.e. Da t a log) e x istenti a l FO (i.e. mon a di c Da t a log) [H ille b r a nd, Ka nell a kis, Ma irson, Va rdi ’95] [C osm a d a kis, Ga ifm a n, Ka nell a kis, Va rdi ’88 ] 2EXPTIME mon a di c predi ca te in existenti a l FO with inequ a lities mon a di c predi ca te in mod a l logi c [Ga ifm a n, Ma irson, Sa giv, Va rdi ’87 ] [O tto ’99 ] EXPTIME mon a di c predi ca te in FO 2 predi ca tes in [K ol a itis, O tto ’98 ] “g u a rded logi c s” [B l u mens a th, O tto, W e y er ’14 ] [B´ a r ´ a n y , ten Ca te, O tto ’1 2] non-element a r y u pper b o u nd 4 / 1 2
Some prior resu lts Boundedness is u nde c id ab le for B o u ndedness is de c id ab le for bin a ry predi ca te in positive mon a di c predi ca te in positi v e existenti a l FO (i.e. Da t a log) e x istenti a l FO (i.e. mon a di c Da t a log) [H ille b r a nd, Ka nell a kis, Ma irson, Va rdi ’95] [C osm a d a kis, Ga ifm a n, Ka nell a kis, Va rdi ’88 ] 2EXPTIME mon a di c predi ca te in existenti a l FO with inequ a lities mon a di c predi ca te in mod a l logi c [Ga ifm a n, Ma irson, Sa giv, Va rdi ’87 ] [O tto ’99 ] EXPTIME mon a di c predi ca te in FO 2 predi ca tes in [K ol a itis, O tto ’98 ] “g u a rded logi c s” [B l u mens a th, O tto, W e y er ’14 ] [B´ a r ´ a n y , ten Ca te, O tto ’1 2] non-element a r y u pper b o u nd o u r c ontri bu tion : element a r y u pper b o u nd (or b etter) 4 / 1 2
Guarded logi c s constr a in qu a ntifi ca tion ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [A ndr ´ ek a , v a n B enthem, FO N´ emeti ’95-’98] ML 5 / 1 2
Guarded logi c s constr a in qu a ntifi ca tion ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [A ndr ´ ek a , v a n B enthem, FO N´ emeti ’95-’98] ML constr a in UNF neg a tion ∃ x ( ψ ( xy )) ¬ ψ ( x ) [ ten Ca te, S egoufin ’11] 5 / 1 2
Guarded logi c s constr a in qu a ntifi ca tion ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GF [A ndr ´ ek a , v a n B enthem, 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] 5 / 1 2
Guarded logi c s constr a in qu a ntifi ca tion ∃ x ( G ( xy ) ∧ ψ ( xy )) ∀ x ( G ( xy ) → ψ ( xy )) GFP FO [A ndr ´ ek a , v a n B enthem, + N´ emeti ’95-’98] L µ GNFP LFP c onstr a in UNFP 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] 6 / 1 2
Guarded logi c s Gu a rded logi c s a re expressive. F or inst a n c e, GNFP ca ptures: mu- ca l c ulus, 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. 7 / 12
Guarded logi c s Gu a rded logi c s a re expressive. F or inst a n c e, GNFP ca ptures: mu- ca l c ulus, 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. G u a rded logi c s h a v e m a n y ni c e model theoreti c properties. GF , UNF , a nd GNF h a v e finite models. GFP , UNFP , a nd GNFP h a v e tree-like models (models of b o u nded tree- w idth). 7 / 12
Guarded logi c s Gu a rded logi c s a re expressive. F or inst a n c e, GNFP ca ptures: mu- ca l c ulus, 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. G u a rded logi c s h a v e m a n y ni c e model theoreti c properties. GF , UNF , a nd GNF h a v e finite models. GFP , UNFP , a nd GNFP h a v e tree-like models (models of b o u nded tree- w idth). G u a rded logi c s h a v e ni c e c omp u t a tion a l properties. Sa tisfi ab ilit y is de c id ab le, a nd is 2 EXPTIME - c omplete (e v en EXPTIME - c omplete for fi x ed- w idth GFP ). 7 / 12
Boundedness for guarded logi c s (We s a y ψ ( x ) is a ns w er-g ua rded if it is of the form G ( x ) ∧ ψ ′ ( x ) .) C oroll a ry to tree-like model property F or ψ in GFP or a nswer-gu a rded GNFP : ψ is b ounded over a ll stru c tures iff ψ is b ounded over tree-like stru c tures. 8 / 12
Boundedness for guarded logi c s (We s a y ψ ( x ) is a ns w er-g ua rded if it is of the form G ( x ) ∧ ψ ′ ( x ) .) C oroll a ry to tree-like model property F or ψ in GFP or a nswer-gu a rded GNFP : ψ is b ounded over a ll stru c tures iff ψ is b ounded over tree-like stru c tures. ⇒ a men ab le to te c hniques using tree a utom a t a 8 / 12
Boundedness for guarded logi c s (We s a y ψ ( x ) is a ns w er-g ua rded if it is of the form G ( x ) ∧ ψ ′ ( x ) .) C oroll a ry to tree-like model property F or ψ in GFP or a nswer-gu a rded GNFP : ψ is b ounded over a ll stru c tures iff ψ is b ounded over tree-like stru c tures. ⇒ a men ab le to te c hniques using tree a utom a t a L ogi c - a utom a t a c onne c tion utilized in B lumens a th et a l. ’14 b ut only yields non-element a ry c omplexity sin c e their proof goes vi a MSO . 8 / 12
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