A step up in expressiveness of decidab le fi x point logi c s Micha - PowerPoint PPT Presentation

A step up in expressiveness of decidab le fi x point logi c s Micha el B enedikt 1 , P ierre B o u rhis 2 , a nd M i c h a el Va nden B oom 1 1 U ni v ersit y of Ox ford 2 CNRS CRIS t AL , U ni v ersit ´ e L ille 1, INRIA L ille LICS 20 16 N e w Y ork, USA 1 / 15

Fixpoint logi c s Fixpoint logi c s ca n 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 w , it is possi b le to R -re ac h some P -element” [ Reach- P ]( w ) 2 / 15

Fixpoint logi c s Fixpoint logi c s ca n 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 w , it is possi b le to R -re ac h some P -element” [ lfp Y , y . ∃ z ( Ryz ∧ ( Pz ∨ Yz ))]( w ) 2 / 15

LFP LFP: extension of first-order logic w ith fi x point form u l a s [ lfp Y , y . ψ ( y , Y )]( w ) for ψ ( y , Y ) positi v e in Y (of a rit y m = ∣ y ∣ ). F or 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 3 / 15

LFP LFP: extension of first-order logic with fixpoint formul a s [ lfp Y , y . ψ ( y , Y )]( w ) for ψ ( y , Y ) positive in Y (of a rity m = ∣ y ∣ ). F or a ll stru c tures A , the formul a ψ indu c 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 unique le a st fixpoint [ 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 S em a nti c s of fi x point oper a tor: A , a ⊧ [ lfp Y , y . ψ ( y , Y )]( w ) iff a ∈ ⋃ α ψ α A 3 / 15

Exa mples “from w , it is possible to R -re ac h some P -element” [ lfp Y , y . ∃ z ( Ryz ∧ ( Pz ∨ Yz ))]( w ) a 1 a 2 a 3 a k a k + 1 4 / 15

Exa mples “from w , it is possible to R -re ac h some P -element” [ lfp Y , y . ∃ z ( Ryz ∧ ( Pz ∨ Yz ))]( w ) a 1 a 2 a 3 a k a k + 1 , i.e. “ ( w , x ) is in the tr a nsitive c losure of R ” “from w , it is possi b le to R -re ac h x ” [ lfp Y , y . ∃ z ( Ryz ∧ ( z = x ∨ Yz ))]( w ) ( F ree first-order v a ri ab le x in the fixpoint predi ca te is ca lled a p a r a meter.) 4 / 15

Some decidab le fr a gments of LFP (fi x point e x tension of FO ) The f a mily of “gu a rded” fixpoint logi c s h a s de c id ab le s a tisfi ab ility. GFP LFP L µ GNFP UNFP G u a rded fixpoint logi c ( GFP ): A ndr ´ ek a , v a n B enthem, N´ emeti ’95-’98; G r¨ a del, Wa l u kie w i c z ’99 U n a r y neg a tion fi x point logi c ( UNFP ): ten Ca te, S ego u fin ’11 G u a rded neg a tion fi x point logi c ( GNFP ): B´ a r ´ a n y , ten Ca te, S ego u fin ’11 5 / 15

Guarded negation fixpoint logi c ( GNFP ) Let σ be a sign a ture of rel a tions a nd c onst a nts. Sy nt ax of GNFP [ σ ] φ ∶∶ = R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃ y ( ψ ( x y )) ∣ G ( x ) ∧ ¬ ψ ( x ) ∣ [ lfp Y , y . G ( y ) ∧ φ ( y , Y , Z )]( t ) where Y only o cc urs positively in φ where R a nd G a re rel a tions in σ or = , a nd t is a tuple over v a ri ab les a nd c onst a nts. 6 / 15

Guarded negation fixpoint logi c ( GNFP ) Let σ be a sign a ture of rel a tions a nd c onst a nts. Sy nt ax of GNFP [ σ ] φ ∶∶ = R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃ y ( ψ ( x y )) ∣ G ( x ) ∧ ¬ ψ ( x ) ∣ [ lfp Y , y . G ( y ) ∧ φ ( y , Y , Z )]( t ) where Y only o cc urs positively in φ where R a nd G a re rel a tions in σ or = , a nd t is a tuple over v a ri ab les a nd c onst a nts. R estri c tions on fixpoint oper a tor: must define a gu a rded rel a tion (tuples in the fixpoint must b e gu a rded b y a n a tom from σ or = ) ca nnot use p a r a meters 6 / 15

Satisfiab ilit y These gu a rded fixpoint logi c s a ll h a ve the tree-like model property (models with tree de c ompositions of b ounded tree-width) ⇒ a men ab le to tree a utom a t a te c hniques 7 / 15

Satisfiab ilit y These gu a rded fixpoint logi c s a ll h a ve the tree-like model property (models with tree de c ompositions of b ounded tree-width) ⇒ a men ab le to tree a utom a t a te c hniques T heorem ( G r¨ a del, Wa lukiewi c z ’99; B´ a r ´ a n y , S ego u fin, ten Ca te ’11; B´ a r ´ a n y , B oj a´ n c zy k ’1 2 ) Sa tisfi ab ilit y a nd finite s a tisfi ab ilit y a re de c id ab le for g u a rded fi x point logi c s ( 2 EXPTIME in gener a l, EXPTIME for fi x ed- w idth form u l a s in GFP ). I de a : R ed u c e to tree a u tom a ton emptiness test. 7 / 15

Exa mples I n GNFP : [ lfp Y , y . ∃ z ( R yz ∧ ( P z ∨ Y z ))]( w ) 8 / 15

Exa mples I n GNFP : [ lfp Y , y . y = y ∧ ∃ z ( R yz ∧ ( P z ∨ Y z ))]( w ) 8 / 15

Exa mples I n GNFP : [ lfp Y , y . y = y ∧ ∃ z ( R yz ∧ ( P z ∨ Y z ))]( w ) N ot in GNFP : [ lfp Y , y . y = y ∧ ∃ z ( R yz ∧ ( z = x ∨ Y z ))]( w ) 8 / 15

Can we go further? GFP LFP L µ GNFP UNFP Re ca ll the restri c tions on the fixpoint oper a tors in GNFP : must define a gu a rded rel a tion ca nnot use p a r a meters W hi c h of these restri c tions a re essenti a l for de c id ab ility? 9 / 15

Can we go further? GFP LFP L µ GNFP UNFP Re ca ll the restri c tions on the fixpoint oper a tors in GNFP : must define a gu a rded rel a tion ca nnot use p a r a meters W hi c h of these restri c tions a re essenti a l for de c id ab ility? Answer: only first one! 9 / 15

GNFP UP GNFP UP : extend GNFP with ungu a rded p a r a meters in fixpoint 1 0 / 15

GNFP UP GNFP UP : extend GNFP with ungu a rded p a r a meters in fixpoint Syntax of GNFP UP [ σ ] φ ∶∶ = R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃ y ( ψ ( x y )) ∣ G ( x ) ∧ ¬ ψ ( x ) ∣ [ lfp Y , y . G ( y ) ∧ φ ( x , y , Y , Z )]( t ) where Y only o cc urs positively in φ where R a nd G a re rel a tions in σ or = , a nd t is a tuple over v a ri ab les a nd c onst a nts. 1 0 / 15

GNFP UP GNFP UP : extend GNFP with ungu a rded p a r a meters in fixpoint Syntax of GNFP UP [ σ ] φ ∶∶ = R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃ y ( ψ ( x y )) ∣ G ( x ) ∧ ¬ ψ ( x ) ∣ [ lfp Y , y . G ( y ) ∧ φ ( x , y , Y , Z )]( t ) where Y only o cc urs positively in φ where R a nd G a re rel a tions in σ or = , a nd t is a tuple over v a ri ab les a nd c onst a nts. Exa mple GNFP UP ca n express the tr a nsitive c losure of a b in a ry rel a tion R using [ lfp Y , y . ∃ z ( R yz ∧ ( z = x ∨ Y z ))]( w ) 1 0 / 15

Expressivity of GNFP UP GFP LFP L µ GNFP UP GNFP UNFP GNFP UP a lso su b sumes C 2 RPQ s ( c onjun c tive 2-w a y regul a r p a th queries) ∃ x yz ( [ R ∗ S ]( x , y ) ∧ [ S ∣ R ]( y , z ) ∧ P ( z ) ) MQ s a nd GQ s [R udolph, K r¨ otzs c h ’13 ; B o u rhis, K r¨ ot z s c h, R u dolph ’15 ] 11 / 15

Satisfiab ilit y for GNFP UP GNFP UP still h a s tree-like models ⇒ still a men ab le to tree a utom a t a te c hniques U nlike other gu a rded logi c s, s a tisfi ab ility testing for φ ∈ GNFP UP is non-element a ry, with running time 2 2 . . .2 f (∣ φ ∣) where the height of the tower depends only on the p a r a meter depth: the num b er of nested p a r a meter c h a nges in the formul a . 1 2 / 15

Satisfiab ilit y for GNFP UP GNFP UP still h a s tree-like models ⇒ still a men ab le to tree a utom a t a te c hniques U nlike other gu a rded logi c s, s a tisfi ab ility testing for φ ∈ GNFP UP is non-element a ry, with running time 2 2 . . .2 f (∣ φ ∣) where the height of the tower depends only on the p a r a meter depth: the num b er of nested p a r a meter c h a nges in the formul a . T heorem Sa tisfi ab ility is de c id ab le for φ ∈ GNFP UP in ( n + 2 ) - EXPTIME , where n is the p a r a meter depth of φ . 1 2 / 15

Skirting undecidab ilit y It is known th a t s a tisfi ab ility is unde c id ab le for GFP (even without fixpoints) when c ert a in rel a tions a re required to b e tr a nsitive. [G r¨ a del ’99, Ga nzinger et a l. ’99 ] 1 3 / 15

Skirting undecidab ilit y It is known th a t s a tisfi ab ility is unde c id ab le for GFP (even without fixpoints) when c ert a in rel a tions a re required to b e tr a nsitive. [G r¨ a del ’99, Ga nzinger et a l. ’99 ] GNFP UP ca n express the tr a nsitive c losure of a b in a ry rel a tion R using [ lfp Y , y . ∃ z ( R yz ∧ ( z = x ∨ Y z ))]( w ) . B ut it ca nnot enfor c e th a t R is tr a nsitive. 1 3 / 15

FO-definab ilit y T heorem It is decid ab le whether [ lfp Y , y . G ( y ) ∧ ψ ( x , y , Y )]( w ) ∈ GNFP UP ca n b e expressed in FO (when ψ does not use a ny a ddition a l fixpoints). I t is de c id ab le whether a C 2 RPQ ca n b e expressed in FO . I de a : A d a pt a utom a t a for GNFP UP , a nd redu c e to a b oundedness question for c ost a utom a t a ( a utom a t a with c ounters). 14 / 15

Conc l u sion We ca n a llow ungu a rded p a r a meters in gu a rded fixpoint logi c s. C ontri bu tions W e showed th a t: ▶ tree a utom a t a te c hniques ca n b e used to a n a lyze GNFP UP ▶ s a tisfi ab ility is de c id ab le for GNFP UP , a nd the key f ac tor imp ac ting the c omplexity is the p a r a meter depth ▶ some b oundedness a nd FO -defin ab ility pro b lems a re de c id ab le for GNFP UP 15 / 15