Query Answering with Transitive and Linear-Ordered Data Antoine Amar - PowerPoint PPT Presentation

Query Answering with Transitive and Linear-Ordered Data Antoine Amar illi 1 , M i c h a el B enedikt 2 , P ierre B o u rhis 3 a nd Micha el Va nden B oom 2 1 LTCI , CNRS , T´ el ´ e c om Pa ris T e c h, U ni v ersit ´ e Pa ris- Sac l ay 2 U ni v ersit y of Ox ford 3 CNRS CRIS t AL , U ni v ersit ´ e L ille 1, INRIA L ille IJCAI 20 16 N e w Y ork, USA 1 / 8

Q uery a nswering pro b lem ( QA ) Given: finite set of initia l f ac ts F 0 , c onstr a ints Σ , b oole a n q u er y Q ( UCQ ). T he query a nswering pro b lem QA ( F 0 , Σ , Q ) a sks: does F 0 ∧ Σ ent a il Q ? 2 / 8

Q uery a nswering pro b lem ( QA ) Given: finite set of initia l f ac ts F 0 , c onstr a ints Σ , b oole a n q u er y Q ( UCQ ). T he query a nswering pro b lem QA ( F 0 , Σ , Q ) a sks: does F 0 ∧ Σ ent a il Q ? E q u i va lentl y : is Q c ert a in gi v en F 0 a nd Σ ? is F 0 ∧ Σ ∧ ¬ Q u ns a tisfi ab le? for a ll sets of f ac ts F ⊇ F 0 s a tisf y ing Σ , does F s a tisf y Q ? 2 / 8

Q uery a nswering pro b lem ( QA ) Given: finite set of initia l f ac ts F 0 , c onstr a ints Σ , b oole a n q u er y Q ( UCQ ). T he query a nswering pro b lem QA ( F 0 , Σ , Q ) a sks: does F 0 ∧ Σ ent a il Q ? E q u i va lentl y : is Q c ert a in gi v en F 0 a nd Σ ? is F 0 ∧ Σ ∧ ¬ Q u ns a tisfi ab le? for a ll sets of f ac ts F ⊇ F 0 s a tisf y ing Σ , does F s a tisf y Q ? E x a mple S ( a , b ) , R ( b , a ) F 0 ∶ ∀ xy ( S ( x , y ) → R ( x , y )) Σ ∶ ∀ x ( R ( x , x ) → ∃ y T ( y )) ∃ x T ( x ) Q ∶ Q is not c ert a in in gener a l... 2 / 8

Q uery a nswering pro b lem ( QA ) Given: finite set of initia l f ac ts F 0 , c onstr a ints Σ , b oole a n q u er y Q ( UCQ ). T he query a nswering pro b lem QA ( F 0 , Σ , Q ) a sks: does F 0 ∧ Σ ent a il Q ? E q u i va lentl y : is Q c ert a in gi v en F 0 a nd Σ ? is F 0 ∧ Σ ∧ ¬ Q u ns a tisfi ab le? for a ll sets of f ac ts F ⊇ F 0 s a tisf y ing Σ , does F s a tisf y Q ? E x a mple S ( a , b ) , R ( b , a ) F 0 ∶ ∀ xy ( S ( x , y ) → R ( x , y )) Σ ∶ ∀ x ( R ( x , x ) → ∃ y T ( y )) ∃ x T ( x ) Q ∶ Q is not c ert a in in gener a l... bu t it is c ert a in w hen R is a tr a nsiti v e rel a tion. 2 / 8

T r a nsitivity in des c ription logi c s Many DLs support transitive r el a tions. QA is de c id ab le for ZIQ , ZOQ , ZOI [Ca l va nese et a l., 200 9 ] H orn- SROIQ [O rti z et a l., 20 11 ] reg u l a r- EL ++ [K r¨ ot z s c h a nd R u dolph, 200 7 ] (sometimes w ith restri c tions on inter ac tion b et w een tr a nsiti v it y & other fe a t u res). QA is u nde c id ab le for ALCOIF ∗ [O rti z et a l., 20 1 0 ] ZOIQ [O rti z , 20 1 0 ] QA is open for SROIQ a nd SHOIQ [O rti z a nd ˇ S imk u s, 20 1 2 ] 3 / 8

QA with tuple gener a ting dependen c ies ( a .k. a . existenti a l rules) TGD : ∀ xy ( φ ( x , y ) → ∃ z ψ ( y , z )) body φ and head ψ are a conjunction of atoms 4 / 8

QA with tuple gener a ting dependen c ies ( a .k. a . existenti a l rules) TGD : ∀ xy ( φ ( x , y ) → ∃ z ψ ( y , z )) body φ and head ψ are a conjunction of atoms F rontier-gu a rded TGD (FGTGD): φ inc l u des a tom u sing a ll of the frontier va ri ab les y ∀ x y 1 y 2 ( S ( x , y 1 ) ∧ S ( x , y 2 ) ∧ R ( y 1 , y 2 ) → ∃ z ( S ( y 2 , z ) ∧ T ( y 1 )) ) 4 / 8

QA with tuple gener a ting dependen c ies ( a .k. a . existenti a l rules) TGD : ∀ xy ( φ ( x , y ) → ∃ z ψ ( y , z )) body φ and head ψ are a conjunction of atoms F rontier-gu a rded TGD (FGTGD): φ inc l u des a tom u sing a ll of the frontier va ri ab les y ∀ x y 1 y 2 ( S ( x , y 1 ) ∧ S ( x , y 2 ) ∧ R ( y 1 , y 2 ) → ∃ z ( S ( y 2 , z ) ∧ T ( y 1 )) ) QA is de c id ab le w ith FGTGD c onstr a ints a nd UCQ . [Ba get et a l., 20 11 ] 4 / 8

QA with tuple gener a ting dependen c ies ( a .k. a . existenti a l rules) TGD : ∀ xy ( φ ( x , y ) → ∃ z ψ ( y , z )) body φ and head ψ are a conjunction of atoms F rontier-gu a rded TGD (FGTGD): φ inc l u des a tom u sing a ll of the frontier va ri ab les y ∀ x y 1 y 2 ( S ( x , y 1 ) ∧ S ( x , y 2 ) ∧ R ( y 1 , y 2 ) → ∃ z ( S ( y 2 , z ) ∧ T ( y 1 )) ) QA is de c id ab le w ith FGTGD c onstr a ints a nd UCQ . [Ba get et a l., 20 11 ] FGTGD s ca nnot e x press tr a nsiti v it y , a nd QA is u nde c id ab le w ith FGTGD s w hen some rel a tions a re req u ired to b e tr a nsiti v e. [G ottlo b et a l., 20 1 3 ] 4 / 8

QA with tuple gener a ting dependen c ies ( a .k. a . existenti a l rules) TGD : ∀ xy ( φ ( x , y ) → ∃ z ψ ( y , z )) body φ and head ψ are a conjunction of atoms F rontier-gu a rded TGD (FGTGD): φ inc l u des a tom u sing a ll of the frontier va ri ab les y ∀ x y 1 y 2 ( S ( x , y 1 ) ∧ S ( x , y 2 ) ∧ R ( y 1 , y 2 ) → ∃ z ( S ( y 2 , z ) ∧ T ( y 1 )) ) QA is de c id ab le w ith FGTGD c onstr a ints a nd UCQ . [Ba get et a l., 20 11 ] FGTGD s ca nnot e x press tr a nsiti v it y , a nd QA is u nde c id ab le w ith FGTGD s w hen some rel a tions a re req u ired to b e tr a nsiti v e. [G ottlo b et a l., 20 1 3 ] H o w ca n w e re c o v er de c id ab ilit y for QA w ith tr a nsiti v e rel a tions? restri c t to (s u bc l a ss of) line a r TGD s [Ba get et a l., 20 15 ] ; dis a llo w the tr a nsiti v e rel a tions a s g u a rds (o u r a ppro ac h) . 4 / 8

O ur a ppro ac h Fix r el a tion a l sign a t u re σ ∶ = σ B ⊔ σ D w here σ D : distinguished b in a ry rel a tions w ith spe c i a l interpret a tions (e.g., tr a nsiti v el y c losed) σ B : ba se rel a tions W e introd uc e c onstr a int l a ng ua ges th a t dis a llo w σ D -rel a tions a s g ua rds: Ba se FGTGD : FGTGD w here g ua rd for frontier va ri ab les is from σ B . ∀ x y 1 y 2 ( R ( x , y 1 ) ∧ R ( x , y 2 ) ∧ S ( y 1 , y 2 ) → ∃ z ( R ( y 2 , z ) ∧ T ( y 1 )) ) 5 / 8

O ur a ppro ac h Fix r el a tion a l sign a t u re σ ∶ = σ B ⊔ σ D w here σ D : distinguished b in a ry rel a tions w ith spe c i a l interpret a tions (e.g., tr a nsiti v el y c losed) σ B : ba se rel a tions W e introd uc e c onstr a int l a ng ua ges th a t dis a llo w σ D -rel a tions a s g ua rds: Ba se FGTGD : FGTGD w here g ua rd for frontier va ri ab les is from σ B . ∀ x y 1 y 2 ( R ( x , y 1 ) ∧ R ( x , y 2 ) ∧ S ( y 1 , y 2 ) → ∃ z ( R ( y 2 , z ) ∧ T ( y 1 )) ) Ba se- c overed FGTGD : Ba se FGTGD w here for e v er y σ D - a tom in the b od y , there is a σ B - a tom in the b od y u sing its va ri ab les. ∀ x y 1 y 2 ( C ( x , y 1 ) ∧ R ( x , y 1 ) ∧ C ( x , y 2 ) ∧ R ( x , y 2 ) ∧ S ( y 1 , y 2 ) → ∃ z ( R ( y 2 , z ) ∧ T ( y 1 )) ) 5 / 8

O ur c ontri b ution We consider three different specia l interpret a tions for rel a tions in σ D : QA tr e ac h R ∈ σ D is tr a nsiti v el y c losed e ac h R + ∈ σ D is the tr a nsiti v e c los u re of R ∈ σ B QA t c QA lin e ac h R ∈ σ D is a line a r order 6 / 8

O ur c ontri b ution We consider three different specia l interpret a tions for rel a tions in σ D : QA tr e ac h R ∈ σ D is tr a nsiti v el y c losed e ac h R + ∈ σ D is the tr a nsiti v e c los u re of R ∈ σ B QA t c QA lin e ac h R ∈ σ D is a line a r order T heorem QA tr a nd QA t c a re de c id ab le w ith ba se FGTGD s a nd UCQ . QA lin is de c id ab le w ith ba se- c o v ered FGTGD s a nd ba se- c o v ered UCQ . 6 / 8

O ur c ontri b ution We consider three different specia l interpret a tions for rel a tions in σ D : QA tr e ac h R ∈ σ D is tr a nsiti v el y c losed e ac h R + ∈ σ D is the tr a nsiti v e c los u re of R ∈ σ B QA t c QA lin e ac h R ∈ σ D is a line a r order T heorem QA tr a nd QA t c a re de c id ab le w ith ba se FGTGD s a nd UCQ . QA lin is de c id ab le w ith ba se- c o v ered FGTGD s a nd ba se- c o v ered UCQ . W e a lso a n a l yz e c om b ined c omple x it y a nd d a t a c omple x it y , a nd sho w th a t slight c h a nges in the restri c tions le a d to u nde c id ab ilit y . 6 / 8

T r a nsitive rel a tions T heorem QA tr ( F 0 , Σ, Q ) is decidab le in 2EXPTIME c om b ined c omple x it y a nd PTIME d a t a c omple x it y for ba se- c o v ered FGTGD s Σ a nd ba se- c o v ered UCQ Q . P roof ide a : R ed uc e in PTIME to tr a dition a l QA pro b lem QA ( F 0 , Σ ′ , Q ) w ith FGTGD s Σ ′ . 7 / 8

T r a nsitive rel a tions T heorem QA tr ( F 0 , Σ, Q ) is decidab le in 2EXPTIME c om b ined c omple x it y a nd PTIME d a t a c omple x it y for ba se- c o v ered FGTGD s Σ a nd ba se- c o v ered UCQ Q . P roof ide a : R ed uc e in PTIME to tr a dition a l QA pro b lem QA ( F 0 , Σ ′ , Q ) w ith FGTGD s Σ ′ . Bad news: w e ca nnot ax iom a ti z e tr a nsiti v it y u sing FGTGD s. 7 / 8

T r a nsitive rel a tions T heorem QA tr ( F 0 , Σ, Q ) is decidab le in 2EXPTIME c om b ined c omple x it y a nd PTIME d a t a c omple x it y for ba se- c o v ered FGTGD s Σ a nd ba se- c o v ered UCQ Q . P roof ide a : R ed uc e in PTIME to tr a dition a l QA pro b lem QA ( F 0 , Σ ′ , Q ) w ith FGTGD s Σ ′ . Bad news: w e ca nnot ax iom a ti z e tr a nsiti v it y u sing FGTGD s. Good news : w e ca n a ppro x im a te tr a nsiti v it y u sing FGTGD c onstr a ints Σ ′ ⊇ Σ . I f F 0 ∧ Σ ′ ∧ ¬ Q is s a tisfi ab le, then it h a s a tree-like witness ( a set of f ac ts w ith a tree de c omposition of some b o u nded tree- w idth). Key technica l res u lt : T his tree-like w itness ca n b e e x tended to a set of f ac ts s a tisf y ing F 0 ∧ Σ ∧ ¬ Q w here R ∈ σ D is tr a nsiti v el y c losed. 7 / 8