-ASP for computing repairs with existential ontologies Jean-Franc - PowerPoint PPT Presentation

∃ -ASP for computing repairs with existential ontologies Jean-Franc ¸ois Baget (1) Zied Bouraoui (2) Farid Nouioua (3) Odile Papini (3) Swan Rocher (4) Eric W¨ urbel (3) (1) INRIA, France. (2) Cardiff University, UK. (3) Aix-Marseille University, France. (4) Montpellier University, France. Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 1 / 31

The aim Framework : Existential rules Inconsistency-tolerant inference Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 2 / 31

The aim Framework : Existential rules Inconsistency-tolerant inference Repairs Unified framework [Baget, Benferhat, Bouraoui, Croitoru, Mugnier, Papini, Rocher, Tabia, KR 2016] Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 2 / 31

The aim Framework : Existential rules Inconsistency-tolerant ∃ -ASP Program inference ∃ -ASP [Garreau, Garcia, Lef` evre, St´ ephan, ONTOLP 2015] Repairs Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 2 / 31

The aim Framework : Existential rules Inconsistency-tolerant ∃ -ASP Program inference ASPeRiX Repairs Answer Sets Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 2 / 31

Overview Premliminaries 1 Existential rules ∃ -ASP From inconsistency-tolerant inferences to ∃ -ASP 2 One-to-one correspondence between answer sets and repairs 3 Conclusion 4 Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 3 / 31

Premliminaries Existential rules Existential rules Knowledge base K = ( F , R , N ) F : Set of facts Existential closure of conjunction of atoms R : Set of existential rules rules of the form B → H where body B and head H are conjunction of atoms. B [ � X , � Y ] → H [ � Y , � X , � � Y : universally quantified, � Z ] Z : existential variable FOL : ∀ � X , ∀ � Y ( φ ( B ) → ( ∃ � Z φ ( H ))) . N : Set of constraints rules of the form B → ⊥ where B is a set of atoms Inconsistent Knowledge base K : ⊥ ∈ Cl ( F , R ∪ N ) Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 4 / 31

Premliminaries Existential rules Existential rules Classes where the skolem chase stops [Baget, Garreau, Mugnier, Rocher. NM2014] mfa 2-Exp-Time Summarizing Results msa U+ msa U msa D Exp-Time msa swa U+ swa U ja U+ swa D ja U ar U+ ja D swa ar U fd U+ ar D ja fd U wa U+ fd D co-NP ar wa U wa D fd co-NP P wa agrd EMETTEUR - NOM DE LA PRESENTATION J.-F. Baget NMR 2014 19 juillet 2014 1 Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 5 / 31

Premliminaries Existential rules Repairs K = ( F , R , N ) with F : set of ground atoms standard repair of K : R ( K ) An Inclusion-maximal subset of F (consistent w.r.t. ( R , N ) ) g + Cl ( X ) : ground positive closure of a set of atoms X . The restriction of Cl ( X , R ) to basic ground atoms closed repair of K : CR ( K ) A set of basic ground atoms g + Cl ( F ′ ) , where F ′ is a standard repair of K . repairs of the closure of K : RC ( K ) A standard repair of ( g + Cl ( F , R ) , R , N ) . Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 6 / 31

Premliminaries ∃ -ASP ∃ -ASP ∃ -ASP Program Π a set of rules of the form : H ← B + , not N − 1 , . . . , not N − k . head H , positive body B + , negative bodies N − : sets of basic atoms i Skolemization of an ∃ -ASP Program Π = Π F ∪ Π R Π F : basic atoms, Π R : ∃ -ASP rules Skolemization of Π : grounding of Π F + skolemization of Π R Answer set An answer set of ∃ -ASP program : an answer set of its skolemization Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 7 / 31

Premliminaries ∃ -ASP Answer sets computing The Computation Tree (see ASPeRiX) • Let F be a set of facts and R be a set of  -ASP rules IN OUT MBT (must be true) Let R  R s.t.  (B)  IN and there is no i s.t. F    (N i )  IN  (N 1 ) F   (H) …    (N 1 )  (N k ) F  (N k ) Repeat for each obtained node by applying existential rules breadth-first. J.-F. Baget NMR 2014 19 juillet 2014 - 22 Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 8 / 31

Premliminaries ∃ -ASP Answer sets computing The Computation Tree (cont) • The computation tree generates a (possibly infinite) tree. • It is complete when no further rule application can add new atoms in IN. • The result of a complete branch is the union of all IN found in that branch. • A branch is OUT-valid when no fact present in a OUT of that branch is also present in the result. … • A branch is MBT-valid when all An answer set of (F, R ) is the result of a facts present in a MBT of that complete, OUT and MBT valid branch of a branch are also present in the computation tree from (F, R ) result. J.-F. Baget NMR 2014 19 juillet 2014 - 23 Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 9 / 31

From inconsistency-tolerant inferences to ∃ -ASP Transformation into ASP Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 10 / 31

From inconsistency-tolerant inferences to ∃ -ASP Transformation into ASP Inconsistency-tolerant ASP Program inference from K = ( F , R , N ) with vocabulary V Skolemization : ( F , R sk , N ) Extension of the vocabulary V → V ′ : For any p ∈ V p i : initial p p : possible p g : ground p s : may be selected p c : chosen p n : forbidden p v : valid p d : display Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 11 / 31

From inconsistency-tolerant inferences to ∃ -ASP Transformation into ASP Computation tree (1) For every p ( � p i ( � t ) ∈ F t ) : initial predicate Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 12 / 31

From inconsistency-tolerant inferences to ∃ -ASP Transformation into ASP Computation tree (1) For every p ( � p i ( � t ) ∈ F t ) : initial predicate For every predicate in V : [ P 1 :] p p ( � X ) ← p i ( � X ) Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 13 / 31

From inconsistency-tolerant inferences to ∃ -ASP Transformation into ASP Computation tree (1) For every p ( � p i ( � t ) ∈ F , t ) : initial predicate For every predicate in V , [ P 1 :] p p ( � X ) ← p i ( � X ) . For every rule in R sk , [ R 1 :] H p ( � X , � Y ) , fct ( Y 1 ) , · · · , fct ( Y k ) ← B p ( � X ) . Jean-Franc ¸ois Baget (1), Zied Bouraoui (2), Farid Nouioua (3), Odile Papini (3), Swan Rocher (4), Eric W¨ ∃ -ASP for computing repairs with existential ontologies urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. 14 / 31