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

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

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-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

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SLIDE 1

∃-ASP for computing repairs with existential

  • ntologies

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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 1 / 31

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SLIDE 2

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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 2 / 31

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SLIDE 3

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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 2 / 31

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SLIDE 4

The aim

Framework : Existential rules Inconsistency-tolerant inference Repairs ∃-ASP Program ∃-ASP [Garreau, Garcia, Lef`

evre, St´ ephan, ONTOLP 2015]

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. ∃-ASP for computing repairs with existential ontologies 2 / 31

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SLIDE 5

The aim

Framework : Existential rules Inconsistency-tolerant inference Repairs ∃-ASP Program Answer Sets ASPeRiX

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. ∃-ASP for computing repairs with existential ontologies 2 / 31

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SLIDE 6

Overview

1

Premliminaries Existential rules ∃-ASP

2

From inconsistency-tolerant inferences to ∃-ASP

3

One-to-one correspondence between answer sets and repairs

4

Conclusion

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. ∃-ASP for computing repairs with existential ontologies 3 / 31

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SLIDE 7

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, Z]

  • X,

Y : universally quantified, Z : existential variable FOL : ∀ X, ∀ Y(φ(B) → (∃ Zφ(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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 4 / 31

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SLIDE 8

Premliminaries Existential rules

Existential rules

Classes where the skolem chase stops [Baget, Garreau, Mugnier, Rocher. NM2014]

EMETTEUR - NOM DE LA PRESENTATION 19 juillet 2014 1 J.-F. Baget NMR 2014

wa fd ar ja swa msa mfa waD fdD arD jaD swaD msaD agrd P co-NP Exp-Time 2-Exp-Time waU fdU arU jaU swaU msaU waU+ fdU+ arU+ jaU+ swaU+ msaU+

Summarizing Results

co-NP

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. ∃-ASP for computing repairs with existential ontologies 5 / 31

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SLIDE 9

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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 6 / 31

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SLIDE 10

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−

i

: sets of basic atoms Skolemization of an ∃-ASP Program Π = ΠF ∪ ΠR ΠF : basic atoms, ΠR : ∃-ASP rules Skolemization of Π : grounding of ΠF + skolemization of ΠR 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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 7 / 31

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SLIDE 11

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

19 juillet 2014 J.-F. Baget NMR 2014

  • 22

IN OUT MBT (must be true) F   Let R  R s.t. (B)  IN and there is no i s.t. (Ni)  IN F  (H) (N1) … (Nk)  F  (N1)  (Nk) Repeat for each obtained node by applying existential rules breadth-first.

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. ∃-ASP for computing repairs with existential ontologies 8 / 31

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SLIDE 12

Premliminaries ∃-ASP

Answer sets computing

The Computation Tree (cont)

19 juillet 2014 J.-F. Baget NMR 2014

  • 23

…

  • 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

facts present in a MBT of that branch are also present in the result. An answer set of (F, R) is the result of a complete, OUT and MBT valid branch of a computation tree from (F, R)

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. ∃-ASP for computing repairs with existential ontologies 9 / 31

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SLIDE 13

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¨ urbel (3) ( (1) INRIA, France. (2) Cardiff University, UK. ∃-ASP for computing repairs with existential ontologies 10 / 31

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SLIDE 14

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Inconsistency-tolerant inference ASP Program from K = (F, R, N) with vocabulary V Skolemization : (F, Rsk, N) Extension of the vocabulary V → V′ : For any p ∈ V pi : initial pp : possible pg : ground ps : may be selected pc : chosen pn : forbidden pv : valid pd : display

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. ∃-ASP for computing repairs with existential ontologies 11 / 31

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SLIDE 15

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (1) For every p( t) ∈ F pi( t) : initial predicate

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. ∃-ASP for computing repairs with existential ontologies 12 / 31

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SLIDE 16

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (1) For every p( t) ∈ F pi( t) : initial predicate For every predicate in V : [P1 :] pp( X) ← pi( X)

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. ∃-ASP for computing repairs with existential ontologies 13 / 31

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SLIDE 17

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (1) For every p( t) ∈ F, pi( t) : initial predicate For every predicate in V, [P1 :] pp( X) ← pi( X). For every rule in Rsk, [R1 :] Hp( X, Y), fct(Y1), · · · , fct(Yk) ← Bp( X).

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. ∃-ASP for computing repairs with existential ontologies 14 / 31

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SLIDE 18

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (1) For every p( t) ∈ F, pi( t) : initial predicate For every predicate in V, [P1 :] pp( X) ← pi( X). For every rule in Rsk, [R1 :] Hp( X, Y), fct(Y1), · · · , fct(Yk) ← Bp( X). For every predicate in V, [P2 :] pg( X) ← pp( X), notfct(X1), · · · notfct(Xk).

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. ∃-ASP for computing repairs with existential ontologies 15 / 31

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SLIDE 19

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (2) : Selection A selection strategy selects some finite subset X of atoms and marks them “s” For instance Select all initial atoms Select all ground atoms

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. ∃-ASP for computing repairs with existential ontologies 16 / 31

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SLIDE 20

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (3) : Choice [P3 :] pc( X) ← ps( X), not pn( X)

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. ∃-ASP for computing repairs with existential ontologies 17 / 31

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SLIDE 21

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (3) : Choice [P3 :] pc( X) ← ps( X), not pn( X)

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. ∃-ASP for computing repairs with existential ontologies 18 / 31

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SLIDE 22

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (3) : Choice

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. ∃-ASP for computing repairs with existential ontologies 19 / 31

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SLIDE 23

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (3) : Choice

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. ∃-ASP for computing repairs with existential ontologies 20 / 31

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SLIDE 24

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (4) : Contexts [P4 :] pv( X, base) ← pc( X).

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. ∃-ASP for computing repairs with existential ontologies 21 / 31

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SLIDE 25

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (4) : Contexts [P4 :] pv( X, base) ← pc( X). [P5 :] pv( X, ctx(p, X)), context(ctx(p, X)) ← ps( X), not pc( X).

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. ∃-ASP for computing repairs with existential ontologies 22 / 31

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SLIDE 26

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (4) : Contexts [P4 :] pv( X, base) ← pc( X). [P5 :] pv( X, ctx(p, X)), context(ctx(p, X)) ← ps( X), not pc( X). [P6 :] pv( X, C) ← pv( X, base), context(C).

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. ∃-ASP for computing repairs with existential ontologies 23 / 31

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SLIDE 27

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (5) : Propagation For every rule in Rsk, B( X) → H( X), [R2 :] Hv( X, C) ← Bv( X, C).

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. ∃-ASP for computing repairs with existential ontologies 24 / 31

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SLIDE 28

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (5) : Propagation For every rule in Rsk, B( X) → H( X), [R2 :] Hv( X, C) ← Bv( X, C). For every constraint in N [C1 :], absurd(C) ← p1

v(

X1, C), · · · , pk

v(

Xk, C).

Yc is consistent iff absurd(base) is not derived Yc ∪ {a} is consistent iff absurd(ctx(a)) is not derived Finally Yc is a maximal consistent subset of Xs when absurd(base) is not derived and for every

  • ther context ctx(a) absurd(ctx(a)) is derived

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. ∃-ASP for computing repairs with existential ontologies 25 / 31

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SLIDE 29

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (6) : Evaluation Case 1 : absurd base context [C2 :] pn( X) ← pc( X), absurd(base). Yc is not consistent and the branch is not OUT-valid

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. ∃-ASP for computing repairs with existential ontologies 26 / 31

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SLIDE 30

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (6) : Evaluation Case 2 : base context not absud, but all the other contexts are absurd [C3 :] pn( X) ← ps( X), not pc( X), pv( X, C), context(C), absurd(C). Yc is consistent and the branch is OUT-valid and MBT-valid

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. ∃-ASP for computing repairs with existential ontologies 27 / 31

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SLIDE 31

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (6) : Evaluation Case 3 : One non-base context is not absurd [C3 :] pn( X) ← ps( X), not pc( X), pv( X, C), context(C), absurd(C). Yc is not maximal consistent and the branch is not MBT-valid

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. ∃-ASP for computing repairs with existential ontologies 28 / 31

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SLIDE 32

From inconsistency-tolerant inferences to ∃-ASP

Transformation into ASP

Computation tree (7) : Display Display strategy : Selects some finite subset of atoms valid in the base context and restricts them to the original vocabulary all initial atoms in the base context all ground atoms in the base context

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. ∃-ASP for computing repairs with existential ontologies 29 / 31

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SLIDE 33

One-to-one correspondence between answer sets and repairs

One-to-one correspondence between answer sets and repairs

Unified framework Semantics

  • 1 or R

computes the set of repairs of K : R(K)

  • 5 or CR

computes the closed repairs of K : CR(K)

  • 7 or RC

computes the repairs of the closure of K : RC(K) SEL1 / DISP1 : Select / Display all atoms SEL2 / DISP2 : Select / Display all ground atoms One-to-one correspondence Let K = (F, R, N) a knowledge base. ρ(AS) denotes the restriction of AS to the original vocabulary V ∃-ASP program SEL DISP ρ(AS(Π)) Π1 SEL1 DISP1 ρ(AS(Π1)) repairs of K : R(K) Π5 SEL1 DISP2 ρ(AS(Π5)) closed repairs of K : CR(K) Π7 SEL2 DISP2 ρ(AS(Π7)) repairs of the closure of K : RC(K)

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. ∃-ASP for computing repairs with existential ontologies 30 / 31

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SLIDE 34

Conclusion

Conclusion

generic encoding in ∃-ASP of repair-based techniques allows for computing the semantics : AR, IAR, ICR, ◦7, ∀, ◦7, ∩

Let q : a query, qv obtained from q by replacing any p(

  • t) by pv(
  • t, base)

AR K | =◦1,∀ q iff ∀AS ∈ AS(Π1), qv ∈ AS. IAR K | =◦1,∩ q iff qv ∈ ∩ASi ∈AS(Π1)ASi. ICR K | =◦5,∩ q iff qv ∈ ∩ASi ∈AS(Π5)ASi.

  • 7, ∀ K |

=◦7,∀ q iff ∀AS ∈ AS(Π7), qv ∈ AS. (Closed to CAR)

  • 7, ∩ K |

=◦7,∩ q iff qv ∈ ∩ASi ∈AS(Π7)ASi. (Closed to ICAR)

future works :

implementation and experimentation extension to repairs minimal w.r.t. cardinality

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. ∃-ASP for computing repairs with existential ontologies 31 / 31