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Realizing Default Logic over Description Logic Knowledge Bases Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner KBS Group, Institute of Information Systems, Vienna University of Technology ECSQARU 2009 July 2, 2009 KBS Knowledge-Based

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Realizing Default Logic over Description Logic Knowledge Bases

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner

KBS Group, Institute of Information Systems, Vienna University of Technology

ECSQARU 2009 — July 2, 2009

Knowledge-Based Systems Group

KBS

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The need of common-sense reasoning on top of

  • ntologies

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

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The need of common-sense reasoning on top of

  • ntologies

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

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The need of common-sense reasoning on top of

  • ntologies

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

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The need of common-sense reasoning on top of

  • ntologies

default reasoning on top of ontologies?

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

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The need of common-sense reasoning on top of

  • ntologies

default reasoning on top of ontologies? integrations of rules and ontologies: cq-programs

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 2/17

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Description Logic Knowledge Bases (DL-KBs)

Syntax and Semantics Name Syntax Semantics

Top/Bottom

⊤/⊥ ∆I/∅

Intersection

C ⊓ D CI ∩ DI

Union

C ⊔ D CI ∪ DI

Negation

¬C ∆I \ CI

Value restriction

∀R.C {a ∈ ∆I | ∀b.(a, b) ∈ RI → b ∈ CI}

Existential quant.

∃R.C {a ∈ ∆I | ∃b.(a, b) ∈ RI ∧ b ∈ CI} Modeling: TBox & ABox

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 3/17

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Description Logic Knowledge Bases (DL-KBs)

Syntax and Semantics Name Syntax Semantics

Top/Bottom

⊤/⊥ ∆I/∅

Intersection

C ⊓ D CI ∩ DI

Union

C ⊔ D CI ∪ DI

Negation

¬C ∆I \ CI

Value restriction

∀R.C {a ∈ ∆I | ∀b.(a, b) ∈ RI → b ∈ CI}

Existential quant.

∃R.C {a ∈ ∆I | ∃b.(a, b) ∈ RI ∧ b ∈ CI} Modeling: TBox & ABox Translation to first-order logic πx(A) = A(x) πx(C ⊓ D) = πx(C) ∧ πx(D) πx(∀R.C) = ∀y.R(x, y) ⊃ πy(C) πx(C ⊔ D) = πx(C) ∨ πx(D) πx(∃R.C) = ∀y.R(x, y) ∧ πy(C)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 3/17

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Default Theories over DL-KBs ∆ = L, D

similar to [Baader and Hollunder, 1995] Default rule α( X)

  • α1(

X1) ∧ · · · ∧ αk( Xk) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

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Default Theories over DL-KBs ∆ = L, D

similar to [Baader and Hollunder, 1995] Default rule α( X)

  • α1(

X1) ∧ · · · ∧ αk( Xk) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z) Semantics: based on the Γ∆ operator

◮ Let S be a set of assertions, then Γ∆(S) is the smallest set that

◮ contains Cn(L) ◮ is deductively closed ◮ if α(

X) ∈ Γ∆(S) and ¬βi( Yi) / ∈ S, then γ( Z) ∈ Γ∆(S)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

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Default Theories over DL-KBs ∆ = L, D

similar to [Baader and Hollunder, 1995] Default rule α( X)

  • α1(

X1) ∧ · · · ∧ αk( Xk) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z) Semantics: based on the Γ∆ operator

◮ Let S be a set of assertions, then Γ∆(S) is the smallest set that

◮ contains Cn(L) ◮ is deductively closed ◮ if α(

X) ∈ Γ∆(S) and ¬βi( Yi) / ∈ S, then γ( Z) ∈ Γ∆(S)

◮ E is an extension of ∆ iff Γ∆(E) = E

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 4/17

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Example

∆ = L, D L = Flier ⊑ ¬NonFlier, Penguin ⊑ Bird, Penguin ⊑ NonFlier, Bird(tweety)

  • D =

Bird(X) : Flier(X) Flier(X)

  • E = Cn(L ∪ {Flier(tweety)})

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 5/17

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Example

∆′ = L′, D L′ = Flier ⊑ ¬NonFlier, Penguin ⊑ Bird, Penguin ⊑ NonFlier, Penguin(tweety)

  • D =

Bird(X) : Flier(X) Flier(X)

  • E′ = Cn(L′)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 6/17

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Conjunctive Query Programs [Eiter et al., 2008a]

◮ (union of) conjunctive queries:

q(X) = {X | AirCraft(X) ∨ (UFO(X) ∧ ¬Hoax(X))}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

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Conjunctive Query Programs [Eiter et al., 2008a]

◮ (union of) conjunctive queries:

q(X) = {X | AirCraft(X) ∨ (UFO(X) ∧ ¬Hoax(X))}

◮ cq-atom:

DL[AirCraft ⊎ isBoeing; AirCraft(X) ∨ (UFO(X), ¬Hoax(X))](X)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

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Conjunctive Query Programs [Eiter et al., 2008a]

◮ (union of) conjunctive queries:

q(X) = {X | AirCraft(X) ∨ (UFO(X) ∧ ¬Hoax(X))}

◮ cq-atom:

DL[AirCraft ⊎ isBoeing; AirCraft(X) ∨ (UFO(X), ¬Hoax(X))](X)

◮ cq-rule:

flying thing(X) ← thing(X), DL[AirCraft ⊎ isBoeing; AirCraft(X) ∨ (UFO(X), ¬Hoax(X))](X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

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Conjunctive Query Programs [Eiter et al., 2008a]

◮ (union of) conjunctive queries:

q(X) = {X | AirCraft(X) ∨ (UFO(X) ∧ ¬Hoax(X))}

◮ cq-atom:

DL[AirCraft ⊎ isBoeing; AirCraft(X) ∨ (UFO(X), ¬Hoax(X))](X)

◮ cq-rule:

flying thing(X) ← thing(X), DL[AirCraft ⊎ isBoeing; AirCraft(X) ∨ (UFO(X), ¬Hoax(X))](X).

◮ cq-program: KB = (L, P) — based on answer set semantics

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 7/17

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Answer Set Semantics of cq-Programs

generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b1, . . . , bm, not c1, . . . , not cn.

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

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Answer Set Semantics of cq-Programs

generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b1, . . . , bm, not c1, . . . , not cn.

◮ Interpretation I ⊆ HBP

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

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Answer Set Semantics of cq-Programs

generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b1, . . . , bm, not c1, . . . , not cn.

◮ Interpretation I ⊆ HBP ◮ I is an answer set of P if I is the least model of the reduct PI

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

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Answer Set Semantics of cq-Programs

generalized [Gelfond and Lifschitz, 1991] P is a set of rules of form a ← b1, . . . , bm, not c1, . . . , not cn.

◮ Interpretation I ⊆ HBP ◮ I is an answer set of P if I is the least model of the reduct PI

PI is constructed by

◮ removing rules r ∈ P such that ci ∈ I ◮ removing all ci and nonmonotonic cq-atoms from remaining rules Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 8/17

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Transformation Ω (inspired by [Eiter et al., 2008b])

D is a set of defaults of form α( X) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

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Transformation Ω (inspired by [Eiter et al., 2008b])

D is a set of defaults of form α( X) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z)

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

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Transformation Ω (inspired by [Eiter et al., 2008b])

D is a set of defaults of form α( X) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z)

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

◮ A rule which simulates the Γ∆ operator

inγ( Z) ← DL[λ; α( X)]( X), not DL[λ; ¬β1,1( Y1,1) ∨ · · · ∨ ¬β1,ℓ1( Y1,ℓ1)]( Y1), . . . not DL[λ; ¬βm,1( Ym,1) ∨ · · · ∨ ¬βm,ℓm( Ym,ℓm)]( Ym).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

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Transformation Ω (inspired by [Eiter et al., 2008b])

D is a set of defaults of form α( X) : β1( Y1), . . . , βi( Yi)

  • βi,1(

Yi,1) ∧ · · · ∧ βi,ℓi( Yi,ℓi), . . . , βm( Ym) γ1( Z1) ∧ · · · ∧ γn( Zn)

  • γ(

Z)

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

◮ A rule which simulates the Γ∆ operator

inγ( Z) ← DL[λ; α( X)]( X), not DL[λ; ¬β1,1( Y1,1) ∨ · · · ∨ ¬β1,ℓ1( Y1,ℓ1)]( Y1), . . . not DL[λ; ¬βm,1( Ym,1) ∨ · · · ∨ ¬βm,ℓm( Ym,ℓm)]( Ym). where λ = (γ∗

i ⊎ inγi, γ∗ i −

∪in¬γi | δ ∈ D); γ∗

i is γi without ¬

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 9/17

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Transformation Ω - Bird Example

inFlier(X) ← DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; Bird(X)](X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 10/17

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Transformation Ω - Bird Example

inFlier(X) ← DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; Bird(X)](X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΩ = {inFlier(tweety)}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 10/17

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Transformation Ω - Bird Example

inFlier(X) ← DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; Bird(X)](X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΩ = {inFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)})

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 10/17

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Transformation Ω - Bird Example

inFlier(X) ← DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; Bird(X)](X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΩ = {inFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)}) Add in¬Flier(X) ← broken wing(X). broken wing(tweety).

  • Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner

Realizing Default Logic over Description Logic 10/17

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Transformation Ω - Bird Example

inFlier(X) ← DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; Bird(X)](X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΩ = {inFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)}) Add in¬Flier(X) ← broken wing(X). broken wing(tweety).

  • Then IΩ = {broken wing(tweety), in¬Flier(tweety)}; EΩ = Cn(L)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 10/17

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Transformation Υ

based on [Cholewinski and Truszczynski, 1994]

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 11/17

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Transformation Υ

based on [Cholewinski and Truszczynski, 1994]

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

◮ Rules that select justifications:

consβi( Yi) ← not consβi( Yi). consβi( Yi) ← not consβi( Yi).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 11/17

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Transformation Υ

based on [Cholewinski and Truszczynski, 1994]

◮ Concluding rules

R = {inγi( Zi) ← inγ( Z) | 1 ≤ i ≤ n}

◮ Rules that select justifications:

consβi( Yi) ← not consβi( Yi). consβi( Yi) ← not consβi( Yi).

◮ A rule which simulates the Γ∆ operator:

inγ( Z) ← DL[λ; α( X)]( X), consβ1( Y1), . . . , consβm( Ym). where λ = (γ∗

i ⊎ inγi, γ∗ i −

∪in¬γi | δ ∈ D) γ∗

i is γi without ¬

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 11/17

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Transformation Υ - cont.

◮ Constraints that check the compliance of our guess with the result

fail ← DL[λ; ¬βi,1( Yi,1) ∨ · · · ∨ ¬βi,ℓi( Yi,ℓi)]( Yi), consβi( Yi), not fail. fail ← not DL[λ; ¬βi,1( Yi,1) ∨ · · · ∨ ¬βi,ℓi( Yi,ℓi)]( Yi), consβi( Yi), not fail.

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 12/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). fail ← not fail, consFlier(X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X).

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). fail ← not fail, consFlier(X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΥ = {inFlier(tweety), consFlier(tweety)}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). fail ← not fail, consFlier(X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΥ = {inFlier(tweety), consFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)})

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). fail ← not fail, consFlier(X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΥ = {inFlier(tweety), consFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)}) L = L ∪ {Penguin(tweety)}

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Transformation Υ - Bird Example

consFlier(X) ← not consFlier(X). consFlier(X) ← not consFlier(X). inFlier(X) ← DL[Flier ⊎ inFlier; Bird(X)](X), consFlier(X). fail ← not fail, consFlier(X), DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). fail ← not fail, consFlier(X), not DL[Flier ⊎ inFlier, Flier − ∪in¬Flier; ¬Flier(X)](X). IΥ = {inFlier(tweety), consFlier(tweety)} ; EΩ = Cn(L ∪ {Flier(tweety)}) L = L ∪ {Penguin(tweety)} , then IΥ = {consFlier(tweety)}; EΩ = Cn(L)

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 13/17

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Comparison

10-2 10-1 100 101 102 103 1 2 3 4 5 6 7 8 9 10 11

evaluation time / secs Number of individuals

Π Ω Υ Split

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 14/17

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Conclusion & Future Work

◮ Conclusions:

◮ Consider default theories over DL-KBs ∆ = L, D ◮ Adapt Reiter’s Default Logic using Γ∆ operator ◮ Two new transformations to cq-programs (Ω and Υ) ◮ Ω and Υ perform better than an old transformation in [Eiter et al.,

2008b]

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 15/17

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Conclusion & Future Work

◮ Conclusions:

◮ Consider default theories over DL-KBs ∆ = L, D ◮ Adapt Reiter’s Default Logic using Γ∆ operator ◮ Two new transformations to cq-programs (Ω and Υ) ◮ Ω and Υ perform better than an old transformation in [Eiter et al.,

2008b]

◮ Future work:

◮ Investigate special default theories (normal/semi-normal defaults) ◮ Implement caching for cq-programs ◮ Interface to different DL-reasoners, eg., Pellet, KAON2 Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 15/17

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References I

Franz Baader and Bernhard Hollunder. Embedding Defaults into Terminological Knowledge Representation Formalisms. Journal of Automated Reasoning, 14(1):149–180, February 1995. P . Cholewinski and M. Truszczynski. Minimal number of permutations sufficient to compute all extensions a finite default theory. unpublished note, 1994. Thomas Eiter, Giovambattista Ianni, Thomas Krennwallner, and Roman Schindlauer. Exploiting conjunctive queries in description logic programs. Annals of Mathematics and Artificial Intelligence: Logic in AI: A Special Issue Dedicated to Victor W. Marek on the Occasion of His 65th birthday, 53(1–4):115–152, August 2008. Published online: 27 January 2009.

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 16/17

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References II

Thomas Eiter, Giovambattista Ianni, Thomas Lukasiewicz, Roman Schindlauer, and Hans Tompits. Combining answer set programming with description logics for the semantic web. Artificial Intelligence, 172(12-13):1495–1539, August 2008. Michael Gelfond and Vladimir Lifschitz. Classical negation in logic programs and deductive databases. New Generation Computing, 9:365–385, 1991.

Minh Dao-Tran, Thomas Eiter, Thomas Krennwallner Realizing Default Logic over Description Logic 17/17