Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Semantic Independence in DL-Programs Thomas Eiter Michael Fink Daria Stepanova Knowledge-Based Systems Group, Institute of Information Systems Vienna University of Technology http://www.kr.tuwien.ac.at/ RR 2012 – September 12, 2012 KBS Knowledge-Based Systems Group 1 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Motivation • DL-program: ontology + rules (loose coupling combination approach); • DL-atoms are evaluated under varying input to ontology; • Evaluation of just one DL-atom under certain ontology input may be costly. ?: Which DL-atoms always have the same value regardless of (updated) ontology? 1 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Motivation • DL-program: ontology + rules (loose coupling combination approach); • DL-atoms are evaluated under varying input to ontology; • Evaluation of just one DL-atom under certain ontology input may be costly. ?: Which DL-atoms always have the same value regardless of (updated) ontology? In this work: Semantic notion of independent DL-atom and its characterization (ontology is viewed as a black box). Applications: • optimization of DL-programs [Eiter et al, 2004]; • inconsistency diagnosis [Puehrer et al, 2010], [Fink et al, 2010]; • DL-program repair, etc. 1 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Overview Motivation Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work 2 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work DL-program: syntax Signature: Σ = �F , P o , P p � , where - F is a set of individuals (constants); - P o = P c ∪ P r , P c ( P r ) is a set of atomic concepts (resp. roles); - P p is a set of predicate symbols of arity ≥ 0. 3 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work DL-program: syntax Signature: Σ = �F , P o , P p � , where - F is a set of individuals (constants); - P o = P c ∪ P r , P c ( P r ) is a set of atomic concepts (resp. roles); - P p is a set of predicate symbols of arity ≥ 0. DL-atom is of the form DL [ S 1 op 1 p 1 , . . . , S m op m p m ; Q ]( t ) , m ≥ 0, where • S i ∈ P c or S i ∈ P r ; • op i ∈ {⊎ , − ∪ , − ∩} ; • p i ∈ P p (unary or binary); • Q ( t ) is a DL-query : • C ⊑ D , C �⊑ D , t = ǫ , where C , D ∈ P c ∪ {⊤ , ⊥} ; • C ( t 1 ) , ¬ C ( t 1 ) , t = t 1 , where C ∈ P c ; • R ( t 1 , t 2 ) , ¬ R ( t 1 , t 2 ) , t = t 1 , t 2 , where R ∈ P r . 3 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work DL-program: syntax Signature: Σ = �F , P o , P p � , where - F is a set of individuals (constants); - P o = P c ∪ P r , P c ( P r ) is a set of atomic concepts (resp. roles); - P p is a set of predicate symbols of arity ≥ 0. DL-atom is of the form DL [ S 1 op 1 p 1 , . . . , S m op m p m ; Q ]( t ) , m ≥ 0, where • S i ∈ P c or S i ∈ P r ; • op i ∈ {⊎ , − ∪ , − ∩} ; • p i ∈ P p (unary or binary); • Q ( t ) is a DL-query : • C ⊑ D , C �⊑ D , t = ǫ , where C , D ∈ P c ∪ {⊤ , ⊥} ; • C ( t 1 ) , ¬ C ( t 1 ) , t = t 1 , where C ∈ P c ; • R ( t 1 , t 2 ) , ¬ R ( t 1 , t 2 ) , t = t 1 , t 2 , where R ∈ P r . DL-program: KB = (Φ , Π) , Φ is a DL ontology, Π is a set of DL-rules: a ← b 1 , . . . b k , not b k + 1 , . . . , not b m , m ≥ k ≥ 0, a is a classical literal; b i is a classical literal or a DL-atom. 3 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work DL-program: semantics Consider KB = (Φ , Π) over Σ = �F , P o , P p � . Interpretation I is a consistent set of ground literals over Σ p = �F , P p � . = Φ ℓ iff ℓ ∈ I ; • for ground literal ℓ : I | • for ground DL-atom a = DL [ S 1 op 1 p 1 , . . . , S m op m p m ; Q ]( c ) : = Φ a I | = Q ( c ) , where τ I ( a )= � m iff Φ ∪ τ I ( a ) | i = 1 A i ( I ) is a DL-update of Φ under I by a : • A i ( I ) = { S i ( e ) | p i ( e ) ∈ I } , for op i = ⊎ ; • A i ( I ) = {¬ S i ( e ) | p i ( e ) ∈ I } , for op i = − ∪ ; • A i ( I ) = {¬ S i ( e ) | p i ( e ) �∈ I } , for − ∩ . I is an answer set of Π iff I is a minimal model of its FLP-reduct Π I FLP . FLP-reduct Π I = b + ( r ) and I �| FLP of Π is a set of ground DL-rules r s.t. I | = b − ( r ) . 4 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work DL-program: Example Example KB = { Φ , Π } . Φ = { Sweet ( apple ) } ; Π = { fruit ( apple ) . vitamin ( X ) ← fruit ( X ) . healthyfood ( X ) ← DL [ Healthy ⊎ vitamin ; Healthy ]( X ) . } • Consider I = { fruit ( apple ) , vitamin ( apple ) , healthyfood ( apple ) } ; • vitamin ( apple ) ∈ I , hence τ I ( a ) = { Healthy ( apple ) } ; • Φ ∪ τ I ( a ) | = Healthy ( apple ) . 5 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Independent DL-atoms Definition A ground DL-atom a is independent if for all satisfiable ontologies Φ , Φ ′ = Φ ′ a . = Φ a iff I ′ | and all interpretations I , I ′ it holds that I | A ground DL-atom a is a contradiction (resp. tautology ), if for all = Φ a satisfiable ontologies Φ and all interpretations I , it holds that I �| = Φ a ). (resp. I | Contradiction: DL [; C �⊑ C ]() ; . . . ? Tautology: DL [; C ⊑ C ]() ; . . . ? 6 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Contradictions When is a DL-atom contradictory in general? Proposition A ground DL-atom a = DL [ λ ; Q ]( t ) is contradictory iff λ = ǫ and Q ( t ) is unsatisfiable, i.e. has one of the forms: • C �⊑ C; • C �⊑ ⊤ ; • ⊥ �⊑ C; • ⊥ �⊑ ⊤ ; • ⊤ ⊑ ⊥ . 7 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Tautologies When is a DL-atom a = DL [ λ ; Q ]( t ) tautologic in general? • Q is tautologic: Q ∈ { C ⊑ ⊤ , ⊥ ⊑ C , C ⊑ C } ; • λ is s.t. a is tautologic. Concept query case distinction: DL [ λ ; Q ]( t ) C � = D . DL [ λ ; C �⊑ D ]() DL [ λ ; ¬ C ]( t ) DL [ λ ; C ⊑ D ]() DL [ λ ; C ]( t ) no tautologies no tautologies no tautologies 8 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Tautologies When is a DL-atom a = DL [ λ ; Q ]( t ) tautologic in general? • Q is tautologic: Q ∈ { C ⊑ ⊤ , ⊥ ⊑ C , C ⊑ C } ; • λ is s.t. a is tautologic. Concept query case distinction: DL [ λ ; Q ]( t ) C � = D . DL [ λ ; C �⊑ D ]() DL [ λ ; ¬ C ]( t ) DL [ λ ; C ⊑ D ]() DL [ λ ; C ]( t ) no tautologies no tautologies no tautologies Example ∩ p , C ′ ⊎ p , C ′ − ∩ q , C − ∪ q ; ¬ C ]( c ) a = DL [ C − I is s.t. p ( c ) �∈ I , q ( c ) �∈ I I is s.t. p ( c ) ∈ I , q ( c ) �∈ I I is s.t. p ( c ) �∈ I , q ( c ) ∈ I I is s.t. p ( c ) ∈ I , q ( c ) ∈ I 8 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Tautologies When is a DL-atom a = DL [ λ ; Q ]( t ) tautologic in general? • Q is tautologic: Q ∈ { C ⊑ ⊤ , ⊥ ⊑ C , C ⊑ C } ; • λ is s.t. a is tautologic. Concept query case distinction: DL [ λ ; Q ]( t ) C � = D . DL [ λ ; C �⊑ D ]() DL [ λ ; ¬ C ]( t ) DL [ λ ; C ⊑ D ]() DL [ λ ; C ]( t ) no tautologies no tautologies no tautologies Example ∩ p , C ′ ⊎ p , C ′ − ∩ q , C − ∪ q ; ¬ C ]( c ) a = DL [ C − I is s.t. p ( c ) �∈ I , q ( c ) �∈ I I is s.t. p ( c ) ∈ I , q ( c ) �∈ I I is s.t. p ( c ) �∈ I , q ( c ) ∈ I I is s.t. p ( c ) ∈ I , q ( c ) ∈ I 8 / 18
Motivation Overview Preliminaries Independent DL-atoms Independence under inclusion Formal results and future work Tautologies When is a DL-atom a = DL [ λ ; Q ]( t ) tautologic in general? • Q is tautologic: Q ∈ { C ⊑ ⊤ , ⊥ ⊑ C , C ⊑ C } ; • λ is s.t. a is tautologic. Concept query case distinction: DL [ λ ; Q ]( t ) C � = D . DL [ λ ; C �⊑ D ]() DL [ λ ; ¬ C ]( t ) DL [ λ ; C ⊑ D ]() DL [ λ ; C ]( t ) no tautologies no tautologies no tautologies Example ∩ p , C ′ ⊎ p , C ′ − ∩ q , C − ∪ q ; ¬ C ]( c ) a = DL [ C − τ I ( a ) = {¬ C ( c ) } I is s.t. p ( c ) �∈ I , q ( c ) �∈ I I is s.t. p ( c ) ∈ I , q ( c ) �∈ I I is s.t. p ( c ) �∈ I , q ( c ) ∈ I I is s.t. p ( c ) ∈ I , q ( c ) ∈ I 8 / 18
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