Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing DL-Lite Elena Botoeva and Diego Calvanese KRDB Research Centre Free University of Bozen-Bolzano I-39100 Bolzano, Italy Montpellier, BNC, August 2012 Botoeva, Calvanese Circumscribing DL-Lite 1/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Description Logic DL - Lite H bool Description Logics (DLs) are decidable fragments of First-Order Logic, used as Knowledge Representation formalisms. DL-Lite H bool is a light-weight DL that asserts • Boolean combinations of atomic concepts A , the domain ∃ P and the range ∃ P − of atomic roles P , • Hierarchy of atomic roles P and their inverses P − , and • ground facts A ( a ), P ( a , b ). Botoeva, Calvanese Circumscribing DL-Lite 2/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Description Logic DL - Lite H bool Description Logics (DLs) are decidable fragments of First-Order Logic, used as Knowledge Representation formalisms. DL-Lite H bool is a light-weight DL that asserts • Boolean combinations of atomic concepts A , the domain ∃ P and the range ∃ P − of atomic roles P , TBox T • Hierarchy of atomic roles P and their inverses P − , and • ground facts A ( a ), P ( a , b ). ABox A Botoeva, Calvanese Circumscribing DL-Lite 2/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Description Logic DL - Lite H bool Description Logics (DLs) are decidable fragments of First-Order Logic, used as Knowledge Representation formalisms. DL-Lite H bool is a light-weight DL that asserts • Boolean combinations of atomic concepts A , the domain ∃ P and the range ∃ P − of atomic roles P , TBox T • Hierarchy of atomic roles P and their inverses P − , and • ground facts A ( a ), P ( a , b ). ABox A Satisfiability check over a DL-Lite H bool KB K = �T , A� can be done in NP in combined complexity and in AC 0 in data complexity. Botoeva, Calvanese Circumscribing DL-Lite 2/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool DL - Lite H bool Knowledge Base Encoding of the ‘Tweety’ example in DL-Lite H bool : TBox T : Bird ⊓ ¬ Abnormal ⊑ Flier ⊑ Penguin Bird Penguin ⊑ Abnormal ABox A : Bird ( tweety ) Botoeva, Calvanese Circumscribing DL-Lite 3/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscription Circumscription is a non-monotonic formalism introduced by John McCarthy. Intuitively, circumscription of a predicate X says that the only objects that satisfy X are those that can be proven to satisfy it . � � Circ ( X ( a ); X ) = ∀ x X ( x ) ≡ x = a Circ ( ¬ X ( a ); X ) = ∀ x ¬ X ( x ) � � � � Circ ( ∀ x Φ( x ) → X ( x ) ; X ) = ∀ x Φ( x ) ≡ X ( x ) � � Circ ( ∀ x X ( x ) → Φ( x ) ; X ) = ∀ x ¬ X ( x ) Botoeva, Calvanese Circumscribing DL-Lite 4/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscription Circumscription is a non-monotonic formalism introduced by John McCarthy. Intuitively, circumscription of a predicate X says that the only objects that satisfy X are those that can be proven to satisfy it . � � Circ ( X ( a ); X ) = ∀ x X ( x ) ≡ x = a Circ ( ¬ X ( a ); X ) = ∀ x ¬ X ( x ) � � � � Circ ( ∀ x Φ( x ) → X ( x ) ; X ) = ∀ x Φ( x ) ≡ X ( x ) � � Circ ( ∀ x X ( x ) → Φ( x ) ; X ) = ∀ x ¬ X ( x ) predicate completion Botoeva, Calvanese Circumscribing DL-Lite 4/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool The Tweety Example Recall TBox T : ⊑ Bird ⊓ ¬ Abnormal Flier Penguin ⊑ Bird ⊑ Penguin Abnormal ABox A : Bird ( tweety ) We have that Circ ( �T , A� ; Abnormal ) | = Flier ( tweety ) Now, let A ′ = A ∪ { Penguin ( tweety ) } . Then Circ ( �T , A ′ � ; Abnormal ) �| = Flier ( tweety ) Note, that �T , A� �| = Flier ( tweety ) Botoeva, Calvanese Circumscribing DL -Lite 5/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscription: Semantics The models of Circ ( K ; X ) are the models of K such that the extension of X cannot be made smaller without losing the property K . Formally, let I and J be two classical interpretations of K . Then we write I ≤ X J if ◮ ∆ I = ∆ J , ◮ Y I = Y J for every Y � = X . ◮ X I ⊆ X J . An interpretation I is a model of Circ ( K ; X ) if ◮ it is a model of K and ◮ it is minimal relative to ≤ X . Botoeva, Calvanese Circumscribing DL-Lite 6/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing DL - Lite H bool • In this paper we show how to compute circumscription of a single predicate (a concept or a role) in a DL -Lite H bool KB. • To simplify presentation, in this talk I show how to circumscribe DL-Lite H core KBs. Given a DL-Lite H core TBox T and a predicate X , we compute Circ ( T ; X ) Then we show how an ABox can be added to the theory. • DL-Lite H core is a sub-logic of DL-Lite H bool with inclusions of the form B 1 ⊑ B 2 B 2 ⊑ ¬ B 2 R 1 ⊑ R 2 R 2 ⊑ ¬ R 2 ( B i denote A , ∃ P , or ∃ P − , R i denote P or P − ). Botoeva, Calvanese Circumscribing DL-Lite 7/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Concept In DL -Lite H core , minimizing an atomic concept A corresponds to predicate completion . Let T be a DL-Lite H core TBox and Pos T ( A ) = { B i ⊑ A } 1 ≤ i ≤ n the set of all inclusions in T where A appears positively (i.e., without negation on the right-hand side of an ISA inclusion). Then Circ ( T ; A ) = T ∪ { B 1 ⊔ · · · ⊔ B n ≡ A } Note that when computing circumscription of A we can forget about negative occurrences of A , i.e., axioms of the form A ⊑ B or B ⊑ ¬ A . Botoeva, Calvanese Circumscribing DL-Lite 8/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role In DL -Lite H core , a role P can occur positively in the following inclusions: ⊑ for a role R R P B 1 ⊑ ∃ P for a concept B 1 ∃ P − ⊑ for a concept B 2 B 2 core TBox T , if Pos T ( P ) = { R i ⊑ P } 1 ≤ i ≤ n s.t. R i � = P − , For a DL-Lite H then this corresponds to the case of predicate completion and Circ ( T ; P ) = T ∪ { R 1 ⊔ · · · ⊔ R n ≡ P } . It remains to consider the other cases and their combinations. Botoeva, Calvanese Circumscribing DL-Lite 9/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role: B 1 ⊑ ∃ P Assume T = { B 1 ⊑ ∃ P } . I ′ : I : P B 1 B 1 P P P P P P P I ′ is a model of Circ ( T ; P ) . I is not a model of Circ ( T ; P ) . One can show that Circ ( T ; P ) = { B 1 ≡ ∃ P , Funct ( P ) }. Botoeva, Calvanese Circumscribing DL -Lite 10/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role: B 2 ⊑ ∃ P − For T = { B 2 ⊑ ∃ P − } , symmetrically to the previous case, Circ ( T ; P ) = { B 2 ≡ ∃ P − , Funct ( P − ) } , and models have the following form: I : B 2 P P Botoeva, Calvanese Circumscribing DL -Lite 11/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role: B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − However, if T = { B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − } , Circ ( T ; P ) �| = B 1 ≡ ∃ P = B 2 ≡ ∃ P − Circ ( T ; P ) �| because I is a model of Circ ( T ; P ) : I : P B 1 P B 2 P P P From now on, we assume T = { B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − } s.t. P / ∈ Σ( B 1 , B 2 ) . Botoeva, Calvanese Circumscribing DL -Lite 12/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role: B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − - 1 First, we restrict the domain and the range of P : P I 1 : B 1 B 2 P P P Botoeva, Calvanese Circumscribing DL -Lite 13/21
Description Logic DL - LiteH Circumscribed DL - LiteH Circumscription Conclusions bool bool Circumscribing a Role: B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − - 1 First, we restrict the domain and the range of P : P I 1 : B 1 B 2 P P P To prohibit such interpretations: �� � ∀ x , y P ( x , y ) ∧ ¬ B 2 ( y ) ∧ ¬ B 1 ( x ) → ⊥ or in the DL syntax ( A LC required) ∃ P . ¬ B 2 ⊓ ¬ B 1 ⊑ ⊥ Botoeva, Calvanese Circumscribing DL -Lite 13/21
Circumscribing a Role: B 1 ⊑ ∃ P , B 2 ⊑ ∃ P − - 2 Second, does Circ ( T ; P ) entail Funct ( P ) , Funct ( P − ) ? I ′ I 2 : 2 : P P B 1 B 1 P B 2 B 2 P P P P P But I ′ I 2 is not a model of Circ ( T ; P ) . 2 is a model of Circ ( T ; P ) . I ′ I 3 : 3 : P B 1 B 1 B 2 B 2 P P P P P P P But I ′ I 3 is not a model of Circ ( T ; P ) . 3 is a model of Circ ( T ; P ) .
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