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Local Closed World Semantics for Description Logics Adila Krisnadhi, Kunal Sengupta and Pascal Hitzler Knowledge Engineering Lab – Kno.e.sis Center – Wright State University – Dayton, Ohio, USA 1. Introduction 2. Grounded Circumscription Semantics of OWL (and DLs in general) adheres to the Open Circumscription employs minimization straightforwardly. World Assumption (OWA): statements that are not logical Circumscription in DLs: augment a KB with a circumscription pattern consisting consequences of a knowledge base (KB) is not considered false. disjoint sets minimized, ﬁxed, varying predicates and a preference relation on However, the Closed World Assumption (CWA) — statements that interpretations to pick minimal models as the preferred models. are not logical consequences of KB are considered false — is Drawbacks of circumscriptive DLs: preferable in certain situations, e.g., when data is retrieved from a undecidable when TBox is non-empty and some role is minimized; database, or data is considered complete w.r.t. application. extensions of minimized predicates may contain non-named individuals; Need languages that adhere to the Local Closed World Assumption Grounded circumscription (GC) simpliﬁes circumscription: (LCWA): both OWA & CWA as modeling features in one language. only consider minimized and non-minimized predicates; Users want to specify that individuals in the extension of a extensions of minimized predicates are restricted to named individuals, hence is predicate (concept/role name) are those which are necessarily decidable if the underlying DL is decidable; required, i.e., predicates are minimized. roles can be minimized without losing decidability even when the TBox is not empty. 3. Key Deﬁnitions 4. Example (1) hasAuthor(paper1, author1) hasAuthor(paper1, author2) For a DL L, GC-L-KB is a pair (K, M ) where K is an L-KB, hasAuthor(paper2, author3) ∀hasAuthor.Author. M ⊆ NC ∪ NR. Let Ind(K) be set of named inviduals in K. Concept/role name W is minimized w.r.t. K if W ∈ M Take (i) (≤2 hasAuthor.Author)(paper1); (ii) ¬hasAuthor(paper, author3) For any models I and J of K, I is smaller than J w.r.t. M iﬀ Classical semantics: neither (i) nor (ii) is a logical consequence ∆I = ∆J and aI = aJ for every aI ∈ ∆J ; Minimize hasAuthor without UNA: (i) is a GC-inference W I ⊆ W J for every W ∈ M ; and Minimize hasAuthor with UNA: (i) and (ii) are GC-inferences there exists a W ∈ M such that W I ⊂ W J A model I of K is a grounded model w.r.t. M iﬀ for A ∈ M ∩ NC , AI ⊆ {bI | b ∈ Ind(K)} 5. Example (2) for r ∈ M ∩ NR, r I ⊆ {(bI , cI ) | b, c ∈ Ind(K)} I is a GC-model of a GC-L-KB (K, M ) iﬀ: Bear(polarBear) I is a grounded model of K w.r.t. M ; ∃isHabitatFor.(Bear EndangeredSpecies)(arcticSea) I is minimal w.r.t. M . Consider EndangeredSpecies(polarBear) A statement/assertion is a GC-logical consequence (GC-inference) In the original circumscriptive DLs: is not a logical consequence of (K, M ) if every GC-model of (K, M ) satisﬁes it. Under GC-semantics: is a (GC)-logical consequence 6. Decidability of GC-KB SatisﬁabilityFor any DL L with nominals, concept disjunction & concept products: if KB-satisﬁability is decidable for L , then so is GC-KB satisﬁability.Proof (sketch): Let (K, M ) be a GC-L-KB such that M = MA ∪ MR where MA = {A1, . . . , An} and MR = {r1, . . . , rm} with n, m ≥ 0 and Ai, rj are minimized concept and role names w.r.t. K. The set G(K,M ) that contains precisely all tuples (X1, . . . , Xn, Y1, . . . , Ym) where all Xi ⊆ Ind(K) and Yj ⊆ Ind(K) × Ind(K) is ﬁnite. Given a tuple G = (X1, . . . , Xn, Y1, . . . , Ym} ∈ GK,M , let KG be K plus the following axioms: Ai ≡ {a} for every a ∈ Xi and 1 ≤ i ≤ n rj ≡ ({a} × {b}) for every pair (a, b) ∈ Yj and 1 ≤ j ≤ m If (K, M ) has a grounded model I then there exists G ∈ G(K,M ) such that KG has a model J which coincides with I on all minimized predicates. If there exists G ∈ G(K,M ) such that KG has a model J , then (K, M ) has a grounded model I which coincides with J on all minimized predicates. The set G(K,M ) = {G ∈ G(K,M ) | KG has a (classical) model} is ﬁnite and membership in G(K,M ) is decidable if L is decidable. There is a pointwise (partial) ordering of tuples in G(K,M ) induced by ⊆ on the tuple components. (K, M ) has a GC-model if and only if G(K,M ) is non-empty where G(K,M ) = {G ∈ G(K,M ) | G is minimal in (G(K,M ), )} The theorem follows since the emptiness of G(K,M ) is decidable. 7. Future/Ongoing Works References Bonatti, P.A., Lutz, C., Wolter, F.: The Tableau algorithm deciding GC-KB-satisﬁability, e.g., for ALC: Complexity of Circumscription in Add two grounding rules (for minimized concepts and minimized roles) to the standard ALC tableau algorithm. Description Logic. Journal of Artiﬁcial Inference problems beyond GC-KB-satisﬁability: Intelligence Research 35, 717–773 (2009) employs the approach of tableau algorithm for circumscriptive DLs Grimm, S., Hitzler, P.: A Preferential additional check for preference clash Tableaux Calculus for Circumscriptive Other future work: investigate exact worst-case complexity of GC-reasoning. ALCO. In Proc. of RR’09. LNCS, vol. 5837, pp. 40–54. Knowledge Engineering Lab. - Kno.e.sis Center - Wright State University - Dayton, Ohio, USA {adila,kunal,pascal}@knoesis.org http://knoesis.wright.edu/faculty/pascal/knoelab.html
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