A Unified Framework for Non-standard Reasoning Services in Description Logics

Non-standard reasoning in Description Logics (DLs) comprises computing a Least Common Subsumer (LCS), a Concept Difference, a Concept Unifier, or an Interpolant Concept, to name a few. Although some reasoning services have been unified already (e.g., LCS and Most Specific Concept), the definition of non-standard problems and the computation that solve them are very different from each other. We propose to unify the definitions of non-standard services as special Second-order sentences in DLs; when the solution concepts are optimal with respect to some preferences, a fixpoint replaces the Second-order quantification. Moreover, we propose to combine the well-known Tableaux calculi for DLs with rules that compute substitutions of Concept Variables. We prove soundness and completeness of the combined calculus and we give a sufficient condition for termination, which covers some non-trivial cases.

[1]  Franz Baader.  Least Common Subsumers and Most Specific Concepts in a Description Logic with Existential Restrictions and Terminological Cycles , 2003, IJCAI.

[2]  Boris Konev,et al.  Forgetting and Uniform Interpolation in Large-Scale Description Logic Terminologies , 2009, IJCAI.

[3]  Paliath Narendran,et al.  Unification of Concept Terms in Description Logics , 2001, Description Logics.

[4]  W. Marsden.  I and J , 2012 .

[5]  Franz Baader,et al.  Computing the Least Common Subsumer w.r.t. a Background Terminology , 2004, Description Logics.

[6]  Alexander Borgida,et al.  Computing Least Common Subsumers in Description Logics , 1992, AAAI.

[7]  Peter Gärdenfors,et al.  Knowledge in Flux , 1988 .

[8]  J. R. Büchi.  On a Decision Method in Restricted Second Order Arithmetic , 1990 .

[9]  Leon Henkin,et al.  Completeness in the theory of types , 1950, Journal of Symbolic Logic.

[10]  Stephan Tobies,et al.  Complexity results and practical algorithms for logics in knowledge representation , 2001, ArXiv.

[11]  Ian Horrocks,et al.  Practical Reasoning for Very Expressive Description Logics , 2000, Log. J. IGPL.

[12]  Freddy Lécué,et al.  Applying Abduction in Semantic Web Service Composition , 2007, IEEE International Conference on Web Services (ICWS 2007).

[13]  M. Rabin.  Decidability of second-order theories and automata on infinite trees , 1968 .

[14]  Ulrike Sattler,et al.  A Case for Abductive Reasoning over Ontologies , 2006, OWLED.

[15]  Francesco M. Donini,et al.  Abductive Matchmaking using Description Logics , 2003, IJCAI.

[16]  Deborah L. McGuinness,et al.  Explaining Subsumption in Description Logics , 1995, IJCAI.

[17]  Azzurra Ragone,et al.  OWL-DL as a Power Tool to Model Negotiation Mechanisms with Incomplete Information , 2007, ISWC/ASWC.

[18]  Stefan Schlobach,et al.  Explaining Subsumption by Optimal Interpolation , 2004, JELIA.

[19]  Ralf Küsters,et al.  Rewriting Concepts Using Terminologies , 2000, KR.

[20]  Frank Wolter,et al.  Undecidability of the unification and admissibility problems for modal and description logics , 2006, TOCL.

[21]  Gunnar Teege.  Making the Difference: A Subtraction Operation for Description Logics , 1994, KR.

[22]  Franz Baader,et al.  Usability Issues in Description Logic Knowledge Base Completion , 2009, ICFCA.

[23]  Ralf Möller,et al.  Formalizing Multimedia Interpretation based on Abduction over Description Logic Aboxes , 2009, Description Logics.

[24]  Jouko A. Väänänen,et al.  Second-Order Logic and Foundations of Mathematics , 2001, Bulletin of Symbolic Logic.

[25]  Francesco M. Donini,et al.  Semantic Matchmaking as Non-Monotonic Reasoning: A Description Logic Approach , 2007, J. Artif. Intell. Res..

[26]  J. Bruijn,et al.  Effective query rewriting with ontologies over DBoxes , 2009, IJCAI 2009.

[27]  Alexander Borgida,et al.  Towards Measuring Similarity in Description Logics , 2005, Description Logics.

[28]  Meghyn Bienvenu,et al.  Complexity of Abduction in the EL Family of Lightweight Description Logics , 2008, KR.

[29]  Francesco M. Donini,et al.  Weighted Description Logics Preference Formulas for Multiattribute Negotiation , 2009, SUM.