Uniform Interpolation of -Ontologies Using Fixpoints

We present a method to compute uniform interpolants with fixpoints for ontologies specified in the description logic \(\mathcal{ALC}\). The aim of uniform interpolation is to reformulate an ontology such that it only uses a specified set of symbols, while preserving consequences that involve these symbols. It is known that in \(\mathcal{ALC}\) uniform interpolants cannot always be finitely represented. Our method computes uniform interpolants for the target language \(\mathcal{ALC}\mu\), which is \(\mathcal{ALC}\) enriched with fixpoint operators, and always computes a finite representation. If the result does not involve fixpoint operators, it is the uniform interpolant in \(\mathcal{ALC}\). The method focuses on eliminating concept symbols and combines resolution-based reasoning with an approach known from the area of second-order quantifier elimination to introduce fixpoint operators when needed. If fixpoint operators are not desired, it is possible to approximate the interpolant.

[1]  Renate A. Schmidt,et al.  The Ackermann approach for modal logic, correspondence theory and second-order reduction , 2012, J. Appl. Log..

[2]  Giuseppe De Giacomo,et al.  Reasoning in Expressive Description Logics with Fixpoints based on Automata on Infinite Trees , 1999, IJCAI.

[3]  Valentin Goranko,et al.  SCAN Is Complete for All Sahlqvist Formulae , 2003, RelMiCS.

[4]  Kewen Wang,et al.  Tableau-based Forgetting in ALC Ontologies , 2010, ECAI 2010.

[5]  W. Ackermann.  Untersuchungen über das Eliminationsproblem der mathematischen Logik , 1935 .

[6]  Dov M. Gabbay,et al.  Second-Order Quantifier Elimination - Foundations, Computational Aspects and Applications , 2008, Studies in logic : Mathematical logic and foundations.

[7]  Marco Hollenberg,et al.  Uniform Interpolation, Automata and the Modal μ-Calculus , 1996, Advances in Modal Logic.

[8]  Nadeschda Nikitina.  Forgetting in General EL Terminologies , 2011, Description Logics.

[9]  Andrzej Szalas,et al.  Second-order reasoning in description logics , 2006, J. Appl. Non Class. Logics.

[10]  Carsten Lutz,et al.  Foundations for Uniform Interpolation and Forgetting in Expressive Description Logics , 2011, IJCAI.

[11]  Frank Wolter,et al.  Semi-qualitative Reasoning about Distances: A Preliminary Report , 2000, JELIA.

[12]  Andreas Herzig,et al.  Uniform Interpolation by Resolution in Modal Logic , 2008, JELIA.

[13]  Dov M. Gabbay,et al.  Quantifier Elimination in Second-Order Predicate Logic , 1992, KR.

[14]  Georg Struth,et al.  Relational and Kleene-Algebraic Methods in Computer Science , 2003, Lecture Notes in Computer Science.

[15]  Boris Motik,et al.  Reasoning over Ontologies with Hidden Content: The Import-by-Query Approach , 2012, J. Artif. Intell. Res..

[16]  G. D’Agostino,et al.  On modal mu-calculus with explicit interpolants , 2006, J. Appl. Log..

[17]  Robert Piro,et al.  EL-Concepts go Second-Order: Greatest Fixpoints and Simulation Quantifiers , 2010, Description Logics.

[18]  A. Szałas,et al.  A Fixpoint Approach to Second-Order Quantifier Elimination with Applications to Correspondence Theory , 1999 .

[19]  Harald Ganzinger,et al.  Resolution Theorem Proving , 2001, Handbook of Automated Reasoning.

[20]  Ian Horrocks,et al.  Consequence-Based Reasoning beyond Horn Ontologies , 2011, IJCAI.