Living Without Beth and Craig: Definitions and Interpolants in Description Logics with Nominals and Role Inclusions

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP transform potentially hard existence problems into deduction problems in the underlying logic. Description Logics with nominals and/or role inclusions do not enjoy the CIP nor PBDP, but interpolants and explicit definitions have many potential applications in ontology engineering and ontology-based data management. In this article we show the following: even without Craig and Beth, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as ALCO, ALCH and ALCHIO. However, living without Craig and Beth makes this problem harder than deduction: we prove that the existence problems become 2ExpTime-complete, thus one exponential harder than validity. The existence of explicit definitions is 2ExpTime-hard even if one asks for a definition of a nominal using any symbol distinct from that nominal, but it becomes ExpTime-complete if one asks for a definition of a concept name using any symbol distinct from that concept name.Comment: We have added results on description logics with role inclusions and an ExpTime-completeness result for the explicit definability of concept names. The title has been modified by adding role inclusions. This paper has been accepted for AAAA 202

Application of Definability to Query Answering over Knowledge Bases

Answering object queries (i.e. instance retrieval) is a central task in ontology based data access (OBDA). Performing this task involves reasoning with respect to a knowledge base K (i.e. ontology) over some description logic (DL) dialect L. As the expressive power of L grows, so does the complexity of reasoning with respect to K. Therefore, eliminating the need to reason with respect to a knowledge base K is desirable. In this work, we propose an optimization to improve performance of answering object queries by eliminating the need to reason with respect to the knowledge base and, instead, utilizing cached query results when possible. In particular given a DL dialect L, an object query C over some knowledge base K and a set of cached query results S={S1, ..., Sn} obtained from evaluating past queries, we rewrite C into an equivalent query D, that can be evaluated with respect to an empty knowledge base, using cached query results S' = {Si1, ..., Sim}, where S' is a subset of S. The new query D is an interpolant for the original query C with respect to K and S. To find D, we leverage a tool for enumerating interpolants of a given sentence with respect to some theory. We describe a procedure that maps a knowledge base K, expressed in terms of a description logic dialect of first order logic, and object query C into an equivalent theory and query that are input into the interpolant enumerating tool, and resulting interpolants into an object query D that can be evaluated over an empty knowledge base. We show the efficacy of our approach through experimental evaluation on a Lehigh University Benchmark (LUBM) data set, as well as on a synthetic data set, LUBMMOD, that we created by augmenting an LUBM ontology with additional axioms

Foundations for Uniform Interpolation and Forgetting in Expressive Description Logics

We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform inter- polants and their existence in terms of bisimula- tions, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of model- theoretic and automata-theoretic methods that, as a by-product, also provides characterizations of and decision procedures for conservative extensions

On Free Description Logics with Definite Descriptions

Definite descriptions are phrases of the form 'the x such that φ', used to refer to single entities in a context. They are often more meaningful to users than individual names alone, in particular when modelling or querying data over ontologies. We investigate free description logics with both individual names and definite descriptions as terms of the language, while also accounting for their possible lack of denotation. We focus on the extensions of ALC and, respectively, εL with nominals, the universal role, and definite descriptions. We show that standard reasoning in these extensions is not harder than in the original languages, and we characterise the expressive power of concepts relative to first-order formulas using a suitable notion of bisimulation. Moreover, we lay the foundations for automated support for definite descriptions generation by studying the complexity of deciding the existence of definite descriptions for an individual under an ontology. Finally, we provide a polynomial-time reduction of reasoning in other free description logic languages based on dual-domain semantics to the case of partial interpretations

Orchestrating a Network of Mereo(topo)logical Theories

Parthood is used widely in ontologies across subject domains. Some modelling guidance can be gleaned from Ontology, yet it offers multiple mereological theories, and even more when combined with topology, i.e., mereotopology. To complicate the landscape, decidable languages put restrictions on the language features, so that only fragments of the mereo(topo)logical theories can be represented, yet during modelling, those full features may be needed to check correctness. We address these issues by specifying a structured network of theories formulated in multiple logics that are glued together by the various linking constructs of the Distributed Ontology Language, \DOL. For the KGEMT mereotopological theory and five sub-theories, together with the DL-based OWL species and first- and second-order logic, this network in \DOL orchestrates 28 ontologies. Further, we propose automated steps toward resolution of language feature conflicts when combining modules, availing of the new `OWL classifier' tool that pinpoints profile violations

Practical uniform interpolation and forgetting for ALC TBoxes with applications to logical difference

Copyright © 2014, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. We develop a clausal resolution-based approach for computing uniform interpolants of TBoxes formulated in the description logic ALC when such uniform interpolants exist. We also present an experimental evaluation of our approach and of its application to the logical difference problem for real-life ALC ontologies. Our results indicate that in many practical cases uniform interpolants exist and that they can be computed with the presented algorithm

Interpolants and Explicit Definitions in Extensions of the Description Logic EL

We show that the vast majority of extensions of the description logic EL do not enjoy the Craig interpolation nor the projective Beth definability property. This is the case, for example, for EL with nominals, EL with the universal role, EL with role hierarchies and transitive roles, and for ELI. It follows in particular that the existence of an explicit definition of a concept or individual name cannot be reduced to subsumption checking via implicit definability. We show that nevertheless the existence of interpolants and explicit definitions can be decided in polynomial time for standard tractable extensions of EL (such as EL++) and in ExpTime for ELI and various extensions. It follows that these existence problems are not harder than subsumption which is in sharp contrast to the situation for expressive DLs. We also obtain tight bounds for the size of interpolants and explicit definitions and the complexity of computing them: single exponential for tractable standard extensions of EL and double exponential for ELI and extensions. We close with a discussion of Horn-DLs such as Horn-ALCI.</jats:p