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We develop a formal framework for comparing different versions of DL-Lite ontologies. Four notions of difference and entailment between ontologies are introduced and their applications in ontology development and maintenance discussed. These notions are obtained by distinguishing between differences that can be observed among concept inclusions, answers to queries over ABoxes, and by taking into account additional context ontologies. We compare these notions, study their meta-properties, and determine the computational complexity of the corresponding reasoning tasks. Moreover, we show that checking difference and entailment can be automated by means of encoding into QBF satisfiability and using off-the-shelf QBF solvers. Finally, we explore the relationship between the notion of forgetting (or uniform interpolation) and our notions of difference between ontologies.

Contents Acknowledgments vii 1 Introduction. 1 2 General preliminaries. 7 2.1 Logics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.1.1 Languages and structures. . . . . . . . . . . . . . . . . . . 7 2.1.2 Syntax. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.1.3 Semantics. . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.4 Translations. . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.5 Derivability in Basic Modal Logic. . . . . . . . . . . . . . . 10 2.2 Bisimulation and the like. . . . . . . . . . . . . . . . . . . . . . . 12 2.3 A brief introduction to the -Calculus. . . . . . . . . . . . . . . . 17 2.4 The family of graded modal logics and their semantics. . . . . . . 22 2.5 Interpolation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.5.1 Uniform interpolation. . . . . . . . . . . . . . . . . . . . . 29 2.5.2 Elementary interpolation. . . . . . . . . . . . . . . . . . . 30 2.6 N

We develop a formal framework for modular ontologies by analysing four notions of conservative extensions and their applications in refining, re-using, merging, and segmenting ontologies. For two members of the DL-Lite family of description logics, we prove important metaproperties of these notions such as robustness under joins, vocabulary extensions, and iterated import of ontologies. The computational complexity of the corresponding reasoning tasks is investigated.

Summary. Modularity of ontologies is currently an active research field, and many different notions of a module have been proposed. In this paper, we review the fundamental principles of modularity and identify formal properties that a robust notion of modularity should satisfy. We explore these properties in detail in the contexts of description logic and classical predicate logic and put them into the perspective of well-known concepts from logic and modular software specification such as interpolation, forgetting and uniform interpolation. We also discuss reasoning problems related to modularity. 1

We develop a formal framework for comparing different versions of DL-Lite ontologies. The main feature of our approach is that we take into account the vocabulary ( = signature) with respect to which one wants to compare ontologies. Five variants of difference and inseparability relations between ontologies are introduced and their respective applications for ontology development and maintenance discussed. These variants are obtained by generalising the notion of conservative extension from mathematical logic and by distinguishing between differences that can be observed among concept inclusions, answers to queries over ABoxes, by taking into account additional context ontologies, and by considering a model-theoretic, language-independent notion of difference. We compare these variants, study their meta-properties, determine the computational complexity of the corresponding reasoning tasks, and present decision algorithms. Moreover, we show that checking inseparability can be automated by means of encoding into QBF satisfiability and using off-the-shelf general purpose QBF solvers. Inseparability relations between ontologies are then used to develop a formal framework for (minimal) module extraction. We demonstrate that different types of minimal modules induced by these inseparability relations can be automatically extracted from real-world medium-size DL-Lite ontologies by composing the tractable syntactic locality-based module extraction algorithm with non-tractable extraction algorithms using the multi-engine QBF solver aqme. Finally, we explore the relationship between uniform interpolation (or forgetting) and inseparability between ontologies.

Every normal modal logic L gives rise to the consequence relation ϕ |=L ψ which holds iff ψ is true in a world of an L-model whenever ϕ is true in that world. We consider the following algorithmic problem for L. Given two modal formulas ϕ1 and ϕ2, decide whether ϕ1∧ϕ2 is a conservative extension of ϕ1 in the sense that whenever ϕ1 ∧ ϕ2 |=L ψ and ψ does not contain propositional variables not occurring in ϕ1, then ϕ1 |=L ψ. We first prove that the conservativeness problem is coNExpTime-hard for all modal logics of unbounded width (which have rooted frames with more than N successors of the root, for any N < ω). Then we show that this problem is (i) coNExpTime-complete for S5 and K, (ii) in ExpSpace for S4 and (iii) ExpSpace-complete for GL.3 (the logic of finite strict linear orders). The proofs for S5 and K use the fact that these logics have uniform interpolation. 1

We discuss how concepts and methods introduced in mathematical logic can be used to support the engineering and deployment of life science ontologies. The required applications of mathematical logic are not straighforward and we argue that such ontologies provide a new and rich family of logical theories that wait to be explored by logicians.