Michael Zakharyaschev

Decidable fragments of first-order temporal logics

Abstract In this paper, we introduce a new fragment of the first-order temporal language, called the monodic fragment, in which all formulas beginning with a temporal operator (Since or Until) have at most one free variable. We show that the satisfiability problem for monodic formulas in various linear time structures can be reduced to the satisfiability problem for a certain fragment of classical first-order logic. This reduction is then used to single out a number of decidable fragments of first-order temporal logics and of two-sorted first-order logics in which one sort is intended for temporal reasoning. Besides standard first-order time structures, we consider also those that have only finite first-order domains, and extend the results mentioned above to temporal logics of finite domains. We prove decidability in three different ways: using decidability of monadic second-order logic over the intended flows of time, by an explicit analysis of structures with natural numbers time, and by a composition method that builds a model from pieces in finitely many steps.

Multi-Dimensional Modal Logic as a Framework for Spatio-Temporal Reasoning

In this paper we advocate the use of multi-dimensional modal logics as a framework for knowledge representation and, in particular, for representing spatio-temporal information. We construct a two-dimensional logic capable of describing topological relationships that change over time. This logic, called PSTL (Propositional Spatio-Temporal Logic) is the Cartesian product of the well-known temporal logic PTL and the modal logic S4u, which is the Lewis system S4 augmented with the universal modality. Although it is an open problem whether the full PSTL is decidable, we show that it contains decidable fragments into which various temporal extensions (both point-based and interval based) of the spatial logic RCC-8 can be embedded. We consider known decidability and complexity results that are relevant to computation with multi-dimensional formalisms and discuss possible directions for further research.

Ontology-Based Data Access: Ontop of Databases

We present the architecture and technologies underpinning the OBDA system Ontop and taking full advantage of storing data in relational databases. We discuss the theoretical foundations of Ontop: the tree-witness query rewriting,

Reasoning over extended ER models

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Modal description logics: modalizing roles

-mappings and optimisations based on database integrity constraints and SQL features. We analyse the performance of Ontop in a series of experiments and demonstrate that, for standard ontologies, queries and data stored in relational databases, Ontop is fast, efficient and produces SQL rewritings of high quality.

Axiomatizing the monodic fragment of first-order temporal logic

We investigate the computational complexity of reasoning over various fragments of the Extended Entity-Relationship (EER) language, which includes a number of constructs: ISA between entities and relationships, disjointness and covering of entities and relationships, cardinality constraints for entities in relationships and their refinements as well as multiplicity constraints for attributes. We extend the known EXPTIME-completeness result for UML class diagrams [5] and show that reasoning over EER diagrams with ISA between relationships is EXPTIME-complete even without relationship covering. Surprisingly, reasoning becomes NP-complete when we drop ISA between relationships (while still allowing all types of constraints on entities). If we further omit disjointness and covering over entities, reasoning becomes polynomial. Our lower complexity bound results are proved by direct reductions, while the upper bounds follow from the correspondences with expressive variants of the description logic DL-Lite, which we establish in this paper. These correspondences also show the usefulness of DL-Lite as a language for reasoning over conceptual models and ontologies.

Logics of metric spaces

We construct a new concept description language intended for representing dynamic and intensional knowledge. The most important feature distinguishing this language from its predecessors in the literature is that it allows applications of modal operators to all kinds of syntactic terms: concepts, roles and formulas. Moreover, the language may contain both local (i.e., state-dependent) and global (i.e., state-independent) concepts, roles and objects. All this provides us with the most complete and natural means for reflecting the dynamic and intensional behaviour of application domains. We construct a satisfiability checking (mosaic-type) algorithm for this language (based on ALC) in (i) arbitrary multimodal frames, (ii) frames with universal accessibility relations (for knowledge) and (iii) frames with transitive, symmetric and euclidean relations (for beliefs). On the other hand, it is shown that the satisfaction problem becomes undecidable if the underlying frames are arbitrary strict linear orders, 〈

On the Products of Linear Modal Logics

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