The aim of this paper is to summarize and analyze some results obtained in 2000-2001 about decidable and undecidable fragments of various first-order temporal logics, give some applications in the field of knowledge representation and reasoning, and attract the attention of the `temporal community' to a number of interesting open problems.

In this paper we show how to extend clausal temporal resolution to the ground eventuality fragment of monodic first-order temporal logic, which has recently been introduced by Hodkinson, Wolter and Zakharyaschev. While a finite Hilbert-like axiomatization of complete monodic first order temporal logic was developed by Wolter and Zakharyaschev, we propose a temporal resolutionbased proof system which reduces the satisfiability problem for ground eventuality monodic first-order temporal formulae to the satisfiability problem for formulae of classical first-order logic.

Let C 2 p denote the class of first order sentences with two variables and with additional quantifiers "there exists exactly (at most, at least) i", for i p, and let C 2 be the union of C 2 p taken over all integers p. We prove that the satisfiability problem for C 2 1 sentences is NEXPTIME-complete. This strengthens the results by E. Grädel, Ph. Kolaitis and M. Vardi [15] who showed that the satisfiability problem for the first order two-variable logic L 2 is NEXPTIME-complete and by E. Grädel, M. Otto and E. Rosen [16] who proved the decidability of C 2 . Our result easily implies that the satisfiability problem for C 2 is in non-deterministic, doubly exponential time. It is interesting that C 2 1 is in NEXPTIME in spite of the fact, that there are sentences whose minimal (and only) models are of doubly exponential size. It is worth noticing, that by a recent result of E. Gradel, M. Otto and E. Rosen [17], extensions of two-variables logic L 2 by a week access to car...

The paper considers the set ML1 of first-order polymodal formulas the modal operators in which can be applied to subformulas of at most one free variable. Using a mosaic technique, we prove a general satisfiability criterion for formulas in ML1, which reduces the modal satisfiability to the classical one. The criterion is then used to single out a number of new, in a sense optimal, decidable fragments of various modal predicate logics.

This paper investigates the expressive power and computational properties of languages designed for speaking about distances. `Distances' can be induced by difAuthors Addresses: Oliver Kutz, Frank Wolter, Institut fur Informatik, Abteilung intelligente Systeme, Universitat Leipzig, Augustus-Platz 10--11, 04109 Leipzig, Germany; Holger Sturm, Fachbereich Philosophie, Universitat Konstanz, 78457 Konstanz, Germany; Nobu-Yuki Suzuki, Department of Mathematics, Faculty of Science, Shizuoka University, Ohya 836, Shizuoka 422-- 8529, Japan; Michael Zakharyaschev, Department of Computer Science, King's College, Strand, London WC2R 2LS, U.K. Emails: {kutz, wolter}@informatik.uni-leipzig.de, holger.sturm@uni-konstanz.de, smnsuzu@ipz.shizuoka.ac.jp, and mz@dcs.kcl.ac.uk Permission to make digital/hard copy of all or part of this material without fee for personal or classroom use provided that the copies are not made or distributed for profit or commercial advantage, the ACM copyright/server notice, the title of the publication, and its date appear, and notice is given that copying is by permission of the ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists requires prior specific permission and/or a fee

It has been shown recently that monodic first-order temporal logic without functional symbols but with equality is incomplete, i.e. the set of the valid formulae of this logic is not recursively enumerable. In this paper we show that an even simpler fragment consisting of monodic monadic two-variable formulae is not recursively enumerable.

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Abstract. We propose a logic for reasoning about metric spaces with the induced topologies. It combines the ‘qualitative ’ interior and closure operators with ‘quantitative’ operators ‘somewhere in the sphere of radius r, ’ including or excluding the boundary. We supply the logic with both the intended metric space semantics and a natural relational semantics, and show that the latter (i) provides finite partial representations of (in general) infinite metric models and (ii) reduces the standard ‘ε-definitions ’ of closure and interior to simple constraints on relations. These features of the relational semantics suggest a finite axiomatisation of the logic and provide means to prove its EXPTIME-completeness (even if the rational numerical parameters are coded in binary). An extension with metric variables satisfying linear rational (in)equalities is proved to be decidable as well. Our logic can be regarded as a ‘well-behaved ’ common denominator of logical systems constructed in temporal, spatial, and similarity-based quantitative and qualitative representation and reasoning. Interpreted on the real line (with its Euclidean metric), it is a natural fragment of decidable temporal logics for specification and verification of real-time systems. On the real plane, it is closely related to quantitative and qualitative formalisms for spatial

We introduce a family of languages intended for representing knowledge and reasoning about metric (and more general distance) spaces. While the simplest language can speak only about distances between individual objects and Boolean relations between sets, the more expressive ones are capable of capturing notions such as `somewhere in (or somewhere out of) the sphere of a certain radius', `everywhere in a certain ring', etc. The computational complexity of the satisfiability problem for formulas in our languages ranges from NP-completeness to undecidability and depends on the class of distance spaces in which they are interpreted. Besides the class of all metric spaces, we consider, for example, the spaces R \Theta R and N \Theta N with their natural metrics. 1 Introduction The concept of `distance between objects' is one of the most fundamental abstractions both in science and in everyday life. Imagine for instance (only imagine) that you are going to buy a house in London. You then i...

In this paper we analyze the decision problem for fragments of first-order extensions of branching time temporal logics such as computational tree logics CTL and CTL or Prior's Ockhamist logic of historical necessity. On the one hand, we show that the one-variable fragments of logics like first-order CT L ---such as the product of propositional CT L with simple propositional modal logic S5, or even the one-variable bundled first-order temporal logic with sole temporal operator `some time in the future'---are undecidable. On the other hand, it is proved that by restricting applications of first-order quantifiers to state (i.e., path-independent) formulas, and applications of temporal operators and path quantifiers to formulas with at most one free variable, we can obtain decidable fragments. The same arguments show decidability of `non-local' propositional CTL , in which truth values of propositional atoms depend on the history as well as the current time. The positive decidability results can serve as a unifying framework for devising expressive and effective time-dependent knowledge representation formalisms, e.g., temporal description or spatio-temporal logics.