The last years have seen two major advances in Knowledge Representation and Reasoning. First, many interesting problems (ranging from Semi-structured Data to Linguistics) were shown to be expressible in logics whose main deductive problems are EXPTIME-complete. Second, experiments in automated reasoning have substantially broadened the meaning of “practical tractability”. Instances of realistic size for PSPACE-complete problems are now within reach for implemented systems. Still, there is a gap between the reasoning services needed by the expressive logics mentioned above and those provided by the current systems. Indeed, the algorithms based on tree-automata, which are used to prove EXPTIME-completeness, require exponential time and space even in simple cases. On the other hand, current algorithms based on tableau methods can take advantage of such cases, but require double exponential time in the worst case. We propose a tableau calculus for the description logic ALC for checking the satisfiability of a concept with respect to a TBox with general axioms, and transform it into the first simple tableaubased decision procedure working in single exponential time. To guarantee the ease of implementation, we also discuss the effects that optimizations (propositional backjumping, simplification, semantic branching, etc.) might have on our complexity result, and introduce a few optimizations ourselves.

. In previous work we gave an approach, based on labelled natural deduction, for formalizing proof systems for a large class of propositional modal logics that includes K, D, T, B, S4, S4:2, KD45, and S5. Here we extend this approach to quantified modal logics, providing formalizations for logics with varying, increasing, decreasing, or constant domains. The result is modular with respect to both properties of the accessibility relation in the Kripke frame and the way domains of individuals change between worlds. Our approach has a modular metatheory too; soundness, completeness and normalization are proved uniformly for every logic in our class. Finally, our work leads to a simple implementation of a modal logic theorem prover in a standard logical framework. 1 Introduction Motivation Modal logic is an active area of research in computer science and artificial intelligence: a large number of modal logics have been studied and new ones are frequently proposed. Each new log...

We study sequent calculi for propositional modal logics, interpreted over coalgebras, with admissibility of cut being the main result. As applications we present a new proof of the (already known) interpolation property for coalition logic and establish the interpolation property for the conditional logics CK and CK Id.

Abstract. We propose so called clausal tableau systems for the common modal logics K, KD, T, KB, KDB and B. There is a measure such that for each tableau rule of these systems the measure of all its denominators is smaller than the measure of its numerator. Basing on these systems, we give a decision procedure for the logics, which uses O(n 2)-space for the logics T, KB, KDB and B, and O(n. log n)-space for the logics K and KD. We also show that the problem of checking satisfiability in T, KB, KDB, or B for formulae with finitely bounded modal-depth is decidable in O(n. log n)-space. We are the first who explicitly establish space requirements for the logics KB, KDB and B. 1

Abstract. We describe the ongoing work on a tactic-based theorem prover for First-Order Modal and Temporal Logics (FOTLs for the temporal ones). In formal methods, especially temporal logics play a determining role; in particular, FOTLs are natural whenever the modeled systems are infinite-state. But reasoning in FOTLs is hard and few approaches have so far proved effective. Here we introduce a family of sequent calculi for first-order modal and temporal logics which is modular in the structure of time; moreover, we present a tactic-based modal/temporal theorem prover enforcing this approach, obtained employing the higher-order logic programming language λProlog. Finally, we show some promising experimental results and raise some open issues. We believe that, together with the Proof Planning approach, our system will eventually be able to improve the state of the art of formal methods through the use of FOTLs. 1

We give two generic proofs for cut elimination in propositional modal logics, interpreted over coalgebras. We first investigate semantic coherence conditions between the axiomatisation of a particular logic and its coalgebraic semantics that guarantee that the cut-rule is admissible in the ensuing sequent calculus. We then independently isolate a purely syntactic property of the set of modal rules that guarantees cut elimination. Apart from the fact that cut elimination holds, our main result is that the syntactic and semantic assumptions are equivalent in case the logic is amenable to coalgebraic semantics. As applications we present a new proof of the (already known) interpolation property for coalition logic and newly establish the interpolation property for the conditional logics CK and CK + ID.

This paper is a summary of the author’s Ph.D. thesis, concerned with automated reasoning in quantified modal and temporal logics. The relevant contributions are: (i) a sound and complete set of sequent calculi for quantified modal logics is devised; (ii) the approach is extended to the quantified temporal logic of linear, discrete time and a framework for doing automated reasoning via Proof Planning in it is developed; (iii) a set of promising experimental results is shown, obtained by applying the framework to the problem of Feature Interactions in telecommunication systems.

. Recently, there has been growing attention on the space requirements of tableau methods (see for example [6], [2], [10]). We have proposed in [9] a method of reducing modal consequence relations to the global and local consequence relation of (polymodal) K. The reductions used there did however not give rise to ecient time complexity bounds. In this note we shall use reduction functions to obtain rather sharp space bounds. These bounds can be applied to ordinary tableau systems, and do not make use of the Mints transform. It has been shown by Hudelmaier ([6]) that satisability in S4 is O(n 2 log n){space computable, while satisability in K and T are O(n log n){space computable. A O(n log n){space bound for K.D has been obtained by Basin, Matthews and Vigano ([2]). Vigano ([14]) has shown that satisability in K4, KD4 and S4 is in O(n 2 log n){ space. Nguyen has reduced these bounds to O(n log n) for K4, K4D and S4 in [11]. Additionally, O(n log n){bounds are shown for ...

In previous work we gave a new proof-theoretical method for establishing upper-bounds on the space complexity of the provability problem of modal and other propositional non-classical logics. Here we extend and rene these results to give an O(n log n)-space decision procedure for the basic positive relevance logic B + . We compute this upper-bound by rst giving a sound and complete, cut-free, labelled sequent system for B + , and then establishing bounds on the application of the rules of this system. Keywords: Relevance Logics, Computational Complexity, Labelled Deduction Systems, Sequent Systems. 1