A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequent-like tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequent-like tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbert-style formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM + , OB + , OK4 + ,...

Abstract. Knowledge representation systems, including ones based on Description Logics (DLs), use explanation facilities to, among others, debug knowledge bases. Until now, such facilities were not available for expressive DLs, whose reasoning is an un-natural refutation-based tableau. We offer a solution based on a sequent calculus that is closely related to the tableau implementation, exploiting its optimisations. The resulting proofs are pruned and then presented as simply as possible using templates. 1

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.

We present the basic framework of a logic of actions and plans defined in terms of modal logic combined with a notion of dependence. The latter is used as a weak causal connection between actions and literals. In this paper we focus on the frame problem and demonstrate how it can be solved in our framework in a simple and monotonic way. We give the semantics, and associate an axiomatics and a decision procedure to it. The decision procedure is based on a sound and complete tableau method with single step rules to treat dependence. We show how it can be used to generate plans. Our solution is formally assessed by a translation of Gelfond and Lifschitz' logic A. We briefly sketch the second part of the paper, showing how we can go beyond A by some examples involving nondeterminism and ramifications.

A general framework for translating logical formulae from one logic into another logic is presented. The framework is instantiated with two different approaches to translating modal logic formulae into predicate logic. The first one, the well known ‘relational’ translation makes the modal logic’s possible worlds structure explicit by introducing a distinguished predicate symbol to represent the accessibility relation. In the second approach, the ‘functional ’ translation method, paths in the possible worlds structure are represented by compositions of functions which map worlds to accessible worlds. On the syntactic level this means that every flexible symbol is parametrized with particular terms denoting whole paths from the initial world to the actual world. The ‘target logic’ for the translation is a first-order many-sorted logic with built in equality. Therefore the ‘source logic’ may also be first-order many-sorted with built in equality. Furthermore flexible function symbols are allowed. The modal operators may be parametrized with arbitrary terms and particular properties of the accessibility relation may be specified within the

We present a sound, complete, modular and lean labelled tableau calculus for many propositional modal logics where the labels contain "free" and "universal" variables. Our "lean" Prolog implementation is not only surprisingly short, but compares favourably with other considerably more complex implementations for modal deduction.

We show how labelled deductive systems can be combined with a logical framework to provide a natural deduction implementation of a large and well-known class of propositional modal logics (including K, D, T , B, S4, S4:2, KD45, S5). Our approach is modular and based on a separation between a base logic and a labelling algebra, which interact through a fixed interface. While the base logic stays fixed, different modal logics are generated by plugging in appropriate algebras. This leads to a hierarchical structuring of modal logics with inheritance of theorems. Moreover, it allows modular correctness proofs, both with respect to soundness and completeness for semantics, and faithfulness and adequacy of the implementation. We also investigate the tradeoffs in possible labelled presentations: We show that a narrow interface between the base logic and the labelling algebra supports modularity and provides an attractive proof-theory (in comparision to, e.g., semantic embedding) but limits th...

Hybridization is a method invented by Arthur Prior for extending the expressive power of modal languages. Although developed in interesting ways by Robert Bull, and by the So a school (notably, George Gargov, Valentin Goranko, Solomon Passy and Tinko Tinchev), the method remains little known. In our view this has deprived temporal logic of a valuable tool. The aim of the paper is to explain why hybridization is useful in temporal logic. We make two major points, the rst technical, the second conceptual. First, we showthathybridization gives rise to well-behaved logics that exhibit an interesting synergy between modal and classical ideas. This synergy, obvious for hybrid languages with full rst-order expressive strength, is demonstrated for a weaker local language capable of de ning the Until operator � we provide a minimal axiomatization, and show that in a wide range of temporally interesting cases extended completeness results can be obtained automatically. Second, we argue that the idea of sorted atomic symbols which underpins the hybrid enterprise can be developed further. To illustrate this, we discuss the advantages and disadvantages of a simple hybrid language which can quantify over paths. 1

Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the proof-theory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to

The pioneering and well-known work of Burrows, Abadi and Needham (the BAN logic) which dominates the area of security protocol analysis is shown to take an approach which is not fully formal and which consequently permits approval of dangerous protocols. Measures to make the BAN logic formal are then proposed. The formalisation is found to be desirable not only for its potential in providing rigorous analysis of security protocols, but also for its readiness for supporting a computer-aided fashion of analysis. 1 Introduction A security protocol such as one for distributing cryptographic keys is essentially a few lines of a specification of a program. Its analysis can therefore be considered as analogous to the correctness verification of such a program. However, unlike the case of running a computer program, where the user naturally bears an intention to follow the instruction so to avoid potential bugs, the main objective of a dishonest user during a run of a security protocol is to ...