This paper offers a synthetic overview of, and original contributions to, the use of logics and formal methods in the analysis of hybrid systems

. This paper presents a prefixed tableaux calculus for Propositional Dynamic Logic with Converse based on a combination of different techniques such as prefixed tableaux for modal logics and model checkers for ¯-calculus. We prove the correctness and completeness of the calculus and illustrate its features. We also discuss the transformation of the tableaux method (naively NEXPTIME) into an EXPTIME algorithm. 1 Introduction Propositional Dynamic Logics (PDLs) are modal logics introduced in [10] to model the evolution of the computation process by describing the properties of states reached by programs during their execution [15, 24, 27]. Over the years, PDLs have been proved to be a valuable formal tool in Computer Science, Logic, Computational Linguistics, and Artificial Intelligence far beyond their original use for program verification (e.g. [4, 12, 14, 15, 24, 23]). In this paper we focus on Converse-PDL (CPDL) [10], obtained from the basic logic PDL by adding the converse operat...

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 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 consider the modal logics KT and S4, the tense logic K t , and the fragment IPC (^;!) of intuitionistic logic. For these logics backward proof search in the standard sequent or tableau calculi does not terminate in general. In terms of the respective Kripke semantics, there are several kinds of non-termination: loops inside a world (KT), innite resp. looping branches (S4, IPC (^;!) ), and innite branching degree (K t ). We give uniform sequent-based calculi that contain specically tailored loop-checks such that the eciency of proof search is not deteriorated. Moreover all these loop-checks are easy to implement and can be combined with optimizations. Note that our calculus for S4 is not a pure contraction-free sequent calculus, but this (theoretical) defect does not hinder its application in practice. 1 Introduction For many non-classical propositional logics, backward proof search in the usual sequent calculi does not terminate in general. For all the logics we consider in th...

In this paper we define two logics, KL n and BL n , and present resolutionbased proof methods for both. KL n is a temporal logic of knowledge. Thus, in addition to the usual connectives of linear discrete temporal logic, it contains a set of unary modal connectives for representing the knowledge possessed by agents. The logic BL n is somewhat similar: it is a temporal logic that contains connectives for representing the beliefs of agents. The proof methods we present for these logics involve two key steps. First, a formula to be tested for unsatisfiability is translated into a normal form. Secondly, a family of resolution rules are used, to deal with the interactions between the various operators of the logics. In addition to a description of the normal form and the proof methods, we present some short worked examples and proposals for future work. 1

In this paper we present a prefixed analytic tableau calculus for a class of normal multimodal logics and we present some results about decidability and undecidability of this class. The class is characterized by axioms of the form [t 1 ] : : : [t n ]' oe [s1 ] : : : [sm ]', called inclusion axioms, where the t i 's and s j 's are constants. This class of logics, called grammar logics, was introduced for the first time by Farinas del Cerro and Penttonen to simulate the behaviour of grammars in modal logics, and includes some well-known modal systems. The prefixed tableau method is used to prove the undecidability of modal systems based on unrestricted, context sensitive, and context free grammars. Moreover, we show that the class of modal logics, based on right-regular grammars, are decidable by means of the filtration methods, by defining an extension of the Fischer-Ladner closure.

This paper presents a prefixed tableaux calculus for Propositional Dynamic Logic with Converse based on a combination of different techniques such as prefixed tableaux for modal logics and model checkers for mu-calculus. We prove the correctness and completeness of the calculus and illustrate its features. We also discuss the transformation of the tableaux method (naively NEXPTIME) into an EXPTIME algorithm.

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. This paper presents a translation-based resolution decision procedure for the multi-modal logic K (m) (\; [; ^) dened over families of relations closed under intersection, union and converse. The relations may satisfy certain additional frame properties. Dierent from previous resolution decision procedures which are based on ordering renements our procedure is based on a selection renement, the derivations of which correspond to derivations of tableaux or sequent proof systems. This procedure has the advantage that it can be used both as a satisability checker and a model builder. We show that tableaux and sequent-style proof systems can be polynomially simulated with our procedure. Furthermore, the nite model property follows for a number of extended modal logics. Keywords: modal logic, automated theorem proving, resolution decision procedures, tableaux proof systems, satisability testing, model generation, simulation, relative proof complexity, relative search space complex...