Stochastic Coalgebraic Logic
Coalgebraic logic is an important research topic in the areas of concurrency theory, semantics, transition systems and modal logics. It provides a general approach to modeling systems, allowing us to apply important results from coalgebras, universal alge
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Advisory Board: G. Ausiello M. Broy C.S. Calude A. Condon D. Harel J. Hartmanis T. Henzinger T. Leighton M. Nivat C. Papadimitriou D. Scott
Ernst-Erich Doberkat
Stochastic Coalgebraic Logic
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Author Prof. Dr. Ernst-Erich Doberkat Lehrstuhl f¨ur Software-Technologie Fakult¨at f¨ur Informatik Technische Universit¨at Dortmund Germany [email protected] Series Editors Prof. Dr. Wilfried Brauer Institut f¨ur Informatik der TUM Boltzmannstr. 3 85748 Garching, Germany [email protected] Prof. Dr. Grzegorz Rozenberg Leiden Institute of Advanced Computer Science University of Leiden Niels Bohrweg 1 2333 CA Leiden, The Netherlands [email protected]
Prof. Dr. Juraj Hromkoviˇc ETH Zentrum Department of Computer Science Swiss Federal Institute of Technology 8092 Z¨urich, Switzerland [email protected] Prof. Dr. Arto Salomaa Turku Centre of Computer Science Lemmink¨aisenkatu 14 A 20520 Turku, Finland [email protected]
ISSN 1431-2654 ISBN 978-3-642-02994-3 e-ISBN 978-3-642-02995-0 DOI 10.1007/978-3-642-02995-0 Springer Heidelberg Dordrecht London New York ACM Computing Classification (1998): F.4.1, G.3 Library of Congress Control Number: 2009939283 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: K¨unkelLopka GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
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Preface
Motivation Modal logics are usually interpreted through Kripke models, branching logics find their interpretation through models which deal with infinite paths. These seemingly structurally different interpretations can be unified by considering coalgebras which model the underlying worlds suitably; the predicates through which the formulas are represented in their interpretation are modelled using natural transformations between functors, to which the functor that underlies the coalgebra contributes. The basic functor is usually based on the power set functor. Adopting this general approach, we see that a fairly general and uniform way of interpreting modal logics and their step twins arises through coalgebras and the generalization of predicates into suitable natural transformations. We will show
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