Exemplaric Expressivity of Modal Logics Bart Jacobs1 and Ana Sokolova2? 1 Institute for Computing and Information Sciences, Radboud University Nijmegen P.O. Box 9010, 6500 GL Nijmegen, The Netherlands. Email:
[email protected] URL: http://www.cs.ru.nl/B.Jacobs 2 Department of Computer Sciences, University of Salzburg Jakob-Haringer-Str. 2, 5020 Salzburg, Austria. Email:
[email protected] URL: http://www.cs.uni-salzburg.at/˜anas May 16, 2008 Abstract. This paper investigates expressivity of modal logics for transition sys- tems, multitransition systems, Markov chains, and Markov processes, as coal- gebras of the powerset, finitely supported multiset, finitely supported distribu- tion, and measure functor, respectively. Expressivity means that logically indis- tinguishable states, satisfying the same formulas, are behaviourally indistinguish- able. The investigation is based on the framework of dual adjunctions between spaces and logics and focuses on a crucial injectivity property. The approach is generic both in the choice of systems and modalities, and in the choice of a “base logic”. Most of these expressivity results are already known, but the applicability of the uniform setting of dual adjunctions to these particular examples is what constitutes the contribution of the paper. 1 Introduction During the last decade, coalgebra [30,17] has become accepted as an abstract frame- work for describing state-based dynamic systems. Fairly quickly it was recognised, first in [26], that modal logic is the natural logic for coalgebras and also that coalgebras pro- vide obvious models for modal logics. Intuitively there is indeed a connection, because modal operators can be interpreted in terms of next or previous states, with respect to some transition system or, more abstractly, coalgebra.