Exptime tableaux for the coalgebraic μ-calculus

The coalgebraic approach to modal logic provides a uniform framework that captures the semantics of a large class of structurally different modal logics, including e.g. graded and probabilistic modal logics and coalition logic. In this paper, we introduc

A compositional approach to defining logics for coalgebras

We present a compositional approach to defining expressive logics for coalgebras of endofunctors on Set. This approach uses a notion of language constructor and an associated notion of semantics to capture one inductive step in the definition of a language for coalgebras and of its semantics. We show that suitable choices for the language constructors and for their associated semantics yield logics which are both adequate and expressive w.r.t. behavioural equivalence. Moreover, we show that type-building operations give rise to corresponding operations both on language constructors and on their associated semantics, thus allowing the derivation of expressive logics for increasingly complex coalgebraic types. Our framework subsumes several existing approaches to defining logics for coalgebras, and at the same time allows the derivation of new logics, with logics for probabilistic systems being the prime example

Modular games for coalgebraic fixed point logics.

We build on existing work on finitary modular coalgebraic logics [C. Cîrstea. A compositional approach to defining logics for coalgebras. Theoretical Computer Science, 327:45–69, 2004; C. Cîrstea and D. Pattinson. Modular construction of complete coalgebraic logics. Theoretical Computer Science, 388:83–108, 2007], which we extend with general fixed points, including CTL- and PDL-like fixed points, and modular evaluation games. These results are generalisations of their correspondents in the modal μ-calculus, as described e.g. in [C. Stirling. Modal and Temporal Properties of Processes. Springer, 2001]. Inspired by recent work of Venema [Y. Venema. Automata and fixed point logic: a coalgebraic perspective. Information and Computation, 204:637–678, 2006], we provide our logics with evaluation games that come equipped with a modular way of building the game boards. We also study a specific class of modular coalgebraic logics that allow for the introduction of an implicit negation operator.© 2008 Published by Elsevier B.V. Open access under CC BY-NC-ND license. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0