Motivation

The seminar deals with recent developments in the area of coalgebraic logic, a branch of logics which combines modal logics with coalgebraic semantics. Modal logic finds its uses when reasoning about behavioural and temporal properties of computation and communication, coalgebras have evolved into a general theory of systems. Consequently, it is natural to combine both areas for a mathematical description of system specification. The seminar is a sequel to the first Dagstuhl seminar 09502 on coalgebraic logic, December 7 - 9, 2009. Some of it has been documented in the special issue in Mathematical Structures in Computer Science, (2011) 21(2).

Following on from Moss's seminal paper of the same title, Coalgebraic Logic is now growing into a successful area. Extending and generalising existing work in modal logic, coalgebraic techniques have been applied successfully to topics such as completeness, expressivity, compositionality, complexity, rule formats for process calculi, contributing numerous results on these classic topics.

The success of coalgebraic logic is based on treating different types of systems in a uniform way by taking the type of a class of systems to be a functor F on for a given base category C. Coalgebraic logic develops techniques which are uniform in (F,C) and thus, for example, allows to transfer results known from Kripke semantics (where F is powerset and C is the category of sets) to other settings. Here probabilistic approaches deserve to be mentioned (where F takes probability distributions and C is a suitable category of measurable or topological spaces). In a number of papers Markov transition systems could be shown to interpret modal logics under different assumptions on the probabilistic structure. In this context, it was shown that logical equivalence, bisimilarity, and behavioral equivalence are equivalent concepts. This approach was recently generalized from general modal logics to coalgebraic logics; these logics are interpreted through coalgebras in which the subprobability functor and the functor suggested by the phenomenon to be modelled form various syntactic alliances. This generalization brings stochastic coalgebraic logic into the mainstream of coalgebraic logics: the problems considered are similar, and one sees a convergence of methods. One of the current challenges is to bring dynamic logics into this framework.

In the probabilistic case above or more generally, one important idea of coalgebraic logic is to provide effective syntax and uniform algorithms (e.g. to check the satisfiability of logical specifications or the bisimilarity of process expressions) which are parametric in the functor F and the category C. In order to achieve this, one needs effective syntactic presentations of (F,C). There has been recent progress in studying the formal properties of different notions of presentation and linking them to algorithms. But we would expect these ideas to have a wider impact on coalgebraic logic and provide one unifying thread running through each of the following topics.

Topics of interest include:

• Category Theoretic Aspects of Coalgebraic Logic,
• Probabilistic Transition Systems,
• Modal Logic and Domain Theory,
• Stone Duality,
• Coalgebraic Logic, Automata Theory, Fixed Point Logics,
• Coalgebraic Logic for Structural Operational Semantics,
• Applied Coalgebraic Logic.

Further topics will be added, depending on the specific interest of the participants.