A Logic for Coalgebraic Simulation

AbstractBuilding on the work of L. Moss on coalgebraic logic, we study in a general setting a class of infinitary modal logics for F-coalgebras, designed to capture simulation and bisimulation. We use work by A. Thijs on coalgebraic modelling of simulation, in terms of relators Γ as extensions of functors. We prove our logics can indeed capture simulation and bisimulation, i.e. the existence of a simulation (or bisimulation) is equivalent to the preservation of (or equivalence with respect to) certain classes of sentences. Moreover, we prove that one can characterize any given coalgebra up to simulation (and, in certain conditions, up to bisimulation) by a single sentence. We show that truth for this logic can be understood as a simulation relation itself, but with respect to a richer functor F moreover, it is the the largest simulation, i.e. the similarity relation between states of the coalgebra and elements of the language. This sheds a new perspective on the classical preservation and characterizability results, and also on logic games. The two kinds of games normally used in logic (“truth games” to define the semantics dynamically, and “similarity games” between two structures) are seen to be the same kind of game at the level of coalgebras: simulation games

(Metric) Bisimulation Games and Real-Valued Modal Logics for Coalgebras

Behavioural equivalences can be characterized via bisimulations, modal logics and spoiler-defender games. In this paper we review these three perspectives in a coalgebraic setting, which allows us to generalize from the particular branching type of a transition system. We are interested in qualitative notions (classical bisimulation) as well as quantitative notions (bisimulation metrics). Our first contribution is to introduce a spoiler-defender bisimulation game for coalgebras in the classical case. Second, we introduce such games for the metric case and furthermore define a real-valued modal coalgebraic logic, from which we can derive the strategy of the spoiler. For this logic we show a quantitative version of the Hennessy-Milner theorem

A Logic with Reverse Modalities for History-preserving Bisimulations

We introduce event identifier logic (EIL) which extends Hennessy-Milner logic by the addition of (1) reverse as well as forward modalities, and (2) identifiers to keep track of events. We show that this logic corresponds to hereditary history-preserving (HH) bisimulation equivalence within a particular true-concurrency model, namely stable configuration structures. We furthermore show how natural sublogics of EIL correspond to coarser equivalences. In particular we provide logical characterisations of weak history-preserving (WH) and history-preserving (H) bisimulation. Logics corresponding to HH and H bisimulation have been given previously, but not to WH bisimulation (when autoconcurrency is allowed), as far as we are aware. We also present characteristic formulas which characterise individual structures with respect to history-preserving equivalences.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407

A New Game Equivalence and its Modal Logic

We revisit the crucial issue of natural game equivalences, and semantics of game logics based on these. We present reasons for investigating finer concepts of game equivalence than equality of standard powers, though staying short of modal bisimulation. Concretely, we propose a more finegrained notion of equality of "basic powers" which record what players can force plus what they leave to others to do, a crucial feature of interaction. This notion is closer to game-theoretic strategic form, as we explain in detail, while remaining amenable to logical analysis. We determine the properties of basic powers via a new representation theorem, find a matching "instantial neighborhood game logic", and show how our analysis can be extended to a new game algebra and dynamic game logic.Comment: In Proceedings TARK 2017, arXiv:1707.0825