Simulations and Bisimulations for Coalgebraic Modal Logics
Simulations serve as a proof tool to compare the behaviour of reactive systems. We define a notion of Λ-simulation for coalgebraic modal logics, parametric in the choice of a set Λ of monotone predicate liftings for a functor T. That is, we obtain a generic notion of simulation that can be flexibly instantiated to a large variety of systems and logics, in particular in settings that semantically go beyond the classical relational setup, such as probabilistic, game-based, or neighbourhood-based systems. We show that this notion is adequate in several ways: i) Λ-simulations preserve truth of positive formulas, ii) for Λ a separating set of monotone predicate liftings, the associated notion of Λ-bisimulation corresponds to T-behavioural equivalence (moreover, this correspondence extends to the respective finite-lookahead counterparts), and iii) Λ-bisimulations remain sound when taken up to difunctional closure. In essence, we arrive at a modular notion of equivalence that, when used with a separating set of monotone predicate liftings, coincides with T-behavioural equivalence regardless of whether T preserves weak pullbacks. That is, for finitary set-based coalgebras, Λ-bisimulation works under strictly more general assumptions than T-bisimulation in the sense of Aczel and Mendler.
KeywordsModal Logic Kripke Frame Positive Formula Coalgebra Structure Monotone Boolean Function
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- 2.Enqvist, S.: Homomorphisms of coalgebras from predicate liftings. In: Heckel, R., Milius, S. (eds.) CALCO 2013. LNCS, vol. 8089, pp. 126–140. Springer, Heidelberg (2013)Google Scholar
- 3.Hansen, H., Kupke, C.: A coalgebraic perspective on monotone modal logic. In: Coalgebraic Methods in Computer Science (CMCS 2004). ENTCS, vol. 106, pp. 121–143. Elsevier (2004)Google Scholar
- 11.Pauly, M.: Bisimulation for general non-normal modal logic (1999) (unpublished Manuscript)Google Scholar
- 12.Pauly, M.: Logic for social software. Ph.D. thesis, Universiteit van Amsterdam (2001)Google Scholar