Truth-as-Simulation: Towards a Coalgebraic Perspective on Logic and Games

Building on the work of L. Moss on coalgebraic logic, I study in a general setting a class of infinitary modal logics for F -coalgebras, designed to capture simulation and bisimulation. For a notion of coalgebraic simulation, I use the work of A. Thijs on modelling simulation in terms of relators Γ (extensions of SET -functors along some family of preoders): simulation is the analogue for relators of the notion of bisimulation for functors. I prove the logics introduced here can indeed capture coalgebraic simulation and bisimulation. Moreover, one can characterize any given coalgebra up to simulation (and, in certain conditions, up to bisimulation) by a single sentence. An interesting feature of this logics is that their notion of truth or satisfaction can be understood as a simulation relation itself, but with respect to a (relator associated to a) richer functor F ; moreover, truth 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. 1991 Mathematics Subject Classification: 03G30, 03G99, 03B45, 03B70, 03C05, 18A15 1991 ACM Computing Classification System: F.4.1, F.3.2, I.6.1