Congruences and Bisimulations for Continuous-Time Stochastic Logic

Continuous stochastic logic (CSL) deals with the verification of systems operating in continuous time, it may be traced to the well known tree logic CTL. The present logic contains a nextas well as an until-operator, and it permits expressing steady-state probabilities. We propose in this paper a probabilistic interpretation of this logic that is based on stochastic relations without making specific assumptions on the underlying distribution: usually some kind of exponential distribution based on a rate function is imposed, but these assumptions are shown to be unnecessarily restrictive. We study the problem of bisimulations in this fairly general context from the viewpoint of congruences for stochastic relations with the goal of finding minimal sets of formulas that permit efficient checking of models. The main contribution is to propose a general context for the interpretation of the logic including the steady-state operator, and to shed some light on the intricate nature of bisimulations. Continuous-time stochastic logic