An implementation relation for cyclic systems with refusals and discrete time

This paper explores a particular type of model, a cyclic model, in which there are sequences of observable actions separated by discrete time intervals, introduces a novel implementation relation and studies some properties of this relation. Implementation relations formalise what it means for an unknown model of the system under test (SUT) to be a correct implementation of a specification. Many implementation relations are variants of the well known ioco implementation relation, and this includes several timed versions of ioco. It transpires that the timed variants of ioco are not suitable for cyclic models. Our implementation relation encapsulates the discrete nature of time in cyclic models and takes into account not only the actions that models can perform but also the ones that they can refuse at each point of time. We prove that our implementation relation is a conservative extension of trace containment and present two alternative characterisations.