Implementation relations and testing for cyclic systems with refusals and discrete time

We present a formalism to represent cyclic models and study different semantic frameworks that support testing. These models combine sequences of observable actions and the passing of (discrete) time and can be used to specify a number of classes of reactive systems, an example being robotic systems. We use implementation relations in order to formally define a notion of correctness of a system under test (SUT) with respect to a specification. As usual, the aim is to devise an extension of the classical ioco implementation relation but available timed variants of ioco are not suitable for cyclic models. This paper thus defines new implementation relations that encapsulate the discrete nature of time and take into account not only the actions that models can perform but also the ones that they can refuse. In addition to defining these relations, we study a number of their properties and provide alternative characterisations, showing that the relations are appropriate conservative extensions of trace containment. Finally, we give test derivation algorithms and prove that they are sound and also are complete in the limit.