Yes / We present a formalism to represent cyclic models and study di erent 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. / Engineering and Physical Sciences Research Council Grant numbers: EP/R025134/2. Ministerio de Economía y Competitividad Grant numbers: RTI2018-093608-B-C31. Comunidad de Madrid Grant numbers: S2018/TCS-4314

Yes / This paper concerns the systematic testing of robotic control software based on state-based models. We focus on cyclic systems that typically receive inputs (values from sensors), perform computations, produce outputs (sent to actuators) and possibly change state. We provide a testing theory for such cyclic systems where time can be represented and probabilities are used to quantify non-deterministic choices, making it possible to model probabilistic algorithms. In addition, refusals, the inability of a system to perform a set of actions, are taken into account. We consider several possible testing scenarios. For example, a tester might only be able to passively observe a sequence of events and so cannot check probabilities, while in another scenario a tester might be able to repeatedly apply a test case and so estimate the probabilities of sequences of events. These different testing scenarios lead to a range of implementation relations (notions of correctness). As a consequence, this paper provides formal definitions of implementation relations that can form the basis of sound automated testing in a range of testing scenarios. We also validate the implementation relations by showing how observers can be used to provide an alternative but equivalent characterisation. / This work has been supported by EPSRC, United Kingdom grant EP/R025134/2 RoboTest: Systematic Model-Based Testing and Simulation of Mobile Autonomous Robots, the Spanish MINECO-FEDER grant PID2021- 122215NB-C31 (AwESOMe) and the Region of Madrid grant S2018/TCS-4314 (FORTE-CM) co-funded by EIE Funds of the European Union.