Abstract 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.

中文翻译：

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具有拒绝和离散时间的循环系统的实现关系和测试

摘要 我们提出了一种形式来表示循环模型并研究支持测试的不同语义框架。这些模型结合了可观察到的动作序列和（离散）时间的流逝，可用于指定多种类型的反应系统，例如机器人系统。我们使用实现关系来正式定义被测系统 (SUT) 相对于规范的正确性概念。像往常一样，目标是设计经典 ioco 实现关系的扩展，但 ioco 的可用定时变体不适用于循环模型。因此，本文定义了新的实现关系，这些关系封装了时间的离散性质，不仅考虑了模型可以执行的操作，还考虑了它们可以拒绝的操作。除了定义这些关系之外，我们还研究了它们的许多特性并提供了替代特征，表明这些关系是痕量遏制的适当保守扩展。最后，我们给出了测试推导算法，并证明它们是合理的，并且在极限上也是完备的。