Divergent quiescent transition systems

Quiescence is a fundamental concept in modelling system behaviour, as it explicitly represents the fact that no output is produced in certain states. The notion of quiescence is also essential to model-based testing: if a particular implementation under test does not provide any output, then the test evaluation algorithm must decide whether or not to allow this behaviour. To explicitly model quiescence in all its glory, we introduce Divergent Quiescent Transition Systems (DQTSs). DQTSs model quiescence using explicit delta-labelled transitions, analogous to Suspension Automata (SAs) in the well-known ioco framework. Whereas SAs have only been defined implicitly, DQTSs for the first time provide a fully-formalised framework for quiescence. Also, while SAs are restricted to convergent systems (i.e., without tau-cycles), we show how quiescence can be treated naturally using a notion of fairness, allowing systems exhibiting divergence to be modelled as well. We study compositionality under the familiar automata-theoretical operations of determinisation, parallel composition and action hiding. We provide a non-trivial algorithm for detecting divergent states, and discuss its complexity. Finally, we show how to use DQTSs in the context of model-based testing, for the first time presenting a full-fledged theory that allows ioco to be applied to divergent systems.