Testing Divergent Transition Systems

We revisit model-based testing for labelled transition systems in the context of specifications that may contain divergent behaviour, i.e., infinite paths of internal computations. The standard approach based on the theory of input-output conformance, known as the ioco-framework, cannot deal with divergences directly, as it restricts specifications to strongly convergent transition systems. Using the model of Quiescent Input Output Transition Systems (QIOTSs), we can handle divergence successfully in the context of quiescence. Quiescence is a fundamental notion that represents the situation that a system is not capable of producing any output, if no prior input is provided, representing lack of productive progress. The correct treatment of this situation is the cornerstone of the success of testing in the context of systems that are input-enabled, i.e., systems that accept all input actions in any state. Our revised treatment of quiescence also allows it to be preserved under determinization of a QIOTS. This last feature allows us to reformulate the standard ioco-based testing theory and algorithms in terms of classical trace-based automata theory, including finite state divergent computations.