Abstract
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.
This research has been partially funded by NWO under grants 612.063.817 (SYRUP), Dn 63-257 (ROCKS) and 12238 (ArRangeer), and by the EU under grant 318490 (SENSATION).
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Notes
- 1.
If \(L^{\mathrm {I}}\mathrel {\cup }L_{\delta }^\mathrm{O}\) is finite, we can replace this requirement by asking that t is finite.
- 2.
Technically, parallel composition was only defined for QIOTSs, and test cases are no QIOTSs. However, the idea can easily be lifted. Moreover, the actual formal definition of the execution of a test case below circumvents this issue by directly defining the results of the parallel composition.
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Acknowledgements
We would like to thank the reviewers for their thorough comments that really helped improve this paper. We thank Gerjan Stokkink for his large contributions to work that provides important ingredients for this paper [2, 14,15,16].
Since this paper is a part of the Festschrift at the occasion of the 60th birthday of Kim Guldstrand Larsen, we like to thank Kim for the many exciting and fruitful discussions we have had, and still have, over all these years in project meetings, at conferences and many other occasions—Kim was never quiescent.
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Brinksma, E., Stoelinga, M.I.A., Timmer, M. (2017). Testing Divergent Transition Systems. In: Aceto, L., Bacci, G., Bacci, G., Ingólfsdóttir, A., Legay, A., Mardare, R. (eds) Models, Algorithms, Logics and Tools. Lecture Notes in Computer Science(), vol 10460. Springer, Cham. https://doi.org/10.1007/978-3-319-63121-9_17
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