Abstract
Parametric Interval Markov Chains (pIMCs) are a specification formalism that extend Markov Chains (MCs) and Interval Markov Chains (IMCs) by taking into account imprecision in the transition probability values: transitions in pIMCs are labeled with parametric intervals of probabilities. In this work, we study the difference between pIMCs and other Markov Chain abstractions models and investigate the two usual semantics for IMCs: once-and-for-all and at-every-step. In particular, we prove that both semantics agree on the maximal/minimal reachability probabilities of a given IMC. We then investigate solutions to several parameter synthesis problems in the context of pIMCs – consistency, qualitative reachability and quantitative reachability – that rely on constraint encodings. Finally, we propose a prototype implementation of our constraint encodings with promising results.
This work is partially supported by the ANR national research programs PACS (ANR-14-CE28-0002) and Coverif (ANR-15-CE25-0002), and the regional programme Atlanstic2020 funded by the French Region Pays de la Loire and the European Regional Development Fund.
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Notes
- 1.
Indeed, when \(0 \le v(f_1) \le v(f_2) \le 1\) is not respected, the interval is inconsistent and therefore empty.
- 2.
\(|\mathcal {L}_1|\) and \(|\mathcal {L}_2|\) are the sizes of \(\mathcal {L}_1\) and \(\mathcal {L}_2\), respectively.
- 3.
All resources, benchmarks, and source code are available online as a Python library at https://github.com/anicet-bart/pimc_pylib.
- 4.
Available online at http://www.prismmodelchecker.com.
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Bart, A., Delahaye, B., Lime, D., Monfroy, É., Truchet, C. (2017). Reachability in Parametric Interval Markov Chains Using Constraints. In: Bertrand, N., Bortolussi, L. (eds) Quantitative Evaluation of Systems. QEST 2017. Lecture Notes in Computer Science(), vol 10503. Springer, Cham. https://doi.org/10.1007/978-3-319-66335-7_11
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