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Sound statistical model checking for MDP using partial order and confluence reduction

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Abstract

Statistical model checking (SMC) is an analysis method that circumvents the state space explosion problem in model-based verification by combining probabilistic simulation with statistical methods that provide clear error bounds. As a simulation-based technique, it can in general only provide sound results if the underlying model is a stochastic process. In verification, however, models are very often variations of nondeterministic transition systems. In classical exhaustive model checking, partial order reduction and confluence reduction allow the removal of spurious nondeterministic choices. In this paper, we show that both can be adapted to detect and discard such choices on-the-fly during simulation, thus extending the applicability of SMC to a subclass of Markov decision processes. We prove their correctness in a uniform way and study their effectiveness and efficiency using an implementation in the modes simulator for the Modest modelling language. The examples we use highlight the different strengths and limitations of the two approaches. We find that runtime may be affected by unnecessary recomputations, and thus also investigate the feasibility of caching results to speed up simulation at the cost of increased memory usage.

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Notes

  1. Definitions 11, 12, 13, and 14 are all based on [5].

  2. We omit the \(\tau \)-loops that need to be added to deadlock states for brevity from now on.

    Fig. 3
    figure 3

    VMDP modelling hosts (\(H\)) that send on a shared medium (\(M\))

    Full size image

  3. We mostly cite [3] in the remainder of this section as it nicely summarises both linear-time approaches presented before [4, 12] in addition to introducing an extension to \(\text {PCTL*}_{\setminus \text {X}}\).

  4. By abuse of language, we use the word “transition” when we actually mean “equivalence class of transitions under \(\equiv \)”.

  5. Our implementation in modes therefore uses large default values for \(k_{\mathrm{max}}\) and \(l\) so the user usually need not worry about these parameters. If SMC aborts, the cause and its location is reported, including how it was detected, which may be that \(k_{\mathrm{max}}\) or \(l\) was exceeded.

  6. For the CSMA/CD models, the state space sizes reported are for the digital clocks semantics of the PTA created by mcpta and thus not directly comparable to the state spaces modes works on.

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Acknowledgments

This work was supported by the Transregional Collaborative Research Centre SFB/TR 14 AVACS, by the NWO-DFG bilateral project ROCKS, by the 7th EU Framework Programme under grant agreements 295261 (MEALS) and 318490 (SENSATION), by NWO under grant 612.063.817 (SYRUP), and by the CAS/SAFEA International Partnership Program for Creative Research Teams.

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  1. Saarland University, Saarbrücken, Germany

    Arnd Hartmanns

  2. University of Twente, Enschede, The Netherlands

    Mark Timmer

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Hartmanns, A., Timmer, M. Sound statistical model checking for MDP using partial order and confluence reduction. Int J Softw Tools Technol Transfer 17, 429–456 (2015). https://doi.org/10.1007/s10009-014-0349-7

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