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Bandwidth and Wavefront Reduction for Static Variable Ordering in Symbolic Reachability Analysis

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NASA Formal Methods (NFM 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9690))

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Abstract

We investigate the use of bandwidth and wavefront reduction algorithms to determine a static BDD variable ordering. The aim is to reduce the size of BDDs arising in symbolic reachability. Previous work showed that minimizing the (weighted) event span of the variable dependency graph yields small BDDs. The bandwidth and wavefront of symmetric matrices are well studied metrics, used in sparse matrix solvers, and many bandwidth and wavefront reduction algorithms are readily available in libraries like Boost and ViennaCL.

In this paper, we transform the dependency matrix to a symmetric matrix and apply various bandwidth and wavefront reduction algorithms, measuring their influence on the (weighted) event span. We show that Sloan’s algorithm, executed on the total graph of the dependency matrix, yields a variable order with minimal event span. We demonstrate this on a large benchmark of Petri nets, Dve, Promela, B, and mcrl2 models. As a result, good static variable orders can now be determined in milliseconds by using standard sparse matrix solvers.

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Notes

  1. 1.

    We restrict ourselves to languages that induce a disjunctive transition relation.

  2. 2.

    Theorem 1 can be easily proven with the triangle inequality theorem.

  3. 3.

    Reproduction instructions at: https://github.com/utwente-fmt/BW-NFM2016

  4. 4.

    There are three side notes. First, \(\mu \) and \(\sigma \) for bandwidth, profile and wavefront are computed per graph type, because the bipartite and total graph have different sizes. Second, Noack1 and Noack2 can only be computed directly on Petri nets (PNML, Fig. 9e), so bandwidth, profile and wavefront are unknown. Third, when FORCE is executed or without reordering, bandwidth, profile and wavefront are not reported. The reason is that our symmetrization approach typically produces high values for those metrics. Event span does not have this problem.

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Authors and Affiliations

  1. Formal Methods and Tools, University of Twente, Enschede, The Netherlands

    Jeroen Meijer & Jaco van de Pol

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  1. Jeroen Meijer
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  2. Jaco van de Pol
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Correspondence to Jeroen Meijer .

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Editors and Affiliations

  1. University of Minnesota, Minneapolis, Minnesota, USA

    Sanjai Rayadurgam

  2. NASA Ames Research Center , Moffett Field, California, USA

    Oksana Tkachuk

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Meijer, J., van de Pol, J. (2016). Bandwidth and Wavefront Reduction for Static Variable Ordering in Symbolic Reachability Analysis. In: Rayadurgam, S., Tkachuk, O. (eds) NASA Formal Methods. NFM 2016. Lecture Notes in Computer Science(), vol 9690. Springer, Cham. https://doi.org/10.1007/978-3-319-40648-0_20

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  • DOI: https://doi.org/10.1007/978-3-319-40648-0_20

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