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Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods

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Abstract

Aiming at automatic verification and analysis techniques for hybrid discrete-continuous systems, we present a novel combination of enclosure methods for ordinary differential equations (ODEs) with the iSAT solver for large Boolean combinations of arithmetic constraints. Improving on our previous work, the contribution of this paper lies in combining iSAT with VNODE-LP, as a state-of-the-art interval solver for ODEs, and with bracketing systems, which exploit monotonicity properties allowing to find enclosures for problems that VNODE-LP alone cannot enclose tightly. We apply the combined iSAT-ODE solver to the analysis of a variety of non-linear hybrid systems by solving predicative encodings of reachability properties and of an inductive stability argument, and evaluate the impact of the different enclosure methods, decision heuristics and their combination. Our experiments include classic benchmarks from the literature, as well as a newly-designed conveyor belt system that combines hybrid behavior of parallel components, a slip-stick friction model with non-linear dynamics and flow invariants and several dimensions of parameterization. In the paper, we also present and evaluate an extension of VNODE-LP tailored to its use as a deduction mechanism within iSAT-ODE, to allow fast re-evaluations of enclosures over arbitrary subranges of the analyzed time span.

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Notes

  1. For simplicity, the valuation of a vector shall be the vector of its valuations.

  2. Models and raw results from [6]: http://www.avacs.org/fileadmin/Benchmarks/Open/iSAT_ODE_SEFM_2011_models.tar.gz. Updated models and raw results: http://www.avacs.org/fileadmin/Benchmarks/Open/iSAT_ODE_SoSyM_2012_models.tar.gz.

  3. Note that we found this trace when validating our encoding of the original model with iSAT-ODE and were surprised to find this obviously unintended trajectory which is compatible with this often assumed semantics of hybrid systems (e.g., in [14]).

  4. Note that in this special case where the solution consists of only one flow, using just the ODE enclosure and showing that all points from its prebox satisfy the initial condition and all points from the last enclosure lie within the unsafe region, would yield an equally strong proof.

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Acknowledgments

We would like to thank Stefan Ratschan, Christian Herde, Tino Teige, Jens Oehlerking, and Corina Mitrohin for discussions on the region-stability-related proof scheme utilized for the experiments in this paper and all colleagues from the transregional research center AVACS, project H1/2 “Constraint-based Verification for Hybrid Systems” for the joint development of the iSAT core. Additionally, we are grateful to the reviewers of [6] for their detailed comments. Especially by insisting on a more thorough experimental evaluation and by pointing out shortcomings in our presentation, the SoSyM reviewers have helped tremendously to improve the quality of this paper. Thank you!

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Authors and Affiliations

  1. Department of Computing Science, Carl von Ossietzky Universität, 26111 , Oldenburg, Germany

    Andreas Eggers & Martin Fränzle

  2. Université d’Orléans, PRISME, 63 av. de Lattre de Tassigny, 18020 , Bourges, France

    Nacim Ramdani

  3. McMaster University, Hamilton, ON, Canada

    Nedialko S. Nedialkov

Authors
  1. Andreas Eggers
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  2. Nacim Ramdani
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  3. Nedialko S. Nedialkov
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  4. Martin Fränzle
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Corresponding author

Correspondence to Andreas Eggers.

Additional information

A preliminary version of this paper appeared in  [6]. This work has been supported by the German Research Council DFG within SFB/TR 14 “http://www.avacs.org”, by the French National Research Agency under contract ANR 2011 INS 006 04 “http://projects.laas.fr/ANR-MAGIC-SPS”, and by the Natural Sciences and Engineering Research Council of Canada.

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Eggers, A., Ramdani, N., Nedialkov, N.S. et al. Improving the SAT modulo ODE approach to hybrid systems analysis by combining different enclosure methods. Softw Syst Model 14, 121–148 (2015). https://doi.org/10.1007/s10270-012-0295-3

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