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Using Symmetry Transformations in Equivariant Dynamical Systems for Their Safety Verification

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Automated Technology for Verification and Analysis (ATVA 2019)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 11781))

Abstract

In this paper, we investigate how symmetry transformations of equivariant dynamical systems can reduce the computation effort for safety verification. Symmetry transformations of equivariant systems map solutions to other solutions. We build upon this result, producing reachsets from other previously computed reachsets. We augment the standard simulation-based verification algorithm with a new procedure that attempts to verify the safety of the system starting from a new initial set of states by transforming previously computed reachsets. This new algorithm required the creation of a new cache-tree data structure for multi-resolution reachtubes. Our implementation has been tested on several benchmarks and has achieved significant improvements in verification time.

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Notes

  1. 1.

    http://www.cas.mcmaster.ca/~nedialk/vnodelp/.

  2. 2.

    http://capd.sourceforge.net/capdDynSys/docs/html/odes_rigorous.html.

  3. 3.

    http://www.scholarpedia.org/article/Equivariant_dynamical_systems.

  4. 4.

    https://github.com/qibolun/DryVR_0.2.

References

  1. Annpureddy, Y., Liu, C., Fainekos, G., Sankaranarayanan, S.: S-TaLiRo: a tool for temporal logic falsification for hybrid systems. In: Abdulla, P.A., Leino, K.R.M. (eds.) TACAS 2011. LNCS, vol. 6605, pp. 254–257. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19835-9_21

    Chapter  MATH  Google Scholar 

  2. Antuña, L., Araiza-Illan, D., Campos, S., Eder, K.: Symmetry reduction enables model checking of more complex emergent behaviours of swarm navigation algorithms. In: Dixon, C., Tuyls, K. (eds.) TAROS 2015. LNCS (LNAI), vol. 9287, pp. 26–37. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-22416-9_4

    Chapter  Google Scholar 

  3. Clarke, E.M., Jha, S.: Symmetry and induction in model checking. In: Computer Science Today: Recent Trends and Developments, pp. 455–470 (1995)

    Chapter  Google Scholar 

  4. Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955)

    MATH  Google Scholar 

  5. Donzé, A.: Breach, a toolbox for verification and parameter synthesis of hybrid systems. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 167–170. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14295-6_17

    Chapter  Google Scholar 

  6. Duggirala, P.S., Mitra, S., Viswanathan, M.: Verification of annotated models from executions. In: EMSOFT (2013)

    Google Scholar 

  7. Emerson, E.A., Sistla, A.P.: Symmetry and model checking. In: Computer Aided Verification, 5th International Conference, CAV 1993, Elounda, Greece, Proceedings, 28 June–1 July 1993, pp. 463–478 (1993)

    Chapter  Google Scholar 

  8. Fan, C., Mitra, S.: Bounded verification with on-the-fly discrepancy computation. In: Finkbeiner, B., Pu, G., Zhang, L. (eds.) ATVA 2015. LNCS, vol. 9364, pp. 446–463. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24953-7_32

    Chapter  MATH  Google Scholar 

  9. Fan, C., Qi, B., Mitra, S.: Data-driven formal reasoning and their applications in safety analysis of vehicle autonomy features. IEEE Design Test 35(3), 31–38 (2018)

    Article  Google Scholar 

  10. Fan, C., Qi, B., Mitra, S., Viswanathan, M.: DryVR: data-driven verification and compositional reasoning for automotive systems. In: Majumdar, R., Kunčak, V. (eds.) CAV 2017. LNCS, vol. 10426, pp. 441–461. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-63387-9_22

    Chapter  Google Scholar 

  11. Fan, C., Qi, B., Mitra, S., Viswanathan, M., Duggirala, P.S.: Automatic reachability analysis for nonlinear hybrid models with C2E2. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016. LNCS, vol. 9779, pp. 531–538. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-41528-4_29

    Chapter  Google Scholar 

  12. Freund, P.G.O.: Introduction to Supersymmetry. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge (1986)

    Book  Google Scholar 

  13. Gérard, L., Slotine, J.J.: Neuronal networks and controlled symmetries, a generic framework. arXiv preprint q-bio/0612049 (2006)

    Google Scholar 

  14. Golubitsky, M., Stewart, I.: The Symmetry Perspective: From Equilibrium to Chaos in Phase Space and Physical Space. Progress in Mathematics, Birkhäuser Basel (2012)

    Google Scholar 

  15. Golubitsky, M., Stewart, I., Török, A.: Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4(1), 78–100 (2005)

    Article  MathSciNet  Google Scholar 

  16. Huang, Z., Fan, C., Mereacre, A., Mitra, S., Kwiatkowska, M.: Invariant verification of nonlinear hybrid automata networks of cardiac cells. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 373–390. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-08867-9_25

    Chapter  Google Scholar 

  17. Ip, C.N., Dill, D.L.: Better verification through symmetry. In: Proceedings of the 11th IFIP WG10.2 International Conference Sponsored by IFIP WG10.2 and in Cooperation with IEEE COMPSOC on Computer Hardware Description Languages and Their Applications, CHDL 1993, pp. 97–111. North-Holland Publishing Co., Amsterdam, The Netherlands (1993)

    Google Scholar 

  18. Koopman, P., Wagner, M.: Challenges in autonomous vehicle testing and validation. SAE Int. J. Transp. Saf. 4(2016–01–0128), 15–24 (2016)

    Article  Google Scholar 

  19. Kwiatkowska, M.Z., Norman, G., Parker, D.: Symmetry reduction for probabilistic model checking. In: 18th International Conference on Computer Aided Verification, CAV 2006, Seattle, WA, USA, 17–20 August 2006, Proceedings, pp. 234–248 (2006)

    Google Scholar 

  20. Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems. Springer, New York (2010). https://doi.org/10.1007/978-0-387-21792-5

    Book  MATH  Google Scholar 

  21. Mehta, P.G., Hagen, G., Banaszuk, A.: Symmetry and symmetry-breaking for a wave equation with feedback. SIAM J. Appl. Dyn. Syst. 6(3), 549–575 (2007)

    Article  MathSciNet  Google Scholar 

  22. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986). https://doi.org/10.1007/978-1-4684-0274-2

    Book  MATH  Google Scholar 

  23. Pham, Q.C., Slotine, J.J.: Stable concurrent synchronization in dynamic system networks. Neural Netw. 20(1), 62–77 (2007)

    Article  Google Scholar 

  24. Prabhakar, P., Duggirala, P.S., Mitra, S., Viswanathan, M.: Hybrid automata-based cegar for rectangular hybrid systems. Form. Methods Syst. Des. 46(2), 105–134 (2015)

    Article  Google Scholar 

  25. Russo, G., Slotine, J.J.E.: Symmetries, stability, and control in nonlinear systems and networks. Phys. Rev. E 84(4), 041929 (2011)

    Article  Google Scholar 

  26. Spong, M.W., Bullo, F.: Controlled symmetries and passive walking. IEEE Trans. Autom. Control 50(7), 1025–1031 (2005)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

The authors are supported by a research grant from The Boeing Company and a research grant from NSF (CPS 1739966). We would like to thank John L. Olson and Arthur S. Younger from The Boeing Company for valuable technical discussions.

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Authors and Affiliations

  1. University of Illinois, Urbana, IL, 61801, USA

    Hussein Sibai, Navid Mokhlesi & Sayan Mitra

Authors
  1. Hussein Sibai
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  2. Navid Mokhlesi
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  3. Sayan Mitra
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Corresponding author

Correspondence to Hussein Sibai .

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Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, Taipei, Taiwan

    Yu-Fang Chen

  2. DENSO AUTOMOTIVE Deutschland GmbH, Eching, Germany

    Chih-Hong Cheng

  3. Institute of Computer Science, TU München, Munich, Germany

    Javier Esparza

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Sibai, H., Mokhlesi, N., Mitra, S. (2019). Using Symmetry Transformations in Equivariant Dynamical Systems for Their Safety Verification. In: Chen, YF., Cheng, CH., Esparza, J. (eds) Automated Technology for Verification and Analysis. ATVA 2019. Lecture Notes in Computer Science(), vol 11781. Springer, Cham. https://doi.org/10.1007/978-3-030-31784-3_6

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  • DOI: https://doi.org/10.1007/978-3-030-31784-3_6

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-31783-6

  • Online ISBN: 978-3-030-31784-3

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