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Formalisation of Probabilistic Testing Semantics in Coq

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The Art of Modelling Computational Systems: A Journey from Logic and Concurrency to Security and Privacy

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11760))

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Abstract

Van Breugel et al. [Theor. Comput. Sci. 333(1–2):171–197, 2005] have given an elegant testing framework that can characterise probabilistic bisimulation, but its completeness proof is highly involved. Deng and Feng [Inf. Comput. 257:58–64, 2017] have simplified that result for finite-state processes. The crucial part in the latter work is an algorithm that can construct enhanced tests. We formalise the algorithm and prove its correctness by maintaining a number of subtle invariants in Coq. To support the formalisation, we develop a reusable library for manipulating finite sets. This sets an early example of formalising probabilistic concurrency theory or quantitative aspects of concurrency theory at large, which is a rich field to be pursued.

Supported by the National Natural Science Foundation of China (61672229, 61832015), the French national research organization ANR (grant ANR-15-CE25-0008), and the Inria-CAS joint project Quasar.

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References

  1. https://softwarefoundations.cis.upenn.edu/lf-current/index.html

  2. https://coq.inria.fr

  3. Audebaud, P., Paulin-Mohring, C.: Proofs of randomized algorithms in Coq. Sci. Comput. Program. 74(8), 568–589 (2009)

    Article  MathSciNet  Google Scholar 

  4. Barthe, G., Crespo, J.M., Grégoire, B., Kunz, C., Zanella Béguelin, S.: Computer-aided cryptographic proofs. In: Beringer, L., Felty, A. (eds.) ITP 2012. LNCS, vol. 7406, pp. 11–27. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-32347-8_2

    Chapter  Google Scholar 

  5. Bertot, Y., Castéran, P.: Interactive Theorem Proving and Program Development. Coq’Art: The Calculus of Inductive Constructions. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-662-07964-5

    Book  MATH  Google Scholar 

  6. Deng, Y.: Semantics of Probabilistic Processes: An Operational Approach. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-662-45198-4

    Book  Google Scholar 

  7. Deng, Y., Feng, Y.: Probabilistic bisimilarity as testing equivalence. Inf. Comput. 257, 58–64 (2017)

    Article  MathSciNet  Google Scholar 

  8. Deng, Y., van Glabbeek, R.: Characterising probabilistic processes logically. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR 2010. LNCS, vol. 6397, pp. 278–293. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-16242-8_20

    Chapter  Google Scholar 

  9. Deng, Y., van Glabbeek, R., Hennessy, M., Morgan, C.: Testing finitary probabilistic processes. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 274–288. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-04081-8_19

    Chapter  Google Scholar 

  10. Deng, Y., Wu, H.: Modal characterisations of probabilistic and fuzzy bisimulations. In: Merz, S., Pang, J. (eds.) ICFEM 2014. LNCS, vol. 8829, pp. 123–138. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11737-9_9

    Chapter  Google Scholar 

  11. Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002)

    Article  MathSciNet  Google Scholar 

  12. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled Markov processes. Inf. Comput. 184(1), 160–200 (2003)

    Article  MathSciNet  Google Scholar 

  13. Gonthier, G.: Formal proof – the four-color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)

    MathSciNet  MATH  Google Scholar 

  14. Gonthier, G., et al.: A machine-checked proof of the odd order theorem. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 163–179. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39634-2_14

    Chapter  Google Scholar 

  15. Gu, R., Shao, Z., Chen, H., Wu, X.N., Kim, J., Sjöberg, V., Costanzo, D.: CertiKOS: an extensible architecture for building certified concurrent OS kernels. In: Proceedings of OSDI 2016, pp. 653–669. USENIX Association (2016)

    Google Scholar 

  16. Hennessy, M.: Exploring probabilistic bisimulations, Part I. Formal Aspects Comput. 24(4–6), 749–768 (2012)

    Article  MathSciNet  Google Scholar 

  17. Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985)

    Article  MathSciNet  Google Scholar 

  18. Hermanns, H., Parma, A., Segala, R., Wachter, B., Zhang, L.: Probabilistic logical characterization. Inf. Comput. 209(2), 154–172 (2011)

    Article  MathSciNet  Google Scholar 

  19. Jones, C.: Probabilistic nondeterminism. Ph.d. thesis, University of Edinburgh (1990)

    Google Scholar 

  20. Kennedy, A., Benton, N., Jensen, J.B., Dagand, P.: Coq: the world’s best macro assembler? In: Proceedings of PPDP 2013, pp. 13–24. ACM (2013)

    Google Scholar 

  21. Krebbers, R.: The C standard formalized in Coq. Ph.d. thesis, Radboud University Nijmegen (2015)

    Google Scholar 

  22. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94, 1–28 (1991)

    Article  MathSciNet  Google Scholar 

  23. Leroy, X., Blazy, S., Kästner, D., Schommer, B., Pister, M., Ferdinand, C.: CompCert - a formally verified optimizing compiler. In: Proceedings of the 8th European Congress on Embedded Real Time Software and Systems. SEE (2016). https://hal.inria.fr/hal-01238879

  24. Milner, R.: Communication and Concurrency. Prentice Hall, Upper Saddle River (1989)

    MATH  Google Scholar 

  25. Park, D.: Concurrency and automata on infinite sequences. In: Deussen, P. (ed.) GI-TCS 1981. LNCS, vol. 104, pp. 167–183. Springer, Heidelberg (1981). https://doi.org/10.1007/BFb0017309

    Chapter  Google Scholar 

  26. Parma, A., Segala, R.: Logical characterizations of bisimulations for discrete probabilistic systems. In: Seidl, H. (ed.) FoSSaCS 2007. LNCS, vol. 4423, pp. 287–301. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71389-0_21

    Chapter  MATH  Google Scholar 

  27. van Breugel, F., Mislove, M.W., Ouaknine, J., Worrell, J.: Domain theory, testing and simulation for labelled Markov processes. Theoret. Comput. Sci. 333(1–2), 171–197 (2005)

    Article  MathSciNet  Google Scholar 

  28. van Glabbeek, R.J., Smolka, S.A., Steffen, B., Tofts, C.M.N.: Reactive, generative, and stratified models of probabilistic processes. In: Proceedings of LICS 1990, pp. 130–141. IEEE Computer Society (1990)

    Google Scholar 

  29. Zhao, J., Nagarakatte, S., Martin, M.M.K., Zdancewic, S.: Formalizing the LLVM intermediate representation for verified program transformations. In: Proceedings of POPL 2012, pp. 427–440. ACM (2012)

    Google Scholar 

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Acknowledgment

We would like to thank Yves Bertot for helpful discussion.

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Authors and Affiliations

  1. Shanghai Key Laboratory of Trustworthy Computing, MOE International Joint Lab of Trustworthy Software, East China Normal University, Shanghai, China

    Yuxin Deng

  2. Univ. Grenoble Alpes, CNRS, Grenoble INP, VERIMAG, 38000, Grenoble, France

    Jean-Francois Monin

Authors
  1. Yuxin Deng
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  2. Jean-Francois Monin
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Corresponding author

Correspondence to Yuxin Deng .

Editor information

Editors and Affiliations

  1. Universidade Federal de Minas Gerais, Belo Horizonte, Brazil

    Mário S. Alvim

  2. University of Athens, Athens, Greece

    Kostas Chatzikokolakis

  3. Federal University of Rio Grande do Norte - UFRN, Natal, Brazil

    Carlos Olarte

  4. CNRS & University Javeriana Cali, Palaiseau, France

    Frank Valencia

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Deng, Y., Monin, JF. (2019). Formalisation of Probabilistic Testing Semantics in Coq. In: Alvim, M., Chatzikokolakis, K., Olarte, C., Valencia, F. (eds) The Art of Modelling Computational Systems: A Journey from Logic and Concurrency to Security and Privacy. Lecture Notes in Computer Science(), vol 11760. Springer, Cham. https://doi.org/10.1007/978-3-030-31175-9_16

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  • DOI: https://doi.org/10.1007/978-3-030-31175-9_16

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-31174-2

  • Online ISBN: 978-3-030-31175-9

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