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Behavioural Pseudometrics for Nondeterministic Probabilistic Systems

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Book cover Dependable Software Engineering: Theories, Tools, and Applications (SETTA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNPSE,volume 9984))

Abstract

For the model of probabilistic labelled transition systems that allow for the co-existence of nondeterminism and probabilities, we present two notions of bisimulation metrics: one is state-based and the other is distribution-based. We provide a sound and complete modal characterisation for each of them, using real-valued modal logics based on Hennessy-Milner logic. The logic for characterising the state-based metric is much simpler than an earlier logic by Desharnais et al. as it uses only two non-expansive operators rather than the general class of non-expansive operators. For the kernels of the two metrics, which correspond to two notions of bisimilarity, we give a comprehensive comparison with some typical distribution-based bisimilarities in the literature.

Y. Deng—Partially supported by the National Natural Science Foundation of China (61672229, 61261130589), Shanghai Municipal Natural Science Foundation (16ZR1409100), and ANR 12IS02001 PACE.

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Notes

  1. 1.

    Notice that we do not claim that negation and testing operators, plus some constant functions, suffice to represent all the non-expansive operators on the unit interval. That claim is too strong to be true. For example, the operator \(f(x)=\frac{1}{2}x\) cannot be represented by those operators.

  2. 2.

    Since we will compare our logic with that in [17], it is better for our semantic interpretation to be consistent with that in the aforementioned work. In the literature, there are also other ways of interpreting conjunction and disjunction in probabilistic settings, see e.g. [3, 29].

  3. 3.

    Although \(\mathbf {d}_{ d}\) can measure the distance between two subdistributions, the Kantorovich lifting of \(\mathbf {d}_{ s }\) can only measure the distance between full distributions or subdistributions of equal mass, which can easily be normalized to full distributions.

References

  1. Castiglioni, V., Gebler, D., Tini, S.: Logical characterization of bisimulation metrics. In: Proceedings of QAPL 2016. EPTCS (2016)

    Google Scholar 

  2. Chatterjee, K., De Alfaro, L., Majumdar, R., Raman, V.: Algorithms for game metrics. Log. Methods Comput. Sci. 6(3:13), 1–27 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Cleaveland, R., Iyer, S.P., Narasimha, M.: Probabilistic temporal logics via the modal mu-calculus. Theor. Comput. Sci. 342(2–3), 316–350 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. D’Argenio, P.R., Sánchez Terraf, P., Wolovick, N.: Bisimulations for non-deterministic labelled Markov processes. Math. Struct. Comput. Sci. 22(1), 43–68 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. De Alfaro, L., Faella, M., Stoelinga, M.: Linear and branching system metrics. IEEE Trans. Softw. Eng. 35(2), 258–273 (2009)

    Article  MATH  Google Scholar 

  6. De Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game relations and metrics. In: Proceedings of LICS 2007, pp. 99–108. IEEE (2007)

    Google Scholar 

  7. De Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game refinement relations and metrics. Log. Methods Comput. Sci. 4(3:7), 1–28 (2008)

    MathSciNet  MATH  Google Scholar 

  8. de Vink, E.P., Rutten, J.J.M.M.: Bisimulation for probabilistic transition systems: a coalgebraic approach. Theor. Comput. Sci. 221(1/2), 271–293 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  9. den Hartog, J.I.: Probabilistic Extensions of Semantical Models. Ph.D. thesis, Free University Amsterdam (2002)

    Google Scholar 

  10. Deng, Y.: Semantics of Probabilistic Processes: An Operational Approach. Springer, Heidelberg (2015)

    Google Scholar 

  11. Deng, Y., Chothia, T., Palamidessi, C., Pang, J.: Metrics for action-labelled quantitative transition systems. ENTCS 153(2), 79–96 (2006)

    Google Scholar 

  12. Deng, Y., Du, W.: Logical, metric, and algorithmic characterisations of probabilistic bisimulation. Technical report CMU-CS-11-110, Carnegie Mellon University, March 2011

    Google Scholar 

  13. Deng, Y., Feng, Y., Dal Lago, U.: On coinduction and quantum lambda calculi. In: Proceedings of CONCUR 2015, pp. 427–440. LIPIcs (2015)

    Google Scholar 

  14. Deng, Y., Hennessy, M.: On the semantics of Markov automata. Inf. Comput. 222, 139–168 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Deng, Y., 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). doi:10.1007/978-3-642-04081-8_19

    Chapter  Google Scholar 

  16. Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Desharnais, J., Jagadeesan, R., Gupta, V., Panangaden, P.: The metric analogue of weak bisimulation for probabilistic processes. In: Proceedings of LICS 2002, pp. 413–422. IEEE (2002)

    Google Scholar 

  18. Eisentraut, C., Godskesen, J.C., Hermanns, H., Song, L., Zhang, L.: Probabilistic bisimulation for realistic schedulers. In: Bjørner, N., de Boer, F. (eds.) FM 2015. LNCS, vol. 9109, pp. 248–264. Springer, Heidelberg (2015). doi:10.1007/978-3-319-19249-9_16

    Chapter  Google Scholar 

  19. Fahrenberg, U., Legay, A.: The quantitative linear-time branching-time spectrum. Theor. Comput. Sci. 538, 54–69 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Feng, Y., Ying, M.: Toward automatic verification of quantum cryptographic protocols. In: Proceedings of CONCUR 2015. LIPIcs, vol. 42, pp. 441–455 (2015)

    Google Scholar 

  21. Feng, Y., Zhang, L.: When equivalence and bisimulation join forces in probabilistic automata. In: Jones, C., Pihlajasaari, P., Sun, J. (eds.) FM 2014. LNCS, vol. 8442, pp. 247–262. Springer, Heidelberg (2014). doi:10.1007/978-3-319-06410-9_18

    Chapter  Google Scholar 

  22. Ferns, N., Panangaden, P., Precup, D.: Bisimulation metrics for continuous Markov decision processes. SIAM J. Comput. 40(6), 1662–1714 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ferns, N., Precup, D., Knight, S.: Bisimulation for Markov decision processes through families of functional expressions. In: Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. A Tribute to Prakash Panangaden: Essays Dedicated to Prakash Panangaden on the Occasion of His 60th Birthday. LNCS, vol. 8464, pp. 319–342. Springer, Heidelberg (2014). doi:10.1007/978-3-319-06880-0_17

    Chapter  Google Scholar 

  24. Gebler, D., Larsen, K.G., Tini, S.: Compositional metric reasoning with probabilistic process calculi. In: Pitts, A. (ed.) FoSSaCS 2015. LNCS, vol. 9034, pp. 230–245. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46678-0_15

    Chapter  Google Scholar 

  25. Giacalone, A., Jou, C., Smolka, S.: Algebraic reasoning for probabilistic concurrent systems. In: Proceedings of IFIP TC2 Working Conference on Programming Concepts and Methods, pp. 443–458 (1990)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Hermanns, H., Krčál, J., Křetínský, J.: Probabilistic bisimulation: naturally on distributions. In: Baldan, P., Gorla, D. (eds.) CONCUR 2014. LNCS, vol. 8704, pp. 249–265. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44584-6_18

    Google Scholar 

  29. Huth, M., Kwiatkowska, M.Z.: Quantitative analysis and model checking. In: Proceedings of the 12th Annual IEEE Symposium on Logic in Computer Science, pp. 111–122. IEEE Computer Society (1997)

    Google Scholar 

  30. Jonsson, B., Larsen, K.G., Yi, W.: Probabilistic extensions of process algebras. In: Handbook of Process Algebra, pp. 685–710. Elsevier, Amsterdam (2001)

    Google Scholar 

  31. Kantorovich, L., Rubinshtein, G.: On a space of totally additive functions. Vestn Len. Univ. 13(7), 52–59 (1958)

    MATH  Google Scholar 

  32. Kozen, D.: Results on the propositional mu-calculus. Theor. Comput. Sci. 27, 333–354 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kwiatkowska, M., Norman, G.: Probabilistic metric semantics for a simple language with recursion. In: Penczek, W., Szałas, A. (eds.) MFCS 1996. LNCS, vol. 1113, pp. 419–430. Springer, Heidelberg (1996). doi:10.1007/3-540-61550-4_167

    Chapter  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Manna, Z., Pnueli, A.: The Temporal Logic of Reactive and Concurrent Systems: Specification. Springer, Heidelberg (1991)

    MATH  Google Scholar 

  36. Norman, G.J.: Metric Semantics for Reactive Probabilistic Systems. Ph.D. thesis, University of Birmingham (1997)

    Google Scholar 

  37. Philippou, A., Lee, I., Sokolsky, O.: Weak bisimulation for probabilistic systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 334–349. Springer, Heidelberg (2000). doi:10.1007/3-540-44618-4_25

    Google Scholar 

  38. Raman, V.: Game Relations, Metrics and Refinements. Ph.D. thesis, University of California (2010)

    Google Scholar 

  39. Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. Ph.D. thesis, MIT (1995)

    Google Scholar 

  40. Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994). doi:10.1007/978-3-540-48654-1_35

    Google Scholar 

  41. Song, L., Deng, Y., Cai, X.: Towards automatic measurement of probabilistic processes. In: Proceedings of QSIC 2007, pp. 50–59. IEEE (2007)

    Google Scholar 

  42. van Breugel, F., Sharma, B., Worrell, J.: Approximating a behavioural pseudometric without discount for probabilistic systems. Log. Methods Comput. Sci. 4(2:2), 1–23 (2008)

    MathSciNet  MATH  Google Scholar 

  43. Breugel, F., Worrell, J.: An algorithm for quantitative verification of probabilistic transition systems. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 336–350. Springer, Heidelberg (2001). doi:10.1007/3-540-44685-0_23

    Google Scholar 

  44. Breugel, F., Worrell, J.: Towards quantitative verification of probabilistic transition systems. In: Orejas, F., Spirakis, P.G., Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 421–432. Springer, Heidelberg (2001). doi:10.1007/3-540-48224-5_35

    Chapter  Google Scholar 

  45. van Breugel, F., Worrell, J.: A behavioural pseudometric for probabilistic transition systems. Theor. Comput. Sci. 331(1), 115–142 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ying, M.: Bisimulation indexes and their applications. Theor. Comput. Sci. 275(1/2), 1–68 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgement

We thank the anonymous referees for their helpful comments.

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Authors and Affiliations

  1. Shanghai Normal University, Shanghai, China

    Wenjie Du

  2. Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai, China

    Yuxin Deng

  3. VU University Amsterdam, Amsterdam, Netherlands

    Daniel Gebler

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  1. Wenjie Du
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  2. Yuxin Deng
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Correspondence to Yuxin Deng .

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Editors and Affiliations

  1. Carl von Ossietzky Universität, Oldenburg, Germany

    Martin Fränzle

  2. University of New Mexico, Albuquerque, New Mexico, USA

    Deepak Kapur

  3. Chinese Academy of Sciences, Beijing, China

    Naijun Zhan

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Du, W., Deng, Y., Gebler, D. (2016). Behavioural Pseudometrics for Nondeterministic Probabilistic Systems. In: Fränzle, M., Kapur, D., Zhan, N. (eds) Dependable Software Engineering: Theories, Tools, and Applications. SETTA 2016. Lecture Notes in Computer Science(), vol 9984. Springer, Cham. https://doi.org/10.1007/978-3-319-47677-3_5

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  • DOI: https://doi.org/10.1007/978-3-319-47677-3_5

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