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Coherent Resolutions of Nondeterminism

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Computer Performance Engineering (EPEW 2019)

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We study the impact that different ways of resolving nondeterminism within probabilistic automata have on the properties of probabilistic behavioral equivalences. Firstly, we provide a uniform definition of structure-preserving and structure-modifying resolutions of nondeterminism, respectively generated by different families of schedulers. Secondly, we exhibit a number of anomalies arising from the excessive power of the various families of schedulers, which affect the discriminating power, the compositionality, and the backward compatibility of probabilistic trace equivalence. Thirdly, we propose to remove those anomalies by enforcing coherency within resolutions of nondeterminism. This ensures that a scheduler cannot select different continuations in equivalent states of an automaton, so that also the states to which they correspond in any resolution of the automaton have equivalent continuations.

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  1. Baier, C., Katoen, J.-P., Hermanns, H., Wolf, V.: Comparative branching-time semantics for Markov chains. Inf. Comput. 200, 149–214 (2005)

    Article  MathSciNet  Google Scholar 

  2. Bernardo, M.: Genesis and evolution of ULTraS: metamodel, metaequivalences, metaresults. In: Boreale, M., Corradini, F., Loreti, M., Pugliese, R. (eds.) Models, Languages, and Tools for Concurrent and Distributed Programming. LNCS, vol. 11665, pp. 92–111. Springer, Cham (2019).

    Chapter  Google Scholar 

  3. Bernardo, M., De Nicola, R., Loreti, M.: Revisiting trace and testing equivalences for nondeterministic and probabilistic processes. Logical Methods Comput. Sci. 10(1:16), 1–42 (2014)

    MathSciNet  MATH  Google Scholar 

  4. Bernardo, M., De Nicola, R., Loreti, M.: Relating strong behavioral equivalences for processes with nondeterminism and probabilities. Theor. Comput. Sci. 546, 63–92 (2014)

    Article  MathSciNet  Google Scholar 

  5. Bernardo, M., Sangiorgi, D., Vignudelli, V.: On the discriminating power of testing equivalences for reactive probabilistic systems: results and open problems. In: Norman, G., Sanders, W. (eds.) QEST 2014. LNCS, vol. 8657, pp. 281–296. Springer, Cham (2014).

    Chapter  MATH  Google Scholar 

  6. Bonchi, F., Sokolova, A., Vignudelli, V.: The theory of traces for systems with nondeterminism and probability. In: Proceedings of the 34th ACM/IEEE Symposium on Logic in Computer Science (LICS 2019), no. (19:62), pp. 1–14. IEEE-CS Press (2019)

    Google Scholar 

  7. Brookes, S.D., Hoare, C.A.R., Roscoe, A.W.: A theory of communicating sequential processes. J. ACM 31, 560–599 (1984)

    Article  MathSciNet  Google Scholar 

  8. De Nicola, R.: Extensional equivalences for transition systems. Acta Informatica 24, 211–237 (1987)

    Article  MathSciNet  Google Scholar 

  9. De Nicola, R., Hennessy, M.: Testing equivalences for processes. Theor. Comput. Sci. 34, 83–133 (1984)

    Article  MathSciNet  Google Scholar 

  10. Deng, Y., van Glabbeek, R., Morgan, C., Zhang, C.: Scalar outcomes suffice for finitary probabilistic testing. In: De Nicola, R. (ed.) ESOP 2007. LNCS, vol. 4421, pp. 363–378. Springer, Heidelberg (2007).

    Chapter  Google Scholar 

  11. Derman, C.: Finite State Markovian Decision Processes. Academic Press, Cambridge (1970)

    MATH  Google Scholar 

  12. Georgievska, S., Andova, S.: Probabilistic may/must testing: retaining probabilities by restricted schedulers. Formal Aspects Comput. 24, 727–748 (2012)

    Article  MathSciNet  Google Scholar 

  13. van Glabbeek, R.J.: The linear time - branching time spectrum I. In: Handbook of Process Algebra, pp. 3–99. Elsevier (2001)

    Google Scholar 

  14. Huynh, D.T., Tian, L.: On some equivalence relations for probabilistic processes. Fundamenta Informaticae 17, 211–234 (1992)

    MathSciNet  MATH  Google Scholar 

  15. Jonsson, B., Ho-Stuart, C., Yi, W.: Testing and refinement for nondeterministic and probabilistic processes. In: Langmaack, H., de Roever, W.-P., Vytopil, J. (eds.) FTRTFT 1994. LNCS, vol. 863, pp. 418–430. Springer, Heidelberg (1994).

    Chapter  Google Scholar 

  16. Jonsson, B., Yi, W.: Compositional testing preorders for probabilistic processes. In: Proceedings of the 10th IEEE Symposium on Logic in Computer Science (LICS 1995), pp. 431–441. IEEE-CS Press (1995)

    Google Scholar 

  17. Jou, C.-C., Smolka, S.A.: Equivalences, congruences, and complete axiomatizations for probabilistic processes. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 367–383. Springer, Heidelberg (1990).

    Chapter  Google Scholar 

  18. Keller, R.M.: Formal verification of parallel programs. Commun. ACM 19, 371–384 (1976)

    Article  MathSciNet  Google Scholar 

  19. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand, New York (1960)

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  22. Segala, R.: Modeling and verification of randomized distributed real-time systems. Ph.D. thesis (1995)

    Google Scholar 

  23. Segala, R.: A compositional trace-based semantics for probabilistic automata. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 234–248. Springer, Heidelberg (1995).

    Chapter  Google Scholar 

  24. Segala, R.: Testing probabilistic automata. In: Montanari, U., Sassone, V. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 299–314. Springer, Heidelberg (1996).

    Chapter  Google Scholar 

  25. 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).

    Chapter  Google Scholar 

  26. Yi, W., Larsen, K.G.: Testing probabilistic and nondeterministic processes. In: Proceedings of the 12th International Symposium on Protocol Specification, Testing and Verification (PSTV 1992), pp. 47–61. North-Holland (1992)

    Google Scholar 

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We would like to thank Valeria Vignudelli for pointing out the property violation illustrated in Fig. 4 and Rob van Glabbeek for the valuable discussions on interpolating and randomized schedulers.

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Correspondence to Marco Bernardo .

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Bernardo, M. (2020). Coherent Resolutions of Nondeterminism. In: Gribaudo, M., Iacono, M., Phung-Duc, T., Razumchik, R. (eds) Computer Performance Engineering. EPEW 2019. Lecture Notes in Computer Science(), vol 12039. Springer, Cham.

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  • Print ISBN: 978-3-030-44410-5

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