A Hierarchy of Equivalences for Probabilistic Processes

  • Manuel Núñez
  • Luis Llana
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5048)


We study several process equivalences on a probabilistic process algebra. First, we define an operational semantics. Afterwards we introduce the notion of passing a test with a probability. We consider three families of tests according to the intended behavior of an external observer: Reactive (sequential tests), generative (branching tests), and limited generative (equitable branching tests). For each of these families we define three predicates over processes and tests (may-pass, must-pass, pass p ) which induce three equivalences. Finally, we relate these nine equivalences and provide either alternative characterizations or fully abstract denotational semantics. These semantic frameworks cover from simple traces to probabilistic acceptance trees.


Operational Semantic Process Algebra Probabilistic Process Semantic Function Denotational Semantic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Baeten, J.C.M., Weijland, W.P.: Process Algebra. In: Cambridge Tracts in Computer Science 18, Cambridge University Press, Cambridge (1990)Google Scholar
  2. 2.
    Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.): Handbook of Process Algebra. North-Holland, Amsterdam (2001)zbMATHGoogle Scholar
  3. 3.
    Cazorla, D., Cuartero, F., Valero, V., Pelayo, F.L., Pardo, J.J.: Algebraic theory of probabilistic and non-deterministic processes. Journal of Logic and Algebraic Programming 55(1–2), 57–103 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cheung, L., Stoelinga, M., Vaandrager, F.: A testing scenario for probabilistic processes. Journal of the ACM 54(6), Article 29 (2007)Google Scholar
  5. 5.
    Christoff, I.: Testing equivalences and fully abstract models for probabilistic processes. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 126–140. Springer, Heidelberg (1990)Google Scholar
  6. 6.
    Cleaveland, R., Dayar, Z., Smolka, S.A., Yuen, S.: Testing preorders for probabilistic processes. Information and Computation 154(2), 93–148 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    de Nicola, R., Hennessy, M.C.B.: Testing equivalences for processes. Theoretical Computer Science 34, 83–133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Deng, Y., van Glabbeek, R., Hennessy, M., Morgan, C., Zhang, C.: Characterising testing preorders for finite probabilistic processes. In: 22nd Annual IEEE Symposium on Logic in Computer Science, LICS 2007, pp. 313–325. IEEE Computer Society Press, Los Alamitos (2007)Google Scholar
  9. 9.
    van Glabbeek, R.: The linear time-branching time spectrum II. In: Best, E. (ed.) CONCUR 1993. LNCS, vol. 715, pp. 66–81. Springer, Heidelberg (1993)Google Scholar
  10. 10.
    van Glabbeek, R.: The linear time-branching time spectrum I. The semantics of concrete, sequential processes. In: Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.) Handbook of process algebra, ch. 1, North-Holland, Amsterdam (2001)Google Scholar
  11. 11.
    van Glabbeek, R., Smolka, S.A., Steffen, B.: Reactive, generative and stratified models of probabilistic processes. Information and Computation 121(1), 59–80 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Gregorio, C., Núñez, M.: Denotational semantics for probabilistic refusal testing. In: PROBMIV 1998. Electronic Notes in Theoretical Computer Science, vol. 22. Elsevier, Amsterdam (1999)Google Scholar
  13. 13.
    Hennessy, M.: Acceptance trees. Journal of the ACM 32(4), 896–928 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hennessy, M.: An algebraic theory of fair asynchronous communicating processes. Theoretical Computer Science 49, 121–143 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Hennessy, M.: Algebraic Theory of Processes. MIT Press, Cambridge (1988)zbMATHGoogle Scholar
  16. 16.
    Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall, Englewood Cliffs (1985)zbMATHGoogle Scholar
  17. 17.
    Huynh, D.T., Tian, L.: On some equivalence relations for probabilistic processes. Fundamenta Informaticae 17, 211–234 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jonsson, B., Yi, W.: Compositional testing preorders for probabilistic processes. In: 10th IEEE Symposium on Logic In Computer Science, pp. 431–443. IEEE Computer Society Press, Los Alamitos (1995)Google Scholar
  19. 19.
    Jonsson, B., Yi, W.: Testing preorders for probabilistic processes can be characterized by simulations. Theoretical Computer Science 282, 33–51 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    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)Google Scholar
  21. 21.
    Kwiatkowska, M., Norman, G.J.: A testing equivalence for reactive probabilistic processes. In: EXPRESS 1998. Electronic Notes in Theoretical Computer Science, vol. 16. Elsevier, Amsterdam (1998)Google Scholar
  22. 22.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94(1), 1–28 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    López, N., Núñez, M.: An overview of probabilistic process algebras and their equivalences. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 89–123. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  24. 24.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  25. 25.
    Milner, R.: Communicating and Mobile Systems: the π-Calculus. Cambridge University Press, Cambridge (1999)zbMATHGoogle Scholar
  26. 26.
    Narayan Kumar, K., Cleaveland, R., Smolka, S.A.: Infinite probabilistic and nonprobabilistic testing. In: Arvind, V., Ramanujam, R. (eds.) FST TCS 1998. LNCS, vol. 1530, pp. 209–220. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  27. 27.
    Natarajan, V., Cleaveland, R.: Divergence and fair testing. In: Fülöp, Z., Gecseg, F. (eds.) ICALP 1995. LNCS, vol. 944, pp. 648–659. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  28. 28.
    Núñez, M.: Algebraic theory of probabilistic processes. Journal of Logic and Algebraic Programming 56(1–2), 117–177 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Núñez, M., de Frutos, D.: Testing semantics for probabilistic LOTOS. In: 8th IFIP WG6.1 Int. Conf. on Formal Description Techniques, FORTE 1995, pp. 365–380. Chapman & Hall, Boca Raton (1995)Google Scholar
  30. 30.
    Núñez, M., de Frutos, D., Llana, L.: Acceptance trees for probabilistic processes. In: Lee, I., Smolka, S.A. (eds.) CONCUR 1995. LNCS, vol. 962, pp. 249–263. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  31. 31.
    Núñez, M., Rupérez, D.: Fair testing through probabilistic testing. In: Formal Description Techniques for Distributed Systems and Communication Protocols (XII), and Protocol Specification, Testing, and Verification (XIX), pp. 135–150. Kluwer Academic Publishers, Dordrecht (1999)Google Scholar
  32. 32.
    Rensink, A., Vogler, W.: Fair testing. Information and Computation 205(2), 125–198 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Segala, R.: Testing probabilistic automata. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 299–314. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  34. 34.
    Stoelinga, M., Vaandrager, F.: A testing scenario for probabilistic automata. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 464–477. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  35. 35.
    Yi, W., Larsen, K.G.: Testing probabilistic and nondeterministic processes. In: 12th IFIP/WG6.1 Int. Symposium on Protocol Specification, Testing and Verification, PSTV 1992, pp. 47–61. North-Holland, Amsterdam (1992)Google Scholar

Copyright information

© IFIP International Federation for Information Processing 2008

Authors and Affiliations

  • Manuel Núñez
    • 1
  • Luis Llana
    • 1
  1. 1.Dept. Sistemas Informáticos y Computación Facultad de InformáticaUniversidad Complutense de MadridMadridSpain

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