Published results show that various models may be obtained by combining parallel composition with probability and with or without non-determinism. In this paper we treat this problem in the setting of process algebra in the form of ACP. First, probabilities are introduced by an operator for the internal probabilistic choice. In this way we obtain the Basic Process Algebra with probabilistic choice prBPA. After-wards, prBPA is extended with parallel composition to ACP π + . We give the axiom system for ACP π + and a complete operational semantics that preserves the interleaving model for the dynamic concurrent processes. Considering the PAR protocol, a communication protocol that can be used in the case of unreliable channels, we investigate the applicability of ACP π + . Using in addition only the priority operator and the preabstraction operator we obtain a recursive specification of the behaviour of the protocol that can be viewed as a Markov chain.


Probability Distribution Function Operational Semantic Axiom System Probabilistic Choice Parallel Composition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Andova, Process algebra with interleaving probabilistic parallel composition, Eindhoven University of Technology, CSR 99-xx, 1999.Google Scholar
  2. 2.
    J.C.M. Baeten, J. A. Bergstra, Global renaming operators in concrete process algebra, Information and Computation 78, pp. 205–245, 1988.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    J.C.M. Baeten, W. P. Weijland, Process algebra, Cambridge University Press, 1990.Google Scholar
  4. 4.
    J.C.M. Baeten, C. Verhoef, Concrete process algebra, Handbook of Logic in Computer Science, volume 4: “Semantic Modelling”, Oxford University Press, 1995.Google Scholar
  5. 5.
    J.C.M. Baeten, J.A. Bergstra, Process algebra with partial choice, Proc. CONCUR’94, Uppsala, B. Jonsson & J. Parrow, eds., LNCS 836, Springer Verlag, pp. 465–480, 1994.Google Scholar
  6. 6.
    J.C.M. Baeten, J.A. Bergstra, S.A. Smolka, Axiomatizing probabilistic processes: ACP with generative probabilities, Information and Computation 121(2), pp. 234–255, Sep. 1995.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    J.A. Bergstra, J.W. Klop, Process algebra for synchronous communication, Information and Control 60, pp. 109–137, 1984.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    P.R. D’Argenio, C. Verhoef, A general conservative extension theorem in process algebra with inequalities, Theoretical Computer Science 177, pp. 351–380, 1997.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    P.R. D’Argenio, H. Hermanns, J.-P. Katoen On generative parallel composition, Preliminary Proc. of PROBMIV’98, Indianopolis, USA, C. Baier & M. Huth & M Kwiatkowska & M. Ryan ed., pp. 105–121, 1998.Google Scholar
  10. 10.
    A. Giacalone, C.-C. Jou, S. A. Smolka, Algebraic reasoning for probabilistic concurrent systems, Proc. Working Conference on Programming Concepts and Methods, IFIP TC 2, Sea of Galilee, Israel, M. Broy & C.B. Jones ed., pp. 443–458, 1990.Google Scholar
  11. 11.
    R.J. van Glabbeek, S. A. Smolka, B. Steffen, C. M. N. Tofts, Reactive, generative and stratified models of probabilistic processes, Proc. of 5th Annual IEEE Symp. on Logic in Computer Science, Philadelphia, PA, pp. 130–141, 1990.Google Scholar
  12. 12.
    H. Hansson, Time and probability in formal design of distributed systems, Ph.D. thesis, DoCS 91/27, University of Uppsala, 1991.Google Scholar
  13. 13.
    C.-C. Jou, S. A. Smolka Equivalences, congruences and complete axiomatizations for probabilistic processes, Proc. CONCUR’ 90, LNCS 458, Springer Verlag, Berlin, pp. 367–383, 1990.Google Scholar
  14. 14.
    K. G. Larsen, A. Skou, Bisimulation through probabilistic testing, Proc. of 16th ACM Symp. on Principles of Programming Languages, Austin, TX, 1989.Google Scholar
  15. 15.
    F.W. Vaandrager, Two simple protocols, In: Applications of Process algebra, Cambridge University Press, J.C.M. Baeten ed., pp. 23–44, 1990.Google Scholar
  16. 16.
    M.Y. Vardi, Automatic verification of probabilistic concurrent finite state programs, Proc. of 26th Symp. on Foundations of Com. Sc., IEEE Comp. Soc. Press, pp. 327–338, 1985.Google Scholar
  17. 17.
    C. Verhoef, A general conservative extension theorem in process algebra, Proc. of PROCOMET’94, IFIP 2 Working Conference, San Miniato, E.-R. Olderog ed., pp. 149–168, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Suzana Andova
    • 1
  1. 1.Department of Computing ScienceEindhoven University of TechnologyEindhovenThe Netherlands

Personalised recommendations