Scalar Outcomes Suffice for Finitary Probabilistic Testing

  • Yuxin Deng
  • Rob van Glabbeek
  • Carroll Morgan
  • Chenyi Zhang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4421)


The question of equivalence has long vexed research in concurrency, leading to many different denotational- and bisimulation-based approaches; a breakthrough occurred with the insight that tests expressed within the concurrent framework itself, based on a special “success action”, yield equivalences that make only inarguable distinctions.

When probability was added, however, it seemed necessary to extend the testing framework beyond a direct probabilistic generalisation in order to remain useful. An attractive possibility was the extension to multiple success actions that yielded vectors of real-valued outcomes.

Here we prove that such vectors are unnecessary when processes are finitary, that is finitely branching and finite-state: single scalar outcomes are just as powerful. Thus for finitary processes we can retain the original, simpler testing approach and its direct connections to other naturally scalar-valued phenomena.


  1. 1.
    Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic and Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford (1994)Google Scholar
  2. 2.
    Aliprantis, C.D., Border, K.C.: Infinite Dimensional Analysis, 2nd edn. Springer, Heidelberg (1999)MATHGoogle Scholar
  3. 3.
    Cattani, S., Segala, R.: Decision algorithms for probabilistic bisimulation. In: Brim, L., et al. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 371–385. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    De Nicola, R., Hennessy, M.: Testing equivalences for processes. Theoretical Computer Science 34, 83–133 (1984)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Hansson, H., Jonsson, B.: A calculus for communicating systems with time and probabilities. In: Proc. of the Real-Time Systems Symposium (RTSS ’90), pp. 278–287. IEEE Computer Society Press, Los Alamitos (1990)Google Scholar
  6. 6.
    He, J., Seidel, K., McIver, A.K.: Probabilistic models for the guarded command language. Science of Computer Programming 28, 171–192 (1997)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Hoare, C.A.R.: Communicating Sequential Processes. Prentice-Hall, Englewood Cliffs (1985)MATHGoogle Scholar
  8. 8.
    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 and ProCoS 1994. LNCS, vol. 863, pp. 418–430. Springer, Heidelberg (1994)Google Scholar
  9. 9.
    Jonsson, B., Yi, W.: Testing preorders for probabilistic processes can be characterized by simulations. Theoretical Computer Science 282(1), 33–51 (2002)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kozen, D.: A probabilistic PDL. Jnl. Comp. Sys. Sciences 30(2), 162–178 (1985)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Matoušek, J.: Lectures on Discrete Geometry. Springer, Heidelberg (2002)Google Scholar
  12. 12.
    McIver, A.K., Morgan, C.C.: Games, probability and the quantitative μ-calculus qMu. In: Baaz, M., Voronkov, A. (eds.) LPAR 2002. LNCS (LNAI), vol. 2514, pp. 292–310. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  13. 13.
    McIver, A.K., Morgan, C.C.: Abstraction, Refinement and Proof for Probabilistic Systems. Tech. Mono. Comp. Sci. Springer, Heidelberg (2005)MATHGoogle Scholar
  14. 14.
    Morgan, C.C., McIver, A.K., Seidel, K.: Probabilistic predicate transformers. ACM Trans. on Programming Languages and Systems 18(3), 325–353 (1996)CrossRefGoogle Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Puterman, M.L.: Markov Decision Processes. Wiley, Chichester (1994)MATHGoogle Scholar
  17. 17.
    Segala, R.: Testing probabilistic automata. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119, pp. 299–314. Springer, Heidelberg (1996)Google Scholar
  18. 18.
    Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)CrossRefGoogle Scholar
  19. 19.
    Stoelinga, M.I.A., Vaandrager, F.W.: A testing scenario for probabilistic automata. In: Baeten, J.C.M., et al. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 407–418. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Vardi, M.Y.: Automatic verification of probabilistic concurrent finite state programs. In: Proc. FOCS ’85, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)Google Scholar
  21. 21.
    Yi, W., Larsen, K.G.: Testing probabilistic and nondeterministic processes. In: Proc. IFIP TC6/WG6.1 Twelfth Intern. Symp. on Protocol Specification, Testing and Verification. IFIP Transactions, vol. C-8, pp. 47–61. North-Holland, Amsterdam (1992)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Yuxin Deng
    • 1
  • Rob van Glabbeek
    • 1
    • 2
  • Carroll Morgan
    • 1
  • Chenyi Zhang
    • 1
    • 2
  1. 1.School of Comp. Sci. and Eng., University of New South Wales, SydneyAustralia
  2. 2.National ICT Australia, Locked Bag 6016, Sydney, NSW 1466Australia

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