Quantitative Relations and Approximate Process Equivalences

  • Alessandra Di Pierro
  • Chris Hankin
  • Herbert Wiklicky
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2761)


We introduce a characterisation of probabilistic transition systems (PTS) in terms of linear operators on some suitably defined vector space representing the set of states. Various notions of process equivalences can then be re-formulated as abstract linear operators related to the concrete PTS semantics via a probabilistic abstract interpretation. These process equivalences can be turned into corresponding approximate notions by identifying processes whose abstract operators “differ” by a given quantity, which can be calculated as the norm of the difference operator. We argue that this number can be given a statistical interpretation in terms of the tests needed to distinguish two behaviours.


Quantitative Relation Abstract Interpretation Label Transition System Process Equivalence Abstract Domain 
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  1. 1.
    Aldini, A., Bravetti, M., Gorrieri, R.: A process algebraic approach for the analysis of probabilistic non-interference. Journal of Computer Security (2003) (to appear)Google Scholar
  2. 2.
    Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.): Handbook of Process Algebra. Elsevier Science, Amsterdam (2001)zbMATHGoogle Scholar
  3. 3.
    Böttcher, A., Silbermann, B.: Introduction to Large Truncated Toeplitz Matrices. Springer, New York (1999)zbMATHGoogle Scholar
  4. 4.
    Campbell, S.L., Meyer, D.: Generalized Inverse of Linear Transformations. Constable and Company, London (1979)Google Scholar
  5. 5.
    Cousot, P., Cousot, R.: Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs by Construction or Approximation of Fixpoints. In: Proceedings of POPL 1977, Los Angeles, pp. 238–252 (1977)Google Scholar
  6. 6.
    Cousot, P., Cousot, R.: Systematic Design of Program Analysis Frameworks. In: Proceedings of POPL 1979, San Antonio, Texas, pp. 269–282 (1979)Google Scholar
  7. 7.
    Cousot, P., Cousot, R.: Abstract Interpretation and Applications to Logic Programs. Journal of Logic Programming 13(2-3), 103–180 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dams, D., Gerth, R., Grumberg, O.: Abstract interpretation of reactive systems. ACM Transactions on Programming Languages and Systems 19(2), 253–291 (1997)CrossRefGoogle Scholar
  9. 9.
    Desharnais, J., Jagadeesan, R., Gupta, V., Panangaden, P.: The metric analogue of weak bisimulation for probabilistic processes. In: Proceedings of LICS 2002, Denmark, July 22- 25, pp. 22–25. IEEE, Los Alamitos (2002)Google Scholar
  10. 10.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Approximate confinement under uniform attacks. In: Hermenegildo, M.V., Puebla, G. (eds.) SAS 2002. LNCS, vol. 2477, p. 310. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  11. 11.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Approximate non-interference. In: Proceedings of CSFW 2002, Cape Breton, June 24- 26, pp. 24–26. IEEE, Los Alamitos (2002)Google Scholar
  12. 12.
    Di Pierro, A., Hankin, C., Wiklicky, H.: Approximate non-interference. Journal of Computer Security, WITS 2002 Issue (2003) (to appear)Google Scholar
  13. 13.
    Di Pierro, A., Wiklicky, H.: Concurrent Constraint Programming: Towards Probabilistic Abstract Interpretation. In: Proceedings of PPDP 2000, Montréal, Canada, pp. 127–138. ACM, New York (2000)Google Scholar
  14. 14.
    Di Pierro, A., Wiklicky, H.: Measuring the precision of abstract interpretations. In: Lau, K.-K. (ed.) LOPSTR 2000. LNCS, vol. 2042, pp. 147–164. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  15. 15.
    Di Pierro, A., Wiklicky, H.: A C*-algebraic approach to the operational semantics of programming languages (2003) (in preparation)Google Scholar
  16. 16.
    Giacalone, A., Jou, C.-C., Smolka, S.A.: Algebraic reasoning for probabilistic concurrent systems. In: Proceedings of the IFIP WG 2.2/2.3 Working Conference on Programming Concepts and Methods, pp. 443–458. North-Holland, Amsterdam (1990)Google Scholar
  17. 17.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics, vol. 207. Springer, New York (2001)zbMATHGoogle Scholar
  18. 18.
    Greub, W.H.: Linear Algebra. Grundlehren der mathematischen Wissenschaften, vol. 97. Springer, New York (1967)zbMATHGoogle Scholar
  19. 19.
    Jonsson, B., Yi, W., Larsen, K.G.: Probabilistic Extensions of Process Algebras. ch. 11, pp. 685–710. Elsevier Science, Amsterdam (2001) see [2]Google Scholar
  20. 20.
    Kozen, D.: Semantics for probabilistic programs. Journal of Computer and System Sciences 22, 328–350 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Milner, R.: A Calculus of Communication Systems. LNCS, vol. 92. Springer, Heidelberg (1980)zbMATHGoogle Scholar
  23. 23.
    Mohar, B., Woess, W.: A survey on spectra of infinite graphs. Bulletin of the London Mathematical Society 21, 209–234 (1988)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Monniaux, D.: Abstract interpretation of probabilistic semantics. In: Palsberg, J. (ed.) SAS 2000. LNCS, vol. 1824, pp. 322–340. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  25. 25.
    Nielson, F., Riis Nielson, H., Hankin, C.: Principles of Program Analysis. Springer, Berlin (1999)zbMATHGoogle Scholar
  26. 26.
    Smolka, S.A., van Glabbeek, R.J., Steffen, B.: Reactive, Generative and Stratified Models of Probabilistic Processes. Information and Computation 121, 59–80 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Schmidt, D.A.: Binary relations for abstraction and refinement. In: Workshop on Refinement and Abstraction, Amagasaki, Japan (November 1999)Google Scholar
  28. 28.
    Shao, J.: Mathematical Statistics. Springer Texts in Statistics. Springer, New York (1999)zbMATHGoogle Scholar
  29. 29.
    Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems. Texts in Applied Mathematics, vol. 6. Springer, Heidelberg (1990)zbMATHGoogle Scholar
  30. 30.
    van Breugel, F., Worrell, J.: Towards quantitative verification of probabilistic transition systems. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 421–432. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  31. 31.
    van Glabbeek, R.J.: The Linear Time – Branching Time Spectrum I. The Semantics of Concrete, Sequential Processes. ch. 1, pp. 3–99. Elsevier Science, Amsterdam (2001) see [2]Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Alessandra Di Pierro
    • 1
  • Chris Hankin
    • 2
  • Herbert Wiklicky
    • 2
  1. 1.Dipartimento di InformaticaUniversitá di PisaItaly
  2. 2.Department of ComputingImperial CollegeLondonUK

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