Skip to main content
SpringerLink
Log in

Parametric probabilistic transition systems for system design and analysis

  • Original Article
  • Published:
Formal Aspects of Computing

Abstract

We develop a model of parametric probabilistic transition Systems (PPTSs), where probabilities associated with transitions may be parameters. We show how to find instances of the parameters that satisfy a given property and instances that either maximize or minimize the probability of reaching a certain state. As an application, we model a probabilistic non-repudiation protocol with a PPTS. The theory we develop allows us to find instances that maximize the probability that the protocol ends in a fair state (no participant has an advantage over the others).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aldini A, Gorrieri R (2002) Security analysis of a probabilistic non-repudiation protocol. In: Proceedings. of PAPM-PROBMIV’02, Springer LNCS 2399, pp 17–36

  2. de Alfaro L (1998) How to specify and verify the long-run average behaviour of probabilistic systems. In: Proceedings of LICS’98, IEEE, pp 454–465

  3. de Alfaro L (1999) Computing minimum and maximum reachability times in probabilistic systems. In: Proceedings. of CONCUR’99, Springer LNCS 1664, pp 66–81

  4. Alur R, Henzinger TA, Vardi MY (1993) Parametric real-time reasoning. In: Proceedings. of STOC’93, ACM, pp 592–601

  5. Alur R, Dill DL (1994) A theory of timed automata. Theoret Comput Sci 126:183–235

    Article  MATH  MathSciNet  Google Scholar 

  6. Amnell T, Behrmann G, Bengtsson J, D’Argenio PR, David A, Fehnker A, Hune T, Jeannet B, Larsen KG, Moeller MO, Pettersson P, Weise C, Yi W (2000) Uppaal-now, next and future. In: Proceedings. of MOVEP’00, Springer LNCS 2067, pp 99–124

  7. Baeten JCM, Bergstra JA, Smolka SA (1995) Axiomatizing probabilistic processes: ACP with generative probabilities. Inform Comput 121:234–255

    Article  MATH  MathSciNet  Google Scholar 

  8. Baier C, Kwiatkowska M (1998) On the verification of qualitative properties of probabilistic processes under fairness constraints. Inform Process Lett 66:71–79

    Article  MATH  MathSciNet  Google Scholar 

  9. Baier C, Kwiatkowska M (1998) Model checking for probabilistic time logic with fairness. Distrib Comput 11:125–155

    Article  Google Scholar 

  10. Beauquier D (2002) Markov decision processes and Buchi automata. Fundam Inform 50:1–13

    MATH  MathSciNet  Google Scholar 

  11. Bellman RE (1957) Dynamic programming. Princeton University Press, Princeton

    Google Scholar 

  12. Bianco A, de Alfaro L (1995) Model checking of probabilistic and deterministic systems. In: Proceedings of 15th Conference on foundations of computer technology and theoretical computer science, Springer LNCS 1026, pp 499–513

  13. Bohnenkamp H, van der Stok P, Hermanns H, Vaandrager F (2003) Costoptimization of the IPv4 zeroconf protocol. In: Proceedings of DSN’03, IEEE, pp 531–540

  14. Chaum DL (1988) The dining cryptographers problem: unconditional sender and recipient untraceability. J Cryptol 1(1):65–75

    Article  MATH  MathSciNet  Google Scholar 

  15. Cleaveland R, Smolka SA, Zwarico A (1992) Testing preorders for probabilistic processes. In: Proceedings of ICALP’92, Springer LNCS 623, pp 708–719

  16. Daws C (2004) Symbolic and parametric model checking of discrete-time markov Chains. In: Proceedings of ICTAC’04, Springer LNCS 3407, pp 280–294

  17. Deng Y, Chothia T, Palamidessi C, Pang J (2006) Metrics for Action-labelled Quantitative Transition Systems. In: Proceedings of QAPL’05, Elsevier ENTCS 153(2), pp 79–96

  18. Desharnais J, Jagadeesan R, Gupta V, Panangaden P (2002) The metric analogue of weak bisimulation for probabilistic processes. In: Proceedings of LICS’02, IEEE, pp 413–422

  19. Desharnais J, Jagadeesan R, Gupta V, Panangaden P (2004) Metrics for labelled Markov processes. Theoret Comput Sci 318(3): 323–354

    Article  MATH  MathSciNet  Google Scholar 

  20. Desharnais J, Edalat A, Panangaden P (1998) A logic characterization of bisimulation for Markov Processes. In: Proceedings of LICS’98, IEEE, 478–487

  21. Desharnais J, Gupta V, Jagadeesan R, Panangaden P (2003) Approximating Labelled Markov Processes. Inform Comput 184:160–200

    Article  MATH  MathSciNet  Google Scholar 

  22. Halmos PR (1950) Measure theory. Springer, Berlin Heidelberg New York

    MATH  Google Scholar 

  23. Hansson H (1994) Time and probability in formal design of distributed systems. Real-time safety critical systems 1, Elsevier, Amsterdam

    Google Scholar 

  24. Hansson H, Jonsson B (1994) A logic for reasoning about time and reliability. Formal Aspects Comput 6:512–535

    Article  MATH  Google Scholar 

  25. Henzinger TA, Ho P-H, Wong-Toi H (1997) HyTech: a model checker for hybrid systems. In: Proceedings of CAV’97, Springer LNCS 1254, pp 460–463

  26. Howard H (1960) Dynamic programming and Markov processes. MIT, Cambridge

    MATH  Google Scholar 

  27. Hune T, Romijn J, Stoelinga M, Vaandrager FW (1960) Linear parametric model checking of timed automata. J Log Algebr Program 52–53: 183–220

  28. Jonsson B, Larsen K (1991) Specification and refinement of probabilistic processes. In: Proceedings of LICS’91, IEEE, pp 266–277

  29. Kemeny J, Snell J, Knapp A (1996) Denumerable Markov chains. D. Van Nostrand Company Inc.,

  30. Lanotte R, Maggiolo-Schettini A, Troina A (2003) Weak bisimulation for probabilistic timed automata and applications to security. In: Proceedings of SEFM’03, IEEE, pp 34–43

  31. Lanotte R, Maggiolo-Schettini A, Troina A (2004) Decidability results for parametric probabilistic transition systems with an application to security. In: Proceedings of SEFM’04, IEEE, pp 114–121

  32. Lanotte R, Maggiolo-Schettini A, Troina A (2005) Automatic analysis of a non-repudiation protocol. In: Proceedings of QAPL’03, Elsevier ENTCS 112, pp 113–129

  33. Larsen K, Skou A (1991) Bisimulation through probabilistic testing. Inform Comput 94:1–28

    Article  MATH  MathSciNet  Google Scholar 

  34. Maler O(1995) A decomposition theorem for probabilistic transition systems. Theoret Comput Sci 145:391–396

    Article  MATH  MathSciNet  Google Scholar 

  35. Markowitch O, Roggeman Y (1999) Probabilistic non-repudiation without trusted third party. In: Proceedings of 2nd Conference on Security in Communication Network

  36. Reed MG, Syverson PF, Goldschlag DM (1998) Anonymous connections and onion routing. IEEE J Select Areas Commun 16(4):482–494

    Article  Google Scholar 

  37. Reiter MK, Rubin AD (1998) Crowds: anonymity for Web transactions. ACM Trans Inform and Syst Security 1(1):66–92

    Article  Google Scholar 

  38. Renegar J (1992) On the computational complexity and geometry of the first-order theory of the reals, Part I: Introduction. Preliminaries. The geometry of semi-algebraic sets. The decision problem for the existential theory of the reals. J Symb Comput 13:255–300

    Article  MATH  MathSciNet  Google Scholar 

  39. Ross S (1983) Stochastic processes. Wiley, New York

    MATH  Google Scholar 

  40. Stark E, Smolka SA (1998) Compositional analysis of expected delays in networks of probabilistic I/O automata. In: Proceedings of LICS 98, IEEE, pp 466–477

  41. Stewart I (1989) Galois theory. Chapman and Hall, London

    MATH  Google Scholar 

  42. Tanimoto T, Nakata A, Hashimoto H, Higashino T (2005) double depth first search based parametric analysis for parametric time–interval automata. IEICE Trans Fundam 11:3007–3021

    Article  Google Scholar 

  43. Tarski A (1951) A Decision method for elementary algebra and geometry, 2nd edn. University of California Press

  44. Williams D (1991) Probability with martingales. Cambridge University press, Cambridge

    MATH  Google Scholar 

  45. Wu SH, Smolka SA, Stark E (1997) Composition and behaviors of probabilistic I/O automata. Theoret Comput Sci 176:1–38

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Scienze della Cultura, Politiche e dell’Informazione, Università dell’Insubria, Via Valleggio 11, 22100, Como, Italy

    Ruggero Lanotte

  2. Dipartimento di Informatica, Universitá di pisa, Largo Pontecorvo 3, 56127, Pisa, Italy

    Andrea Maggiolo-Schettini & Angelo Troina

Authors
  1. Ruggero Lanotte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrea Maggiolo-Schettini
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Angelo Troina
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ruggero Lanotte.

Additional information

A preliminary version of this paper was presented at SEFM’04 [LMT04].

05 April 2006

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lanotte, R., Maggiolo-Schettini, A. & Troina, A. Parametric probabilistic transition systems for system design and analysis. Form Asp Comp 19, 93–109 (2007). https://doi.org/10.1007/s00165-006-0015-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00165-006-0015-2

Keywords

Search

Navigation