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Probabilistic Automata: System Types, Parallel Composition and Comparison

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Part of the Lecture Notes in Computer Science book series (LNCS,volume 2925)

Abstract

We survey various notions of probabilistic automata and probabilistic bisimulation, accumulating in an expressiveness hierarchy of probabilistic system types. The aim of this paper is twofold: On the one hand it provides an overview of existing types of probabilistic systems and, on the other hand, it explains the relationship between these models. We overview probabilistic systems with discrete probabilities only. The expressiveness order used to built the hierarchy is defined via the existence of mappings between the corresponding system types that preserve and reflect bisimilarity. Additionally, we discuss parallel composition for the presented types of systems, augmenting the map of probabilistic automata with closedness under this compositional operator.

Keywords

  • probabilistic automata (transition systems)
  • probabilistic bisimulation
  • preservation and reflection of bisimulation
  • non-determinism
  • parallel composition

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References

  1. Alur, R., Henzinger, T.A.: Reactive modules. Formal Methods in System Design 15, 7–48 (1999); A preliminary version appeared in the Proceedings of the 11th Annual Symposium on Logic in Computer Science (LICS), pp. 207–218. IEEE Computer Society Press, Los Alamitos (1996)

    Google Scholar 

  2. de Alfaro, L., Henzinger, T.A., Jhala, R.: Compositional methods for probabilistic systems. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 351–365. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  3. de Alfaro, L.: Formal verification of probabilistic systems, Ph.D. thesis, Stanford University (1997)

    Google Scholar 

  4. de Alfaro, L.: Stochastic transition systems. In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 423–438. Springer, Heidelberg (1998)

    CrossRef  Google Scholar 

  5. Andova, S.: Process algebra with probabilistic choice. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, p. 111. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  6. Andova, S.: Probabilistic process algebra. Ph.D. thesis, Eindhoven University of Technology (2002)

    Google Scholar 

  7. Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026. Springer, Heidelberg (1995)

    Google Scholar 

  8. Baier, C.: Polynomial time algorithms for testing probabilistic bisimulation and simulation. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 38–49. Springer, Heidelberg (1996)

    Google Scholar 

  9. Baier, C.: On algorithmic verification methods for probabilistic systems, Habilitationsschrift, FMI, Universitaet Mannheim (1998)

    Google Scholar 

  10. Baeten, J.C.M., Bergstra, J.A., Smolka, S.A.: Axiomatizing probabilistic processes. ACP with generative probabilities, Information and Computation 121(2), 234–255 (1995)

    MATH  MathSciNet  Google Scholar 

  11. Blute, R., Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. In: LICS 1997, pp. 149–158 (1997)

    Google Scholar 

  12. Baier, C., D’Argenio, P.R., Hermanns, H., Katoen, J.-P.: How to cook a probabilistic process calculus (unpublished) (1999)

    Google Scholar 

  13. Baier, C., Engelen, B., Majster-Cederbaum, M.: Deciding bisimilarity and similarity for probabilistic processes. Journal of Computer and System Sciences 60, 187–231 (1999)

    CrossRef  MathSciNet  Google Scholar 

  14. Bernardo, M.: Theory and application of extended Markovian process algebra, Ph.D. thesis, University of Bologna (1999)

    Google Scholar 

  15. Bernardo, M., Gorrieri, R.: A tutorial on EMPA: A theory of concurrent processes with nondeterminism, priorities, probabilities and time. Theoretical Computer Science 202(1), 1–54 (1998)

    MATH  CrossRef  MathSciNet  Google Scholar 

  16. Brinksma, E., Hermanns, H.: Process algebra and markov chains. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) EEF School 2000 and FMPA 2000. LNCS, vol. 2090, pp. 183–232. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  17. Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model checking continuous-time Markov chains by transient analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  18. Brookes, S.D., Hoare, C.A.R., Roscoe, A.W.: A theory of communicating sequential processes. Journal of the ACM 31, 560–599 (1984)

    MATH  CrossRef  MathSciNet  Google Scholar 

  19. Bergstra, J.A., Klop, J.W.: Algebra of communicating processes with abstraction. Theoretical Computer Science 37, 77–121 (1985)

    MATH  CrossRef  MathSciNet  Google Scholar 

  20. Baier, C., Kwiatkowska, M.Z.: Domain equations for probabilistic processes. In: 4th Workshop on Expressiveness in Concurrency (EXPRESS 1997), Santa Margherita. Electronic Notes in Theoretical Computer Science, vol. 7 (1997)

    Google Scholar 

  21. Baier, C., Kwiatkowska, M.Z.: Domain equations for probabilistic processes. Mathematical Structures in Computer Science 10, 665–717 (2000)

    MATH  CrossRef  MathSciNet  Google Scholar 

  22. Benveniste, A., Levy, B.C., Fabre, E., Le Guernic, P.: A calculus of stochastic systems for the specification, simulation, and hidden state estimation of mixed stochastic/non-stochastic systems. Theoretical Computer Science 152, 171–217 (1995)

    MATH  CrossRef  MathSciNet  Google Scholar 

  23. Bloom, B., Meyer, A.R.: A remark on bisimulation between probabilistic processes. In: Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 363, pp. 26–40. Springer, Heidelberg (1989)

    Google Scholar 

  24. Baier, C., Stoelinga, M.I.A.: Norm functions for probabilistic bisimulations with delays. In: Tiuryn, J. (ed.) FOSSACS 2000. LNCS, vol. 1784, pp. 1–16. Springer, Heidelberg (2000)

    CrossRef  Google Scholar 

  25. Bandini, E., Segala, R.: Axiomatizations for probabilistic bisimulation. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 370–381. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  26. Bartels, F., Sokolova, A., de Vink, E.P.: A hierarchy of probabilistic system types. In: Peter Gumm, H. (ed.) Electronic Notes in Theoretical Computer Science, vol. 82. Elsevier, Amsterdam (2003)

    Google Scholar 

  27. Buchholz, P.: Markovian process algebra: Composition and equivalence. In: Proc. of PAPM 1994, Erlangen, Germany, pp. 11–30 (1994)

    Google Scholar 

  28. Christoff, L., Christoff, I.: Efficient algorithms for verification of equivalences for probabilistic processes. In: Larsen, K.G., Skou, A. (eds.) CAV 1991. LNCS, vol. 575. Springer, Heidelberg (1992)

    Google Scholar 

  29. 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 

  30. Cleaveland, R., Smolka, S.A., Zwarico, A.: Testing preorders for probabilistic processes. In: Kuich, W. (ed.) ICALP 1992. LNCS, vol. 623, pp. 708–719. Springer, Heidelberg (1992)

    Google Scholar 

  31. Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM (JACM) 42, 857–907 (1995)

    MATH  CrossRef  MathSciNet  Google Scholar 

  32. D’Argenio, P.R.: Algebras and automata for timed and stochastic system, Ph.D. thesis, University of Twente (1999)

    Google Scholar 

  33. Desharnais, J., Edalat, A., Panangaden, P.: A logical characterization of bisimulation for labeled Markov processes. In: Proc. LICS 1998, Indianapolis, pp. 478–487 (1998)

    Google Scholar 

  34. Derman, C.: Finite state Markovian decision proceses. Academic Press, London (1970)

    Google Scholar 

  35. D’Argenio, P., Hermanns, H., Katoen, J.-P.: On generative parallel composition. In: Proc. PROBMIV 1998. ENTCS, vol. 22, pp. 105–122 (1998)

    Google Scholar 

  36. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  37. D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reduction and refinement strategies for probabilistic analysis. In: Hermanns, H., Segala, R. (eds.) PAPMPROBMIV 2002. LNCS. Springer, Heidelberg (2002)

    Google Scholar 

  38. Giacalone, A., Jou, C., Smolka, S.: Algebraic reasoning for probabilistic concurrent systems. In: Broy, M., Jones, C.B. (eds.) Proc. of the Working Conf. on Programming Concepts and Methods, 1990, North Holland, pp. 443–458 (1990)

    Google Scholar 

  39. van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative, and stratified models of probabilistic processes. Information and Computation 121, 59–80 (1995)

    MATH  CrossRef  MathSciNet  Google Scholar 

  40. van Glabbeek, R.J., Smolka, S.A., Steffen, B., Tofts, C.M.N.: Reactive, generative, and stratified models of probabilistic processes. Logic in Computer Science, 130–141 (1990)

    Google Scholar 

  41. Hansson, H.A.: Time and probability in formal design of distributed systems, Ph.D. thesis, Uppsala University, Department of Computer Systems (1991); Also appeared in Real-Time Safety Critical Systems, vol. 1. Elsevier, Amsterdam (1994)

    Google Scholar 

  42. den Hartog, J.I.: Probabilistic extensions of semantical models, Ph.D. thesis, Vrije Universiteit Amsterdam (2002)

    Google Scholar 

  43. Haverkort, B.R.: Markovian models for performance and dependability evaluation. In: Brinksma, E., Hermanns, H., Katoen, J.-P. (eds.) EEF School 2000 and FMPA 2000. LNCS, vol. 2090, pp. 38–84. Springer, Heidelberg (2001)

    CrossRef  Google Scholar 

  44. Hermanns, H.: Interactive Markov chains, Ph.D. thesis, Universi’́at Erlangen- N’́urnberg (1998); (Revised version appeared) Hermanns, H. (ed.): Interactive Markov Chains. LNCS, vol. 2428, p. 35. Springer, Heidelberg (2002)

    MATH  CrossRef  Google Scholar 

  45. Hillston, J.: A compositional approach to performance modelling, Ph.D. thesis, University of Edinburgh (1994); Also appeared in the CPHC/BCS Distinguished Dissertation Series, Cambridge University Press, Cambridge (1996)

    Google Scholar 

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

    MATH  CrossRef  Google Scholar 

  47. Hoare, C.A.R.: Communicating Sequential Processes. Prentice Hall, Englewood Cliffs (1985)

    MATH  Google Scholar 

  48. Howard, R.A.: Dynamic probabilistic systems. John Wiley & Sons, Inc., New York (1971)

    Google Scholar 

  49. den Hartog, J.I., de Vink, E.P.: Mixing up nondeterminism and probability: A preliminary report. In: Baier, C., Huth, M., Kwiatkowska, M., Ryan, M. (eds.) Proc. PROBMIV 1998. ENTCS, vol. 22 (1998)

    Google Scholar 

  50. den Hartog, J.I., de Vink, E.P.: Verifying probabilistic programs using a Hoare-like logic. International Journal of Foundations of Computer Science 13, 315–340 (2002)

    MATH  CrossRef  MathSciNet  Google Scholar 

  51. Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: Proceedings of Sixth Annual IEEE Symposium on Logic in Computer Science LICS 1991. IEEE, Los Alamitos (1991)

    Google Scholar 

  52. Jonsson, B., Larsen, K.G., Yi, W.: Probabilistic extensions of process algebras. In: Handbook of Process Algebras. Elsevier, North Holland (2001)

    Google Scholar 

  53. Jacobs, B.P.F., Rutten, J.J.M.M.: A tutorial on (co)algebras and (co)induction. Bulletin of the EATCS 62, 222–259 (1996)

    Google Scholar 

  54. Jou, C.-C., Smolka, S.A.: Equivalences, congruences and complete axiomatizations for probabilistic processes. In: Baeten, J.C.M., Klop, J.W. (eds.) Proceedings of CONCUR 1990, pp. 367–383. Springer, Heidelberg (1990)

    Google Scholar 

  55. Jonnson, B., Yi, W.: Testing preorders for probabilistic processes can be characterized by simulations. Theoretical Computer Science 282, 33–51 (2002)

    CrossRef  MathSciNet  Google Scholar 

  56. Kwiatkowska, M.Z., Norman, G.J.: A testing equivalence for reactive probabilistic processes. In: EXPRESS 1998 Fifth International Workshop on Expressiveness in Concurrency, ENTCS, vol. 16(2) (1998)

    Google Scholar 

  57. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York (1976)

    MATH  Google Scholar 

  58. 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)

    CrossRef  Google Scholar 

  59. Lowe, G.: Probabilistic and prioritized models of timed CSP. Theoretical Computer Science 138, 315–352 (1995)

    MATH  CrossRef  MathSciNet  Google Scholar 

  60. Lewis, L.R., Papadimitriou, C.H.: Elements of the theory of computation. Prentice-Hall, Englewood Cliffs (1981)

    MATH  Google Scholar 

  61. Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)

    MATH  CrossRef  MathSciNet  Google Scholar 

  62. Larsen, K.G., Skou, A.: Compositional verification of probabilistic processes. In: Cleaveland, R. (ed.) CONCUR 1992. LNCS, vol. 630, pp. 456–471. Springer, Heidelberg (1992)

    CrossRef  Google Scholar 

  63. Lynch, N.A., Tuttle, M.: Hierarchical completeness proofs for distributed algorithms. In: Proceedings of the 6th Annual ACM Symposium on Principles of Distributed Computing (1987)

    Google Scholar 

  64. MacLane, S.: Categories for the working mathematician. Springer, Heidelberg (1971)

    Google Scholar 

  65. Milner, R.: Calculi for synchrony and asynchrony. Theoretical Computer Science 25, 267–310 (1983)

    MATH  CrossRef  MathSciNet  Google Scholar 

  66. Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  67. Morgan, C., McIver, A., Seidel, K., Sanders, J.W.: Refinement oriented probability for CSP. Formal aspects of computing 8, 617–647 (1996)

    MATH  CrossRef  Google Scholar 

  68. Moss, L.S.: Coalgebraic logic. Annals of Pure and Applied Logic 96, 277–317 (1999)

    MATH  CrossRef  MathSciNet  Google Scholar 

  69. Norman, G.: Metric semantics for reactive probabilistic processes, Ph.D. thesis, School of Computer Science, University of Birmingham (1997)

    Google Scholar 

  70. Plateau, B., Atif, K.: Stochastic automata network for modeling parallel systems. IEEE Trans. on Software Engineering 17, 1093–1108 (1991)

    CrossRef  MathSciNet  Google Scholar 

  71. Pnueli, A., Zuck, L.: erification of multiprocess probabilistic protocols. Distributed Computing 1(1), 53–72 (1986)

    MATH  CrossRef  Google Scholar 

  72. Pnueli, A., Zuck, L.: Probabilistic verification. Information and Computation 103, 1–29 (1993)

    MATH  CrossRef  MathSciNet  Google Scholar 

  73. Rabin, M.O.: Probabilistic automata. Information and Control 6, 230–245 (1963)

    CrossRef  Google Scholar 

  74. Rutten, J.J.M.M.: Universal coalgebra: A theory of systems. Theoretical Computer Science 249, 3–80 (2000)

    MATH  CrossRef  MathSciNet  Google Scholar 

  75. Stark, E.W., Cleaveland, W.R., Smolka, S.A.: A process-algebraic language for probabilistic I/O automata. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 193–207. Springer, Heidelberg (2003)

    CrossRef  Google Scholar 

  76. Segala, R.: Modeling and verification of randomized distributed real-time systems, Ph.D. thesis, MIT (1995)

    Google Scholar 

  77. Seidel, K.: Probabilistic communicating processes. Theoretical Computer Science 152, 219–249 (1995)

    MATH  CrossRef  MathSciNet  Google Scholar 

  78. 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)

    CrossRef  Google Scholar 

  79. Smolka, S.A., Steffen, B.U.: Priority as extremal probability. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR 1990. LNCS, vol. 458, pp. 456–466. Springer, Heidelberg (1990)

    Google Scholar 

  80. Stoelinga, M.I.A.: Alea jacta est: verification of probabilistic, real-time and parametric systems, Ph.D. thesis, University of Nijmegen, The Netherlands (2002)

    Google Scholar 

  81. Stoelinga, M.I.: An introduction to probabilistic automata. EATCS bulletin 78 (2002)

    Google Scholar 

  82. Stoelinga, M.I.A., Vaandrager, F.W.: Root contention in IEEE 1394. In: Katoen, J.-P. (ed.) AMAST-ARTS 1999, ARTS 1999, and AMAST-WS 1999. LNCS, vol. 1601, pp. 53–75. Springer, Heidelberg (1999)

    CrossRef  Google Scholar 

  83. Stoelinga, M.I.A., Vaandrager, F.W.: 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)

    CrossRef  Google Scholar 

  84. Vardi, M.Y.: Automatic verification of probabilistic concurrent finite state programs. In: Proc. FOCS 1995, Portland, Oregon, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)

    Google Scholar 

  85. de Vink, E.P.: On a functor for probabilistic bisimulation and the preservation of weak pullbacks, Tech. Report IR–444, Vrije Universiteit Amsterdam (1998)

    Google Scholar 

  86. de Vink, E.P., Rutten, J.J.M.M.: Bisimulation for probabilistic transition systems: a coalgebraic approach. Theoretical Computer Science 221, 271–293 (1999)

    MATH  CrossRef  MathSciNet  Google Scholar 

  87. Wu, S.-H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theoretical Computer Science 176, 1–38 (1997)

    MATH  CrossRef  MathSciNet  Google Scholar 

  88. Yi, W., Larsen, K.G.: Testing preorders for probabilistic and non-deterministic processes. In: Protocol Specification, Testing and Verification, Florida, USA, vol. 12, pp. 47–61 (1992)

    Google Scholar 

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Sokolova, A., de Vink, E.P. (2004). Probabilistic Automata: System Types, Parallel Composition and Comparison. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, JP., Siegle, M. (eds) Validation of Stochastic Systems. Lecture Notes in Computer Science, vol 2925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24611-4_1

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  • DOI: https://doi.org/10.1007/978-3-540-24611-4_1

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