A Theory for the Semantics of Stochastic and Non-deterministic Continuous Systems

  • Carlos E. Budde
  • Pedro R. D’Argenio
  • Pedro Sánchez Terraf
  • Nicolás Wolovick

Abstract

The description of complex systems involving physical or biological components usually requires to model complex continuous behavior induced by variables such as time, distance, speed, temperature, alkalinity of a solution, etc. Often, such variables can be quantified probabilistically to better understand the behavior of the complex systems. For example, the arrival time of events may be considered a Poisson process or the weight of an individual may be assumed to be distributed according to a log-normal distribution. However, it is also common that the uncertainty on how these variables behave makes us prefer to leave out the choice of a particular probability and rather model it as a purely non-deterministic decision, as it is the case when a system is intended to be deployed in a variety of very different computer or network architectures. Therefore, the semantics of these systems needs to be represented by a variant of probabilistic automata that involves continuous domains on the state space and the transition relation.

In this paper, we provide a survey on the theory of such kind of models. We present the theory of the so-called labeled Markov processes (LMP) and its extension with internal non-determinism (NLMP). We show that in these complex domains, the bisimulation relation can be understood in different manners. We show the relation between the different bisimulations and try to understand their expressiveness through examples. We also study variants of Hennessy-Milner logic that provides logical characterizations of some of these bisimulations.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ash, R., Doléans-Dade, C.: Probability & Measure Theory. Academic Press (2000)Google Scholar
  2. 2.
    Baier, C., Katoen, J.: Principles of Model Checking. The MIT Press (2008)Google Scholar
  3. 3.
    Bohnenkamp, H., D’Argenio, P., Hermanns, H., Katoen, J.P.: MoDeST: A compositional modeling formalism for real-time and stochastic systems. IEEE Trans. Softw. Eng. 32(10), 812–830 (2006)CrossRefGoogle Scholar
  4. 4.
    Bouchard-Côté, A., Ferns, N., Panangaden, P., Precup, D.: An approximation algorithm for labelled markov processes: towards realistic approximation. In: Proc. of QEST 2005, pp. 54–62. IEEE Computer Society (2005)Google Scholar
  5. 5.
    Bravetti, M.: Specification and Analysis of Stochastic Real-Time Systems. Ph.D. thesis, Università di Bologna, Padova, Venezia (2002)Google Scholar
  6. 6.
    Bravetti, M., D’Argenio, P.R.: Tutte le algebre insieme: Concepts, discussions and relations of stochastic process algebras with general distributions. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 44–88. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    Budde, C.: No determinismo completamente medible en procesos probabilísticos continuos. Master’s thesis, FaMAF, Universidad Nacional de Córdoba (2012)Google Scholar
  8. 8.
    Cattani, S.: Trace-based Process Algebras for Real-Time Probabilistic Systems. Ph.D. thesis, University of Birmingham (2005)Google Scholar
  9. 9.
    Cattani, S., Segala, R., Kwiatkowska, M., Norman, G.: Stochastic transition systems for continuous state spaces and non-determinism. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 125–139. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  10. 10.
    Celayes, P.: Procesos de Markov Etiquetados sobre Espacios de Borel Estándar. Master’s thesis, FaMAF, Universidad Nacional de Córdoba (2006)Google Scholar
  11. 11.
    Chaput, P., Danos, V., Panangaden, P., Plotkin, G.: Approximating labelled markov processes again! In: Kurz, A., Lenisa, M., Tarlecki, A. (eds.) CALCO 2009. LNCS, vol. 5728, pp. 145–156. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press (1999)Google Scholar
  13. 13.
    Danos, V., Desharnais, J.: Labelled markov processes: Stronger and faster approximations. In: Proc. of 18th LICS, pp. 341–350. IEEE Computer Society (2003)Google Scholar
  14. 14.
    Danos, V., Desharnais, J., Laviolette, F., Panangaden, P.: Bisimulation and cocongruence for probabilistic systems. Inf. & Comp. 204, 503–523 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    D’Argenio, P.: Algebras and Automata for Timed and Stochastic Systems. Ph.D. thesis, Department of Computer Science, University of Twente (1999)Google Scholar
  16. 16.
    D’Argenio, P., Katoen, J.P.: A theory of stochastic systems, Part I: Stochastic automata, and Part II: Process algebra. Inf. & Comp. 203(1), 1–38, 39–74 (2005)Google Scholar
  17. 17.
    D’Argenio, P., Sánchez Terraf, P., Wolovick, N.: Bisimulations for non-deterministic labelled Markov processes. Mathematical. Structures in Comp. Sci. 22(1), 43–68 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    D’Argenio, P., Wolovick, N., Sánchez Terraf, P., Celayes, P.: Nondeterministic labeled Markov processes: Bisimulations and logical characterization. In: Proc. of QEST 2009, pp. 11–20. IEEE Computer Society (2009)Google Scholar
  19. 19.
    Delahaye, B., Katoen, J.-P., Larsen, K.G., Legay, A., Pedersen, M.L., Sher, F., Wąsowski, A.: Abstract probabilistic automata. In: Jhala, R., Schmidt, D. (eds.) VMCAI 2011. LNCS, vol. 6538, pp. 324–339. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Desharnais, J.: Labeled Markov Process. Ph.D. thesis, McGill University (1999)Google Scholar
  21. 21.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. & Comp. 179(2), 163–193 (2002)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Desharnais, J., Laviolette, F., Turgeon, A.: A logical duality for underspecified probabilistic systems. Inf. Comput. 209(5), 850–871 (2011)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Approximating labelled markov processes. Inf. Comput. 184(1), 160–200 (2003)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled markov processes. Theor. Comput. Sci. 318(3), 323–354 (2004)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Doberkat, E.E.: Kleisli morphisms and randomized congruences for the Giry monad. Journal of Pure and Applied Algebra 211(3), 638–664 (2007)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Doberkat, E.E.: Stochastic relations. Foundations for Markov transition systems. Studies in Informatics Series. Chapman & Hall/CRC (2007)Google Scholar
  27. 27.
    Fränzle, M., Hahn, E., Hermanns, H., Wolovick, N., Zhang, L.: Measurability and safety verification for stochastic hybrid systems. In: Caccamo, M., Frazzoli, E., Grosu, R. (eds.) Proc. of HSCC 2011, pp. 43–52. ACM (2011)Google Scholar
  28. 28.
    Giry, M.: A categorical approach to probability theory. In: Categorical Aspects of Topology and Analysis. LNM, vol. 915, pp. 68–85. Springer (1981)Google Scholar
  29. 29.
    van Glabeek, R.: The linear time–branching time spectrum I. The semantics of concrete, sequential processes. In: Bergstra, J.A., Ponse, A., Smolka, S.A. (eds.) Handbook of Process Algebra, pp. 3–99. North-Holland (2001)Google Scholar
  30. 30.
    Baró Graf, H., Hermanns, H., Kulshrestha, J., Peter, J., Vahldiek, A., Vasudevan, A.: A verified- wireless safety critical hard real-time design. In: Proc. of WOWMOM 2011, pp. 1–9. IEEE (2011)Google Scholar
  31. 31.
    Hahn, E., Hartmanns, A., Hermanns, H., Katoen, J.P.: A compositional modeling and analysis framework for stochastic hybrid systems. Formal Methods in System Design 43(2), 191–232 (2013)CrossRefMATHGoogle Scholar
  32. 32.
    Hartmanns, A., Hermanns, H.: Modelling and decentralised runtime control of self-stabilising power micro grids. In: Margaria, T., Steffen, B. (eds.) ISoLA 2012, Part I. LNCS, vol. 7609, pp. 420–439. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  33. 33.
    Hennessy, M., Milner, R.: Algebraic laws for nondeterminism and concurrency. J. ACM 32(1), 137–161 (1985)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Kechris, A.: Classical Descriptive Set Theory. Graduate Texts in Mathematics, vol. 156. Springer (1995)Google Scholar
  35. 35.
    Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Inf. & Comp. 94(1), 1–28 (1991)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer (2005)Google Scholar
  37. 37.
    Milner, R.: Communication and Concurrency. Prentice Hall (1989)Google Scholar
  38. 38.
    Naimpally, S.: What is a hit-and-miss topology? Topological Comment 8(1) (2003)Google Scholar
  39. 39.
    Panangaden, P.: Labelled Markov Processes. Imperial College Press (2009)Google Scholar
  40. 40.
    Parma, A., Segala, R.: Logical characterizations of bisimulations for discrete probabilistic systems. In: Seidl, H. (ed.) FOSSACS 2007. LNCS, vol. 4423, pp. 287–301. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  41. 41.
    Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley-Interscience (1994)Google Scholar
  42. 42.
    Sánchez Terraf, P.: Unprovability of the logical characterization of bisimulation. Inf. & Comp. 209(7), 1048–1056 (2011)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Sánchez Terraf, P.: Bisimilarity is not Borel. CoRR arXiv:1211.0967 (2012)Google Scholar
  44. 44.
    Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. Ph.D. thesis, Massachusetts Institute of Technology (1995)Google Scholar
  45. 45.
    Strulo, B.: Process Algebra for Discrete Event Simulation. Ph.D. thesis, Department of Computing, Imperial College, University of London (1993)Google Scholar
  46. 46.
    Vardi, M.: Automatic verification of probabilistic concurrent finite-state programs. In: 26th FOCS, pp. 327–338. IEEE (1985)Google Scholar
  47. 47.
    Viglizzo, I.: Coalgebras on Measurable Spaces. Ph.D. thesis, Department of Mathematics, Indiana University (2005)Google Scholar
  48. 48.
    Wolovick, N.: Continuous probability and nondeterminism in labeled transition systems. Ph.D. thesis, Universidad Nacional de Córdoba (2012)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Carlos E. Budde
    • 1
    • 2
  • Pedro R. D’Argenio
    • 1
    • 2
  • Pedro Sánchez Terraf
    • 1
    • 2
  • Nicolás Wolovick
    • 1
  1. 1.FaMAFUniversidad Nacional de CórdobaArgentina
  2. 2.CONICETCórdobaArgentina

Personalised recommendations