Input/Output Stochastic Automata

Compositionality and Determinism
  • Pedro R. D’Argenio
  • Matias David Lee
  • Raúl E. Monti
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9884)


Stochastic automata provide a way to symbolically model systems in which the occurrence time of events may respond to any continuous random variable. We introduce here an input/output variant of stochastic automata that, once the model is closed —i.e., all synchronizations are resolved—, the resulting automaton does not contain non-deterministic choices. This is important since fully probabilistic models are amenable to simulation in the general case and to much more efficient analysis if restricted to Markov models. We present here a theoretical introduction to input/output stochastic automata (IOSA) for which we (i) provide a concrete semantics in terms of non-deterministic labeled Markov processes (NLMP), (ii) prove that bisimulation is a congruence for parallel composition both in NLMP and IOSA, (iii) show that parallel composition commutes in the symbolic and concrete level, and (iv) provide a proof that a closed IOSA is indeed deterministic.



We thank Pedro Sánchez Terraf for the help provided in measure theory, and Carlos E. Budde for early discussions on IOSAs.


  1. 1.
    Ash, R., Doléans-Dade, C.: Probability and Measure Theory. Academic Press, Cambridge (2000)zbMATHGoogle Scholar
  2. 2.
    Bogdoll, J., Ferrer Fioriti, L.M., Hartmanns, A., Hermanns, H.: Partial order methods for statistical model checking and simulation. In: Bruni, R., Dingel, J. (eds.) FORTE 2011 and FMOODS 2011. LNCS, vol. 6722, pp. 59–74. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Bohnenkamp, H.C., D’Argenio, P.R., Hermanns, H., Katoen, J.: MODEST: a compositional modeling formalism for hard and softly timed systems. IEEE Trans. Softw. Eng. 32(10), 812–830 (2006)CrossRefGoogle Scholar
  4. 4.
    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
  5. 5.
    Budde, C.E., D’Argenio, P.R., Hermanns, H.: Rare event simulation with fully automated importance splitting. In: Beltrán, M., Knottenbelt, W.J., Bradley, J.T. (eds.) Computer Performance Engineering. LNCS, vol. 9272, pp. 275–290. Springer International Publishing, Switzerland (2015)CrossRefGoogle Scholar
  6. 6.
    Crouzen, P.: Modularity and Determinism in Compositional Markov Models. Ph.D. thesis, Universität des Saarlandes, Saarbrücken (2014)Google Scholar
  7. 7.
    D’Argenio, P.R.: Algebras and Automata for Timed and Stochastic Systems. Ph.D. thesis, University of Twente, Enschede (1999)Google Scholar
  8. 8.
    D’Argenio, P.R., Katoen, J.P.: A theory of stochastic systems part I: stochastic automata. Inf. Comput. 203(1), 1–38 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    D’Argenio, P.R., Legay, A., Sedwards, S., Traonouez, L.: Smart sampling for lightweight verification of Markov decision processes. STTT 17(4), 469–484 (2015)CrossRefGoogle Scholar
  10. 10.
    D’Argenio, P.R., Sánchez Terraf, P., Wolovick, N.: Bisimulations for non-deterministic labelled Markov processes. Math. Struct. Comput. Sci. 22(1), 43–68 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. Inf. Comput. 179(2), 163–193 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Doberkat, E.E., Sánchez Terraf, P.: Stochastic non-determinism and effectivity functions. J. Logic Comput. (2015, to appear). doi: 10.1093/logcom/exv049
  13. 13.
    Gburek, D., Baier, C., Klüppelholz, S.: Composition of stochastic transition systems based on spans and couplings. In: ICALP 2016. LIPICS (2016, to appear)Google Scholar
  14. 14.
    Giry, M.: A categorical approach to probability theory. In: Banaschewski, B. (ed.) Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics, vol. 915, pp. 68–85. Springer, Heidelberg (1982)CrossRefGoogle Scholar
  15. 15.
    van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative and stratified models of probabilistic processes. Inf. Comput. 121(1), 59–80 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Hartmanns, A., Timmer, M.: Sound statistical model checking for MDP using partial order and confluence reduction. STTT 17(4), 429–456 (2015)CrossRefGoogle Scholar
  17. 17.
    Hermanns, H.: Interactive Markov Chains: and the Quest for Quantified Quality. Springer, Heidelberg (2002)CrossRefzbMATHGoogle Scholar
  18. 18.
    Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. Inf. Comput. 94(1), 1–28 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Milner, R.: Communication and Concurrency. Prentice-Hall Inc., Upper Saddle River (1989)zbMATHGoogle Scholar
  20. 20.
    Viglizzo, I.: Coalgebras on Measurable Spaces. Ph.D. thesis, Indiana University, Argentina (2005)Google Scholar
  21. 21.
    Wolovick, N.: Continuous Probability and Nondeterminism in Labeled Transition Systems. Ph.D. thesis, Universidad Nacional de Córdoba, Argentina (2012)Google Scholar
  22. 22.
    Wu, S., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1–2), 1–38 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Yi, W.: Real-time behaviour of asynchronous agents. In: Baeten, J.C.M., Klop, J.W. (eds.) CONCUR ’90 Theories of Concurrency: Unification and Extension. LNCS, vol. 458, pp. 502–520. Springer, Heidelberg (1990)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Pedro R. D’Argenio
    • 1
  • Matias David Lee
    • 2
  • Raúl E. Monti
    • 1
  1. 1.CONICETUniversidad Nacional de CórdobaCórdobaArgentina
  2. 2.LIP, Université de Lyon, CNRS, ENS de Lyon, Inria, UCBLLyonFrance

Personalised recommendations