Abstract
Behavioral equivalences and preorders are fundamental notions to formalize indistinguishability of transition systems and provide means to abstraction and refinement. We survey a collection of models used to represent concurrent probabilistic real systems, the behavioral equivalences and preorders they are equipped with and the corresponding decision algorithms. These algorithms follow the standard refinement approach and they improve their complexity by taking advantage of the efficient algorithms developed in the optimization community to solve optimization and flow problems.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Ahuja, R.K., Magnanti, T.J., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall (1993)
Andova, S., Willemse, T.A.C.: Branching bisimulation for probabilistic systems: Characteristics and decidability. TCS 356(3), 325–355 (2006)
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.K.: Model-checking continuous-time Markov chains. ACM Transactions on Computational Logic 1(1), 162–170 (2000)
Baier, C., Engelen, B., Majster-Cederbaum, M.: Deciding bisimilarity and similarity for probabilistic processes. J. Computer and Systems Science 60(1), 187–231 (2000)
Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.P.: Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)
Baier, C., Hermanns, H.: Weak bisimulation for fully probabilistic processes. In: Grumberg, O. (ed.) CAV 1997. LNCS, vol. 1254, pp. 119–130. Springer, Heidelberg (1997)
Baier, C., Hermanns, H., Katoen, J.P., Haverkort, B.R.: Efficient computation of time-bounded reachability probabilities in uniform continuous-time Markov decision processes. TCS 345(1), 2–26 (2005)
Baier, C., Katoen, J.P.: Principles of Model Checking. The MIT Press (2008)
Baier, C., Katoen, J.P., Hermanns, H., Wolf, V.: Comparative branching-time semantics for Markov chains. I&C 200(2), 149–214 (2005)
Bellman, R.: A Markovian decision process. Indiana University Mathematics Journal 6, 679–684 (1957)
Bertsekas, D.P.: Dynamic Programming and Optimal Control. Athena Scientific (2005)
Bertsimas, D., Tsitsiklis, J.N.: Introduction to Linear Optimization. Athena Scientific (1997)
Cattani, S., Segala, R.: Decision algorithms for probabilistic bisimulation. In: Brim, L., Jančar, P., Křetínský, M., Kučera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 371–385. Springer, Heidelberg (2002)
Clarke, E.M., Grumberg, O., Long, D.E.: Model checking and abstraction. ACM Transactions on Programming Languages and Systems 16(5), 1512–1542 (1994)
Crafa, S., Ranzato, F.: Probabilistic bisimulation and simulation algorithms by abstract interpretation. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part II. LNCS, vol. 6756, pp. 295–306. Springer, Heidelberg (2011)
Deng, Y.: Axiomatisations and Types for Probabilistic and Mobile Processes. Ph.D. thesis, École des Mines de Paris (2005)
Deng, Y., Hennessy, M.: On the semantics of Markov automata. I&C 222, 139–168 (2012)
Desharnais, J.: Labelled Markov Processes. Ph.D. thesis, McGill University (1999)
Eisentraut, C., Hermanns, H., Katoen, J.-P., Zhang, L.: A semantics for every GSPN. In: Colom, J.-M., Desel, J. (eds.) PETRI NETS 2013. LNCS, vol. 7927, pp. 90–109. Springer, Heidelberg (2013)
Eisentraut, C., Hermanns, H., Krämer, J., Turrini, A., Zhang, L.: Deciding bisimilarities on distributions. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 72–88. Springer, Heidelberg (2013)
Eisentraut, C., Hermanns, H., Schuster, J., Turrini, A., Zhang, L.: The quest for minimal quotients for probabilistic automata. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 16–31. Springer, Heidelberg (2013)
Eisentraut, C., Hermanns, H., Zhang, L.: Concurrency and composition in a stochastic world. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 21–39. Springer, Heidelberg (2010)
Eisentraut, C., Hermanns, H., Zhang, L.: On probabilistic automata in continuous time. In: LICS, pp. 342–351 (2010)
Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comp. 18(1), 30–55 (1989)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. ACM 35(4), 921–940 (1988)
Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)
Hashemi, V., Hermanns, H., Turrini, A.: On the efficiency of deciding probabilistic automata weak bisimulation. ECEASST 66 (2013)
Hermanns, H.: Interactive Markov Chains. LNCS, vol. 2428. Springer, Heidelberg (2002)
Hermanns, H., Turrini, A.: Deciding probabilistic automata weak bisimulation in polynomial time. In: FSTTCS, pp. 435–447 (2012)
Hermanns, H., Turrini, A.: Cost preserving bisimulations for probabilistic automata. In: D’Argenio, P.R., Melgratti, H. (eds.) CONCUR 2013 – Concurrency Theory. LNCS, vol. 8052, pp. 349–363. Springer, Heidelberg (2013)
Howard, R.A.: Dynamic Programming and Markov Processes. John Wiley and Sons, Inc. (1960)
Howard, R.A.: Dynamic Probabilistic Systems: Semi-Markov and Decision Processes, vol. II. Dover Publications (2007)
Jansen, D.N., Song, L., Zhang, L.: Revisiting weak simulation for substochastic Markov chains. In: Joshi, K., Siegle, M., Stoelinga, M., D’Argenio, P.R. (eds.) QEST 2013. LNCS, vol. 8054, pp. 209–224. Springer, Heidelberg (2013)
Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: LICS, pp. 266–277 (1991)
Kanellakis, P.C., Smolka, S.A.: CCS expressions, finite state processes, and three problems of equivalence. I&C 86(1), 43–68 (1990)
Katoen, J.-P., Kemna, T., Zapreev, I., Jansen, D.N.: Bisimulation minimisation mostly speeds up probabilistic model checking. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 87–101. Springer, Heidelberg (2007)
Knast, R.: Continuous-time probabilistic automata. Information and Control 15(4), 335–352 (1969)
Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing (preliminary report). In: POPL, pp. 344–352 (1989)
Milner, R.: Communication and Concurrency. Prentice-Hall International, Englewood Cleiffs (1989)
Milner, R.: Communicating and Mobile Systems: the π-calculus. Cambridge University Press (1999)
Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. on Computing 16(6), 973–989 (1987)
Peterson, M.: An Introduction to Decision Theory. Cambridge University Press (2009)
Philippou, A., Lee, I., Sokolsky, O.: Weak bisimulation for probabilistic systems. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 334–349. Springer, Heidelberg (2000)
Puterman, M.L.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. Wiley Series in Probability and Statistics, vol. (594). John Wiley & Sons, Inc. (2005)
Sack, J., Zhang, L.: A general framework for probabilistic characterizing formulae. In: Kuncak, V., Rybalchenko, A. (eds.) VMCAI 2012. LNCS, vol. 7148, pp. 396–411. Springer, Heidelberg (2012)
Schuster, J., Siegle, M.: Markov automata: Deciding weak bisimulation by means of “non-naïvely” vanishing states. I&C (to appear, 2014), http://dx.doi.org/10.1016/j.ic.2014.02.001
Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. Ph.D. thesis, MIT (1995)
Segala, R.: Probability and nondeterminism in operational models of concurrency. In: Baier, C., Hermanns, H. (eds.) CONCUR 2006. LNCS, vol. 4137, pp. 64–78. Springer, Heidelberg (2006)
Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)
Segala, R., Lynch, N.A.: Probabilistic simulations for probabilistic processes. Nordic J. Computing 2(2), 250–273 (1995)
Segala, R., Turrini, A.: Comparative analysis of bisimulation relations on alternating and non-alternating probabilistic models. In: QEST, pp. 44–53 (2005)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press (1994)
Todd, M.J.: The many facets of linear programming. Mathematical Programming 91(3), 417–436 (2002)
Wolovick, N., Johr, S.: A characterization of meaningful schedulers for continuous-time Markov decision processes. In: Asarin, E., Bouyer, P. (eds.) FORMATS 2006. LNCS, vol. 4202, pp. 352–367. Springer, Heidelberg (2006)
Zhang, L.: Decision Algorithm for Probabilistic Simulations. Ph.D. thesis, Saarland University (2008)
Zhang, L., Hermanns, H.: Deciding simulations on probabilistic automata. In: Namjoshi, K.S., Yoneda, T., Higashino, T., Okamura, Y. (eds.) ATVA 2007. LNCS, vol. 4762, pp. 207–222. Springer, Heidelberg (2007)
Zhang, L., Hermanns, H., Eisenbrand, F., Jansen, D.N.: Flow faster: Efficient decision algorithms for probabilistic simulations. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 155–169. Springer, Heidelberg (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Gebler, D., Hashemi, V., Turrini, A. (2014). Computing Behavioral Relations for Probabilistic Concurrent Systems. In: Remke, A., Stoelinga, M. (eds) Stochastic Model Checking. Rigorous Dependability Analysis Using Model Checking Techniques for Stochastic Systems. ROCKS 2012. Lecture Notes in Computer Science, vol 8453. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-45489-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-662-45489-3_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-45488-6
Online ISBN: 978-3-662-45489-3
eBook Packages: Computer ScienceComputer Science (R0)