Abstract
In the verification of concurrent systems involving probabilities, the aim is to find out the maximum/minimum probability that a given event occurs (examples of such events being “the system reaches a failure state”,“a message is delivered”). Such extremal probabilities are obtained by quantifying over all the possible ways in which the processes may be interleaved. Interleaving choices are considered a particular case of nondeterministic behaviour. Such behaviour is dealt with by considering schedulers that resolve the nondeterministic choices. Each scheduler determines a Markov chain for which actual probabilities can be calculated. In the recent literature on distributed systems, particular attention has been paid to the fact that, in order to obtain accurate results, the analysis must rely on partial information schedulers, instead of full-history dependent schedulers used in the setting of Markov decision processes. In this paper, we present undecidability results for distributed schedulers. These schedulers were devised in previous works, and aim to capture the fact that each process has partial information about the actual state of the system. Some of the undecidability results we present are particularly impressive: in the setting of total information the same problems are inexpensive and, indeed, they are used as preprocessing steps in more general model checking algorithms.
Supported by ANPCyT project PICT 26135 and CONICET project PIP 6391.
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References
Aumann, Y.: Efficient asynchronous consensus with the weak adversary scheduler. In: PODC, pp. 209–218 (1997)
Bianco, A., de Alfaro, L.: Model checking of probabalistic and nondeterministic systems. In: FSTTCS, pp. 499–513 (1995)
Canetti, R., Cheung, L., Kirli Kaynar, D., Lynch, N.A., Pereira, O.: Compositional security for Task-PIOAs. In: CSF, pp. 125–139. IEEE CS, Los Alamitos (2007)
Chatzikokolakis, K., Norman, G., Parker, D.: Bisimulation for demonic schedulers. In: FOSSACS, pp. 318–332 (2009)
Cheung, L.: Reconciling Nondeterministic and Probabilistic Choices. PhD thesis, Radboud Universiteit Nijmegen (2006)
Cheung, L., Lynch, N., Segala, R., Vaandrager, F.: Switched Probabilistic PIOA: Parallel composition via distributed scheduling. Theor. Comput. Sci. 365(1-2), 83–108 (2006)
Ciesinski, F., Baier, C.: LiQuor: A tool for qualitative and quantitative linear time analysis of reactive systems. In: QEST 2006, pp. 131–132. IEEE CS, Los Alamitos (2006)
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)
de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University (1997), Technical report STAN-CS-TR-98-1601
PRISM development team. Prism case studies, http://www.prismmodelchecker.org/casestudies/index.php
Dijkstra, E.W.: Self-stabilizing systems in spite of distributed control. Commun. ACM 17(11), 643–644 (1974)
Giro, S., D’Argenio, P.R.: Quantitative model checking revisited: neither decidable nor approximable. In: Raskin, J.-F., Thiagarajan, P.S. (eds.) FORMATS 2007. LNCS, vol. 4763, pp. 179–194. Springer, Heidelberg (2007)
Giro, S., D’Argenio, P.R.: On the expressive power of schedulers in distributed probabilistic systems. In: Proc. of QAPL 2009 (2009). Extended version to appear in ENTCS, cs.famaf.unc.edu.ar/~sgiro/QAPL09-ext.pdf
Giro, S., D’Argenio, P.R.: On the verification of probabilistic i/o automata with unspecified rates. In: SAC 2009: Proceedings of the 2009 ACM symposium on Applied Computing, pp. 582–586. ACM, New York (2009)
Giro, S.: On the automatic verification of Distributed Probabilistic Automata with Partial Information. PhD thesis, Universidad Nacional de Córdoba (to appear)
van Glabbeek, R.J., Smolka, S.A., Steffen, B.: Reactive, generative, and stratified models of probabilistic processes. Information and Computation 121, 59–80 (1995)
Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A tool for automatic verification of probabilistic systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)
Lynch, N.A., Tuttle, M.R.: An introduction to input/output automata. CWI Quarterly 2(3), 219–246 (1989)
Madani, O., Hanks, S., Condon, A.: On the undecidability of probabilistic planning and related stochastic optimization problems. Artif. Intell. 147(1-2), 5–34 (2003)
Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Laboratory for Computer Science, MIT (1995)
Sipser, M.: Introduction to the Theory of Computation, 2nd edn., pp. 199–205. Thomson Course Technology (2005)
Vardi, M.Y.: Automatic verification of probabilistic concurrent finite state programs. In: Procs. of 26th FOCS, pp. 327–338. IEEE Press, Los Alamitos (1985)
Wu, S.-H., Smolka, S.A., Stark, E.W.: Composition and behaviors of probabilistic I/O automata. Theor. Comput. Sci. 176(1-2), 1–38 (1997)
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Giro, S. (2009). Undecidability Results for Distributed Probabilistic Systems. In: Oliveira, M.V.M., Woodcock, J. (eds) Formal Methods: Foundations and Applications. SBMF 2009. Lecture Notes in Computer Science, vol 5902. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-10452-7_15
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DOI: https://doi.org/10.1007/978-3-642-10452-7_15
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