Abstract
Quantitative model checking computes the probability values of a given property quantifying over all possible schedulers. It turns out that maximum and minimum probabilities calculated in such a way are overestimations on models of distributed systems in which components are loosely coupled and share little information with each other (and hence arbitrary schedulers may result too powerful). Therefore, we focus on the quantitative model checking problem restricted to distributed schedulers that are obtained only as a combination of local schedulers (i.e. the schedulers of each component) and show that this problem is undecidable. In fact, we show that there is no algorithm that can compute an approximation to the maximum probability of reaching a state within a given bound when restricted to distributed schedulers.
Supported by the CONICET/CNRS Cooperation project “Métodos para la Verificación de Programas Concurrentes con aspectos Aleatorios y Temporizados”, ANPCyT project PICT 26135 and CONICET project PIP 6391. The first author was partially supported by the LSIS, UMR CNRS 6168, France.
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References
Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) Foundations of Software Technology and Theoretical Computer Science. LNCS, vol. 1026, pp. 288–299. Springer, Heidelberg (1995)
Cheung, L.: Reconciling Nondeterministic and Probabilistic Choices. PhD thesis, Radboud Universiteit Nijmegen (2006)
Cheung, L., Lynch, N., Segala, R., Vaandrager, F.W.: Switched Probabilistic PIOA: Parallel composition via distributed scheduling. Theoretical Computer Science 365(1-2), 83–108 (2006)
Ciesinski, F., Baier, C.: Liquor: A tool for qualitative and quantitative linear time analysis of reactive systems. In: Proc. of QEST 2006, pp. 131–132. IEEE Computer Society Press, Los Alamitos (2006)
de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University (1997)
de Alfaro, L.: The verification of probabilistic systems under memoryless partial-information policies is hard. In: Proc. of PROBMIV 1999. Tech. Rep. CSR-99-8, pp. 19–32. Univ. of Birmingham (1999)
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)
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 and ETAPS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)
Jeannet, B., D’Argenio, P.R., Larsen, K.G.: Rapture: A tool for verifying Markov Decision Processes. In: I. Cerna, editor, Tools Day 2002, Brno, Czech Republic, Technical Report. Faculty of Informatics, Masaryk University Brno (2002)
Kemeny, J.G., Snell, J.L., Knapp, A.W.: Denumerable Markov Chains. Van Nostrand Company (1966)
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)
Press, W.H., Flannery, B.P., Teukolsky, S.A., Vetterling, W.T.: Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. pp. 343–347. Cambridge University Press, Cambridge (1992)
Puterman, M.: Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley, Chichester (1994)
Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology (1995)
Sokolova, A., de Vink, E.P.: Probabilistic automata: system types, parallel composition and comparison. In: Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P., Siegle, M. (eds.) Validation of Stochastic Systems. LNCS, vol. 2925, pp. 1–43. Springer, Heidelberg (2004)
Stoelinga, M.: Alea jacta est: Verification of Probabilistic, Real-time and Parametric Systems. PhD thesis, Katholieke Universiteit Nijmegen (2002)
Tripakis, S.: Undecidable problems of decentralized observation and control. In: Proc. 40th IEEE Conference on Decision and Control, vol. 5, pp. 4104–4109 (2001)
Varacca, D., Nielsen, M.: Probabilistic Petri nets and mazurkiewicz equivalence 2003 (Unpublished draft)
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Giro, S., D’Argenio, P.R. (2007). Quantitative Model Checking Revisited: Neither Decidable Nor Approximable. In: Raskin, JF., Thiagarajan, P.S. (eds) Formal Modeling and Analysis of Timed Systems. FORMATS 2007. Lecture Notes in Computer Science, vol 4763. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75454-1_14
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DOI: https://doi.org/10.1007/978-3-540-75454-1_14
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