Abstract
Probabilistic bisimilarity is a fundamental notion of equivalence on labelled Markov chains. It has a natural generalisation to a probabilistic bisimilarity pseudometric, whose definition involves the Kantorovich metric on probability distributions. The pseudometric has discounted and undiscounted variants, according to whether one discounts the future in observing discrepancies between states.
This paper is concerned with the complexity of computing probabilistic bisimilarity and the probabilistic bisimilarity pseudometric on labelled Markov chains. We show that the problem of computing probabilistic bisimilarity is P-hard by reduction from the monotone circuit value problem. We also show that the discounted pseudometric is rational and can be computed exactly in polynomial time using the network simplex algorithm and the continued fraction algorithm. In the undiscounted case we show that the pseudometric is again rational and can be computed exactly in polynomial time using the ellipsoid algorithm. Finally, using the notion of couplings on Markov chains, we show that the pseudometric can be used to compute bounds on the variational distance of trace distributions, which is NP-hard to compute directly.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows – theory, algorithms and applications (1993)
Baier, C.: Polynomial Time Algorithms for Testing Probabilistic Bisimulation and Simulation. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 50–61. Springer, Heidelberg (1996)
Baier, C., Katoen, J.-P.: Principles of Model Checking (2008)
Balcázar, J.L., Gabarró, J., Sántha, M.: Deciding bisimilarity is P-complete. FAC 4(6A), 638–648 (1992)
van Breugel, F., Hermida, C., Makkai, M., Worrell, J.: Recursively defined metric spaces without contraction. TCS 380(1-2), 171–197 (2007)
van Breugel, F., Sharma, B., Worrell, J.: Approximating a behavioural pseudometric without discount for probabilistic systems. LMCS 4(2:2) (2008)
van Breugel, F., Worrell, J.: Approximating and computing behavioural distances in probabilistic transition systems. TCS 360(1-3), 373–385 (2006)
Cai, X., Gu, Y.: Measuring Anonymity. In: Bao, F., Li, H., Wang, G. (eds.) ISPEC 2009. LNCS, vol. 5451, pp. 183–194. Springer, Heidelberg (2009)
Chatterjee, K., de Alfaro, L., Majumdar, R., Raman, V.: Algorithms for game metrics. In: FSTTCS, pp. 107–118 (2008)
Comanici, G., Precup, D.: Basis function discovery using spectral clustering and bisimulation metrics. In: AAAI, pp. 325–330 (2011)
Derisavi, S., Hermanns, H., Sanders, W.H.: Optimal state-space lumping in Markov chains. IPL 87(6), 309–315 (2003)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labelled Markov processes. TCS 318(3), 323–354 (2004)
Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing 39(6), 2531–2597 (2010)
Fu, H.: The complexity of deciding a behavioural pseudometric on probabilistic automata. Technical Report AIB-2011-26, RWTH Aachen (2011)
Giacalone, A., Jou, C.-C., Smolka, S.A.: Algebraic reasoning for probabilistic concurrent systems. In: PROCOMET, pp. 443–458 (1990)
Greenlaw, R., James Hoover, H., Ruzzo, W.L.: Limits to Parallel Computation: P-Completeness Theory (1995)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization (1988)
Hillston, J.: A Compositional Approach to Performance Modelling (1996)
Katoen, J.-P., Kemna, T., Zapreev, I.S., 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)
Kemeny, J.G., Laurie Snell, J.: Finite Markov Chains (1960)
Larsen, K.G., Skou, A.: Bisimulation through probabilistic testing. I&C 94(1), 1–28 (1991)
Lyngsø, R.B., Pedersen, C.N.S.: The consensus string problem and the complexity of comparing hidden Markov models. JCSS 65(3), 545–569 (2002)
Milner, R.: Communication and Concurrency (1989)
Mitzenmacher, M., Upfal, E.: Probability and Computing (2005)
Panangaden, P.: Labelled Markov Processes (2009)
Sawa, Z., Jančar, P.: Behavioural equivalences on finite-state systems are PTIME-hard. Computers and Artificial Intelligence 24(5), 513–528 (2005)
Schrijver, A.: Theory of Linear and Integer Programming (1986)
Thorsley, D., Klavins, E.: Approximating stochastic biochemical processes with Wasserstein pseudometrics. IET Systems Biology 4(3), 193–211 (2010)
Torán, J.: On the hardness of graph isomorphism. SIAM Journal on Computing 33(5), 1093–1108 (2004)
Tzeng, W.-G.: A polynomial-time algorithm for the equivalence of probabilistic automata. SIAM Journal on Computing 21(2), 216–227 (1992)
Tzeng, W.-G.: On path equivalence of nondeterministic finite automata. IPL 58(1), 43–46 (1996)
Valmari, A., Franceschinis, G.: Simple O(m logn) Time Markov Chain Lumping. In: Esparza, J., Majumdar, R. (eds.) TACAS 2010. LNCS, vol. 6015, pp. 38–52. Springer, Heidelberg (2010)
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Chen, D., van Breugel, F., Worrell, J. (2012). On the Complexity of Computing Probabilistic Bisimilarity. In: Birkedal, L. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2012. Lecture Notes in Computer Science, vol 7213. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28729-9_29
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DOI: https://doi.org/10.1007/978-3-642-28729-9_29
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