Abstract
We consider the problem of approximating and computing a bisimilarity pseudometric on the state space of a probabilistic automaton. We show that the distances are rational and that both approximation and exact computation of distances are in PPAD. In the proofs, a generalization of the classical game-theoretic characterization of bisimilarity, given in terms of simple stochastic games, plays a central role.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms and Applications. Prentice Hall, Upper Saddle River (1993)
Bacci, G., Bacci, G., Larsen, K.G., Mardare, R.: On-the-fly exact computation of bisimilarity distances. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 1–15. Springer, Heidelberg (2013)
Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)
de Bakker, J.W., de Vink, E.P.: Control Flow Semantics. MIT Press, Cambridge (1996)
Banach, S.: Sur les opérations dans les ensembles abstraits et leurs applications aux equations intégrales. Fundamenta Mathematicae 3, 133–181 (1922)
Bartels, F., Sokolova, A., de Vink, E.P.: A hierarchy of probabilistic system types. In: Gumm, H.P. (ed.) CMCS 2003, Warsaw, Poland. Electronic Notes in Theoretical Computer Science, vol. 82(1), pp. 57–75. Elsevier (April 2003)
Blute, R., Desharnais, J., Edalat, A., Panangaden, P.: Bisimulation for labelled Markov processes. In: LICS 1997, Warsaw, Poland, pp. 149–158. IEEE (June/July 1997)
van Breugel, F.: A note on the Hutchinson metric (July 1999)
van Breugel, F.: An introduction to metric semantics: Operational and denotational models for programming and specification languages. Theoretical Computer Science 258(1/2), 1–98 (2001)
van Breugel, F.: On behavioural pseudometrics and closure ordinals. Information Processing Letters 112(18), 715–718 (2012)
van Breugel, F., Hermida, C., Makkai, M., Worrell, J.: Recursively defined metric spaces without contraction. Theoretical Computer Science 380(1/2), 143–163 (2007)
van Breugel, F., Worrell, J.B.: Towards quantitative verification of probabilistic systems. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 421–432. Springer, Heidelberg (2001)
Canny, J.: Some algebraic and geometric computations in PSPACE. In: Simon, J. (ed.) STOC 1988, Chicago, IL, USA, pp. 460–467. ACM (May 1988)
Chatterjee, K., de Alfaro, L., Majumdar, R., Raman, V.: Algorithms for game metrics. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2008, Bangalore, India. Leibniz International Proceedings in Informatics, vol. 2, pp. 107–118. Schloss Dagstuhl – Leibniz-Zentrum fuer Informatik (December 2008)
Chen, D., van Breugel, F., Worrell, J.: On the complexity of computing probabilistic bisimilarity. In: Birkedal, L. (ed.) FOSSACS 2012. LNCS, vol. 7213, pp. 437–451. Springer, Heidelberg (2012)
Chen, X., Deng, X.: Settling the complexity of two-player Nash equilibrium. In: FOCS 2006, Berkeley, CA, USA, pp. 261–272. IEEE (October 2006)
Chvátal, V.: Linear Programming. W.H. Freeman and Company, New York (1983)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)
Condon, A.: The complexity of stochastic games. Information and Computation 96(2), 203–224 (1992)
Daskalakis, C., Papadimitriou, C.: Continuous local search. In: Choffrut, C., Lengauer, T. (eds.) SODA 2011, San Francisco, CA, USA, pp. 790–804. SIAM (January 2011)
de Alfaro, L., Majumdar, R., Raman, V., Stoelinga, M.: Game relations and metrics. In: LICS 2007, Wroclaw, Poland, pp. 99–108. IEEE (July 2007)
Deng, Y., Chothia, T., Palamidessi, C., Pang, J.: Metrics for action-labelled quantitative transition systems. In: Cerone, A., Wiklicky, H. (eds.) QAPL 2005, Edinburgh, Scotland. Electronic Notes in Theoretical Computer Science, vol. 153(2), pp. 79–96. Elsevier (March 2005)
Desharnais, J., Edalat, A., Panangaden, P.: A logical characterization of bisimulation for labelled Markov processes. In: LICS 1998, Indianapolis, IN, USA, pp. 478–487. IEEE (July 1998)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: Metrics for labeled Markov systems. In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 258–273. Springer, Heidelberg (1999)
Desharnais, J., Gupta, V., Jagadeesan, R., Panangaden, P.: The metric analogue of weak bisimulation for probabilistic processes. In: LICS 2002, Copenhagen, Denmark, pp. 413–422. IEEE (July 2002)
Desharnais, J., Laviolette, F., Tracol, M.: Approximate analysis of probabilistic processes: logic, simulation and games. In: QEST 2008, Saint-Malo, France, pp. 264–273. IEEE (September 2008)
Edgar, G.A.: Integral, Probability, and Fractal Measures. Springer, New York (1998)
Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points (extended abstract). In: FOCS 2007, Providence, RI, USA, pp. 113–123. IEEE (October 2007)
Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM Journal on Computing 39(6), 2531–2597 (2010)
Ferns, N., Panangaden, P., Precup, D.: Metrics for Markov decision processes with infinite state spaces. In: UAI 2005, Edinburgh, Scotland, pp. 201–208. AUAI Press (July 2005)
Fu, H.: Computing game metrics on Markov decision processes. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part II. LNCS, vol. 7392, pp. 227–238. Springer, Heidelberg (2012)
Fu, H.: Personal communication (January 2013)
Giacalone, A., Jou, C.-C., Smolka, S.: Algebraic reasoning for probabilistic concurrent systems. In: PROCOMET 1990, Sea of Gallilee, Israel, pp. 443–458. North-Holland (April 1990)
Har-Peled, S.: Geometric Approximation Algorithms. American Mathematical Society, Providence (2011)
Hutchinson, J.: Fractals and self similarity. Indiana University Mathematics Journal 30(5), 713–747 (1981)
Jacobs, B.: Introduction to Coalgebra (2012)
Jonsson, B., Larsen, K.G.: Specification and refinement of probabilistic processes. In: LICS 1991, Amsterdam, The Netherlands, pp. 266–277. IEEE (July 1991)
Juba, B.: On the hardness of simple stochastic games. Master’s thesis, Carnegie Mellon University, Pittsburgh, PA, USA (May 2005)
Kantorovich, L.: On the transfer of masses. Doklady Akademii Nauk 5(1), 1–4 (1942) (in Russian); Translated in Management Science 5(1), 1–4 (October 1958)
Kantorovich, L.V., Rubinstein, G.S.: On the space of completely additive functions. Vestnik Leningradskogo Universiteta 3(2), 52–59 (1958) (in Russian)
Kintali, S., Poplawski, L.J., Rajaraman, R., Sundaram, R., Teng, S.-H.: Reducibility among fractional stability problems. In: FOCS 2009, Atlanta, GA, USA, pp. 283–292. IEEE (October 2009)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. In: POPL 1989, Austin, TX, USA, pp. 344–352. ACM (January 1989)
Lemke, C., Howson, J.J.T.: Equilibrium points of bimatrix games. Journal of the Society for Industrial and Applied Mathematics 12(2), 413–423 (1964)
Liggett, T.M., Lippman, S.A.: Stochastic games with perfect information and time average payoff. SIAM Review 11(4), 604–607 (1969)
Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilitic Analysis. Cambridge University Press, Cambridge (2005)
Panangaden, P.: Labelled Markov Processes. Imperial College Press, London (2009)
Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Sciences 48(3), 498–532 (1994)
Rutten, J.: Relators and metric bisimulations. In: Jacobs, B., Moss, L., Reichel, H., Rutten, J. (eds.) CMCS 1998, Lisbon, Portugal. Electronic Notes in Theoretical Computer Science, vol. 11, pp. 252–258. Elsevier (March 1998)
Scarf, H.: The approximation of fixed points of a continuous mapping. Journal of the Society for Industrial and Applied Mathematics 15(5), 1328–1343 (1967)
Schrijver, A.: Theory of Linear and Integer Programming. John Wiley & Sons, Chichester (1986)
Segala, R.: Modeling and Verification of Randomized Distributed Real-Time Systems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA (June 1995)
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)
Serre, D.: Matrices: Theory and Applications. Springer, New York (2010)
Shapley, L.S.: Stochastic games. Proceedings of the Academy of Sciences 39(10), 1095–1100 (1953)
Stirling, C.: Modal and Temporal Properties of Processes. Springer, New York (2001)
Tarski, A.: A decision method for elementary algebra and geometry. University of California Press, Berkeley (1951)
Tarski, A.: A lattice-theoretic fixed point theorem and its applications. Pacific Journal of Mathematics 5(2), 285–309 (1955)
Tracol, M., Desharnais, J., Zhioua, A.: Computing distances between probabilistic automata. In: Massink, M., Norman, G. (eds.) QAPL 2011, Saarbrücken, Germany. Electronic Proceedings in Theoretical Computer Science, vol. 57, pp. 148–162. Elsevier (April 2011)
Worrell, J.: Coinduction for recursive data types: partial orders, metric spaces and Omega-categories. In: Reichel, H. (ed.) CMCS 2000, Berlin, Germany. Electronic Notes in Theoretical Computer Science, vol. 33, pp. 337–356. Elsevier (March 2000)
Yannakakis, M.: The analysis of local search problems and their heuristics. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 298–311. Springer, Heidelberg (1990)
Yannakakis, M.: Computational Aspects of Equilibria. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 2–13. Springer, Heidelberg (2009)
Zwick, U., Paterson, M.: The complexity of mean payoff games. In: Li, M., Du, D.-Z. (eds.) COCOON 1995. LNCS, vol. 959, pp. 1–10. Springer, Heidelberg (1995)
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van Breugel, F., Worrell, J. (2014). The Complexity of Computing a Bisimilarity Pseudometric on Probabilistic Automata. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds) Horizons of the Mind. A Tribute to Prakash Panangaden. Lecture Notes in Computer Science, vol 8464. Springer, Cham. https://doi.org/10.1007/978-3-319-06880-0_10
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