Abstract
Several types of weighted-automata models and formalisms to specify and verify constraints on accumulated weights have been studied in the past. The lack of monotonicity for weight functions with positive and negative values as well as for ratios of the accumulated values of non-negative weight functions renders many verification problems to be undecidable or computationally hard. Our contribution comprises polynomial-time algorithms for computing ratio and weight quantiles in Markov chains, which provide optimal bounds guaranteed almost surely or with positive probability on, e.g., cost-utility ratios or the energy conversion efficiency.
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The authors are supported by the DFG through the collaborative research centre HAEC (SFB 912), the Excellence Initiative by the German Federal and State Governments (cluster of excellence cfAED and Institutional Strategy), the Graduiertenkolleg QuantLA (1763), Deutsche Telekom Stiftung, the EU-FP-7 grant MEALS (295261).
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Andova, S., Hermanns, H., Katoen, J.P.: Discrete-time rewards model-checked. In: Larsen, K.G., Niebert, P. (eds.) FORMATS 2003. LNCS, vol. 2791, pp. 88–104. Springer, Heidelberg (2004)
Baier, C., Daum, M., Dubslaff, C., Klein, J., Klüppelholz, S.: Energy-utility quantiles. In: Badger, J.M., Rozier, K.Y. (eds.) NFM 2014. LNCS, vol. 8430, pp. 285–299. Springer, Heidelberg (2014)
Baier, C., Dubslaff, C., Klein, J., Klüppelholz, S., Wunderlich, S.: Probabilistic model checking for energy-utility analysis. In: van Breugel, F., Kashefi, E., Palamidessi, C., Rutten, J. (eds.) Horizons of the Mind. LNCS, vol. 8464, pp. 96–123. Springer, Heidelberg (2014)
Baier, C., Dubslaff, C., Klüppelholz, S.: Trade-off analysis meets probabilistic model checking. In: CSL-LICS 2014, pp. 1:1–1:10. ACM (2014)
Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)
Baier, C., Klein, J., Klüppelholz, S., Wunderlich, S.: Weight monitoring with linear temporal logic: complexity and decidability. In: CSL-LICS 2014, pp. 11:1–11:10. ACM (2014)
Boker, U., Chatterjee, K., Henzinger, T.A., Kupferman, O.: Temporal specifications with accumulative values. In: LICS 2011, pp. 43–52. IEEE Computer Society (2011)
Bozzelli, L., Ganty, P.: Complexity analysis of the backward coverability algorithm for VASS. In: Delzanno, G., Potapov, I. (eds.) RP 2011. LNCS, vol. 6945, pp. 96–109. Springer, Heidelberg (2011)
Brázdil, T., Brozek, V., Chatterjee, K., Forejt, V., Kucera, A.: Two views on multiple mean-payoff objectives in Markov decision processes. Logical Methods Comput. Sci. 10(1), 1–29 (2014)
Brázdil, T., Brozek, V., Etessami, K., Kucera, A., Wojtczak, D.: One-counter Markov decision processes. In: SODA 2010, pp. 863–874. SIAM (2010)
Brázdil, T., Esparza, J., Kiefer, S., Kucera, A.: Analyzing probabilistic pushdown automata. Formal Methods Syst. Des. 43(2), 124–163 (2013)
Brázdil, T., Kiefer, S., Kucera, A., Novotný, P., Katoen, J.-P.: Zero-reachability in probabilistic multi-counter automata. In: CSL-LICS 2014. ACM (2014)
Cardoza, E., Lipton, R., Meyer, A.R.: Exponential space complete problems for Petri nets and commutative semigroups (preliminary report). In: STOC 1976, pp. 50–54. ACM (1976)
Chatterjee, K., Doyen, L.: Energy and mean-payoff parity Markov decision processes. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 206–218. Springer, Heidelberg (2011)
Chatterjee, K., Doyen, L.: Energy parity games. Theoret. Comput. Sci. 458, 49–60 (2012)
Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (2000)
de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University, Department of Computer Science (1997)
Etessami, K., Kwiatkowska, M., Vardi, M.Y., Yannakakis, M.: Multi-objective model checking of Markov decision processes. Logical Methods Comput. Sci. 4(4), 1–21 (2008)
Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. J. ACM 56(1), 1:1–1:66 (2009)
Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39(6), 2531–2597 (2010)
Freedman, D.: Markov Chains. Springer, New York (1983)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1993)
Haase, C., Kiefer, S.: The odds of staying on budget. In: Halldórsson, M.M., Iwama, K., Kobayashi, N., Speckmann, B. (eds.) ICALP 2015. LNCS, vol. 9135, pp. 234–246. Springer, Heidelberg (2015)
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)
Juhl, L., Guldstrand Larsen, K., Raskin, J.-F.: Optimal bounds for multiweighted and parametrised energy games. In: Liu, Z., Woodcock, J., Zhu, H. (eds.) He Festschrift. LNCS, vol. 8051, pp. 244–255. Springer, Heidelberg (2013)
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)
Katoen, J.-P., Zapreev, I., Hahn, E., Hermanns, H., Jansen, D.: The ins and outs of the probabilistic model checker MRMC. Perform. Eval. 68(2), 90–104 (2011)
Krähmann, D., Schubert, J., Baier, C., Dubslaff, C.: Ratio and weight quantiles. Technical report, Technische Universität Dresden (2015). http://wwwtcs.inf.tu-dresden.de/ALGI/PUB/MFCS15/
Rackoff, C.: The covering and boundedness problems for vector addition systems. Theoret. Comput. Sci. 6(2), 223–231 (1978)
Randour, M., Raskin, J.-F., Sankur, O.: Percentile queries in multi-dimensional Markov decision processes. In: CAV 2015. LNCS. Springer, Heidelberg (2015, to appear)
Schubert, J.: Weight and ratio objectives in annotated Markov chains. Master’s thesis, TU Dresden (2015)
Ummels, M., Baier, C.: Computing quantiles in Markov reward models. In: Pfenning, F. (ed.) FOSSACS 2013 (ETAPS 2013). LNCS, vol. 7794, pp. 353–368. Springer, Heidelberg (2013)
von Essen, C., Jobstmann, B.: Synthesizing systems with optimal average-case behavior for ratio objectives. In: iWIGP 2011. EPTCS, vol. 50, pp. 17–32 (2011)
Acknowledgements
We thank Stefan Kiefer for pointing us to the continued-fraction method and its application [20].
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Krähmann, D., Schubert, J., Baier, C., Dubslaff, C. (2015). Ratio and Weight Quantiles. In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9234. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48057-1_27
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DOI: https://doi.org/10.1007/978-3-662-48057-1_27
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