Ratio and Weight Quantiles

  • Daniel KrähmannEmail author
  • Jana Schubert
  • Christel Baier
  • Clemens Dubslaff
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9234)


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.


Markov Chain Weight Function Polynomial Time Weight Objective Negative Cycle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We thank Stefan Kiefer for pointing us to the continued-fraction method and its application [20].


  1. 1.
    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) CrossRefGoogle Scholar
  2. 2.
    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) CrossRefGoogle Scholar
  3. 3.
    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) Google Scholar
  4. 4.
    Baier, C., Dubslaff, C., Klüppelholz, S.: Trade-off analysis meets probabilistic model checking. In: CSL-LICS 2014, pp. 1:1–1:10. ACM (2014)Google Scholar
  5. 5.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. MIT Press, Cambridge (2008)zbMATHGoogle Scholar
  6. 6.
    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)Google Scholar
  7. 7.
    Boker, U., Chatterjee, K., Henzinger, T.A., Kupferman, O.: Temporal specifications with accumulative values. In: LICS 2011, pp. 43–52. IEEE Computer Society (2011)Google Scholar
  8. 8.
    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) CrossRefGoogle Scholar
  9. 9.
    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)CrossRefGoogle Scholar
  10. 10.
    Brázdil, T., Brozek, V., Etessami, K., Kucera, A., Wojtczak, D.: One-counter Markov decision processes. In: SODA 2010, pp. 863–874. SIAM (2010)Google Scholar
  11. 11.
    Brázdil, T., Esparza, J., Kiefer, S., Kucera, A.: Analyzing probabilistic pushdown automata. Formal Methods Syst. Des. 43(2), 124–163 (2013)CrossRefzbMATHGoogle Scholar
  12. 12.
    Brázdil, T., Kiefer, S., Kucera, A., Novotný, P., Katoen, J.-P.: Zero-reachability in probabilistic multi-counter automata. In: CSL-LICS 2014. ACM (2014)Google Scholar
  13. 13.
    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)Google Scholar
  14. 14.
    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) CrossRefGoogle Scholar
  15. 15.
    Chatterjee, K., Doyen, L.: Energy parity games. Theoret. Comput. Sci. 458, 49–60 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Clarke, E., Grumberg, O., Peled, D.: Model Checking. MIT Press, Cambridge (2000)Google Scholar
  17. 17.
    de Alfaro, L.: Formal Verification of Probabilistic Systems. PhD thesis, Stanford University, Department of Computer Science (1997)Google Scholar
  18. 18.
    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)MathSciNetGoogle Scholar
  19. 19.
    Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. J. ACM 56(1), 1:1–1:66 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Etessami, K., Yannakakis, M.: On the complexity of Nash equilibria and other fixed points. SIAM J. Comput. 39(6), 2531–2597 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Freedman, D.: Markov Chains. Springer, New York (1983)CrossRefGoogle Scholar
  22. 22.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1993)CrossRefzbMATHGoogle Scholar
  23. 23.
    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) CrossRefGoogle Scholar
  24. 24.
    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) CrossRefGoogle Scholar
  25. 25.
    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) CrossRefGoogle Scholar
  26. 26.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2002)CrossRefzbMATHGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Krähmann, D., Schubert, J., Baier, C., Dubslaff, C.: Ratio and weight quantiles. Technical report, Technische Universität Dresden (2015).
  29. 29.
    Rackoff, C.: The covering and boundedness problems for vector addition systems. Theoret. Comput. Sci. 6(2), 223–231 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Randour, M., Raskin, J.-F., Sankur, O.: Percentile queries in multi-dimensional Markov decision processes. In: CAV 2015. LNCS. Springer, Heidelberg (2015, to appear)Google Scholar
  31. 31.
    Schubert, J.: Weight and ratio objectives in annotated Markov chains. Master’s thesis, TU Dresden (2015)Google Scholar
  32. 32.
    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) CrossRefGoogle Scholar
  33. 33.
    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)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Daniel Krähmann
    • 1
    Email author
  • Jana Schubert
    • 1
  • Christel Baier
    • 1
  • Clemens Dubslaff
    • 1
  1. 1.Faculty of Computer ScienceTechnische Universität DresdenDresdenGermany

Personalised recommendations