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Runtime Analysis of Probabilistic Programs with Unbounded Recursion

  • Tomáš Brázdil
  • Stefan Kiefer
  • Antonín Kučera
  • Ivana Hutařová Vařeková
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6756)

Abstract

We study the runtime in probabilistic programs with unbounded recursion. As underlying formal model for such programs we use probabilistic pushdown automata (pPDA) which exactly correspond to recursive Markov chains. We show that every pPDA can be transformed into a stateless pPDA (called “pBPA”) whose runtime and further properties are closely related to those of the original pPDA. This result substantially simplifies the analysis of runtime and other pPDA properties. We prove that for every pPDA the probability of performing a long run decreases exponentially in the length of the run, if and only if the expected runtime in the pPDA is finite. If the expectation is infinite, then the probability decreases “polynomially”. We show that these bounds are asymptotically tight. Our tail bounds on the runtime are generic, i.e., applicable to any probabilistic program with unbounded recursion. An intuitive interpretation is that in pPDA the runtime is exponentially unlikely to deviate from its expected value.

Keywords

IEEE Computer Society Termination Time Probabilistic Program Runtime Analysis Recursive Program 
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.

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References

  1. 1.
    Bini, D., Latouche, G., Meini, B.: Numerical methods for Structured Markov Chains. Oxford University Press, Oxford (2005)CrossRefzbMATHGoogle Scholar
  2. 2.
    Brázdil, T.: Verification of Probabilistic Recursive Sequential Programs. PhD thesis, Masaryk University, Faculty of Informatics (2007)Google Scholar
  3. 3.
    Brázdil, T., Brožek, V., Holeček, J., Kučera, A.: Discounted properties of probabilistic pushdown automata. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 230–242. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Brázdil, T., Brožek, V., Etessami, K.: One-counter stochastic games. In: Proceedings of ST&TCS 2010. Leibniz International Proceedings in Informatics, vol. 8, pp. 108–119. Schloss Dagstuhl–Leibniz-Zentrum für Informatik (2010)Google Scholar
  5. 5.
    Brázdil, T., Brožek, V., Etessami, K., Kučera, A., Wojtczak, D.: One-counter Markov decision processes. In: Proceedings of SODA 2010, pp. 863–874. SIAM, Philadelphia (2010)Google Scholar
  6. 6.
    Brázdil, T., Esparza, J., Kučera, A.: Analysis and prediction of the long-run behavior of probabilistic sequential programs with recursion. In: Proceedings of FOCS 2005, pp. 521–530. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  7. 7.
    Brázdil, T., Kiefer, S., Kučera, A.: Efficient analysis of probabilistic programs with an unbounded counter. CoRR, abs/1102.2529 (2011)Google Scholar
  8. 8.
    Brázdil, T., Kiefer, S., Kučera, A., Hutařová Vařeková, I.: Runtime analysis of probabilistic programs with unbounded recursion. CoRR, abs/1007.1710 (2010)Google Scholar
  9. 9.
    Canny, J.: Some algebraic and geometric computations in PSPACE. In: Proceedings of STOC 1988, pp. 460–467. ACM Press, New York (1988)Google Scholar
  10. 10.
    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Esparza, J., Kiefer, S., Luttenberger, M.: Convergence thresholds of Newton’s method for monotone polynomial equations. In: STACS 2008, pp. 289–300 (2008)Google Scholar
  12. 12.
    Esparza, J., Kiefer, S., Luttenberger, M.: Computing the least fixed point of positive polynomial systems. SIAM Journal on Computing 39(6), 2282–2335 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Esparza, J., Kučera, A., Mayr, R.: Model-checking probabilistic pushdown automata. In: Proceedings of LICS 2004, pp. 12–21. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  14. 14.
    Esparza, J., Kučera, A., Mayr, R.: Quantitative analysis of probabilistic pushdown automata: Expectations and variances. In: Proceedings of LICS 2005, pp. 117–126. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  15. 15.
    Etessami, K., Wojtczak, D., Yannakakis, M.: Quasi-birth-death processes, tree-like QBDs, probabilistic 1-counter automata, and pushdown systems. In: Proceedings of 5th Int. Conf. on Quantitative Evaluation of Systems (QEST 2008). IEEE Computer Society Press, Los Alamitos (2008)Google Scholar
  16. 16.
    Etessami, K., Yannakakis, M.: Algorithmic verification of recursive probabilistic systems. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 253–270. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Etessami, K., Yannakakis, M.: Checking LTL properties of recursive Markov chains. In: Proceedings of 2nd Int. Conf. on Quantitative Evaluation of Systems (QEST 2005), pp. 155–165. IEEE Computer Society Press, Los Alamitos (2005)Google Scholar
  18. 18.
    Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. Journal of the Association for Computing Machinery 56 (2009)Google Scholar
  19. 19.
    Harris, T.E.: The Theory of Branching Processes. Springer, Heidelberg (1963)CrossRefzbMATHGoogle Scholar
  20. 20.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  21. 21.
    Kiefer, S., Luttenberger, M., Esparza, J.: On the convergence of Newton’s method for monotone systems of polynomial equations. In: STOC 2007, pp. 217–226 (2007)Google Scholar
  22. 22.
    Latouche, G., Ramaswami, V.: Introduction to Matrix Analytic Methods in Stochastic Modeling. ASA-SIAM series on statistics and applied probability (1999)Google Scholar
  23. 23.
    Manning, C., Schütze, H.: Foundations of Statistical Natural Language Processing. The MIT Press, Cambridge (1999)zbMATHGoogle Scholar
  24. 24.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  25. 25.
    Pakes, A.G.: Some limit theorems for the total progeny of a branching process. Advances in Applied Probability 3(1), 176–192 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Quine, M.P., Szczotka, W.: Generalisations of the Bienayme-Galton-Watson branching process via its representation as an embedded random walk. The Annals of Applied Probability 4(4), 1206–1222 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tomáš Brázdil
    • 1
  • Stefan Kiefer
    • 2
  • Antonín Kučera
    • 1
  • Ivana Hutařová Vařeková
    • 1
  1. 1.Faculty of InformaticsMasaryk UniversityCzech Republic
  2. 2.Department of Computer ScienceUniversity of OxfordUnited Kingdom

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