Advertisement

Stochastic Context-Free Grammars, Regular Languages, and Newton’s Method

  • Kousha Etessami
  • Alistair Stewart
  • Mihalis Yannakakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7966)

Abstract

We study the problem of computing the probability that a given stochastic context-free grammar (SCFG), G, generates a string in a given regular language L(D) (given by a DFA, D). This basic problem has a number of applications in statistical natural language processing, and it is also a key necessary step towards quantitative ω-regular model checking of stochastic context-free processes (equivalently, 1-exit recursive Markov chains, or stateless probabilistic pushdown processes).

We show that the probability that G generates a string in L(D) can be computed to within arbitrary desired precision in polynomial time (in the standard Turing model of computation), under a rather mild assumption about the SCFG, G, and with no extra assumption about D. We show that this assumption is satisfied for SCFG’s whose rule probabilities are learned via the well-known inside-outside (EM) algorithm for maximum-likelihood estimation (a standard method for constructing SCFGs in statistical NLP and biological sequence analysis). Thus, for these SCFGs the algorithm always runs in P-time.

Keywords

Model Check Regular Language Critical Depth Rule Probability Model Check Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Corazza, A., De Mori, R., Gretter, D., Satta, G.: Computation of probabilities for an island-driven parser. IEEE Trans. PAMI 13(9), 936–950 (1991)CrossRefGoogle Scholar
  2. 2.
    Durbin, R., Eddy, S.R., Krogh, A., Mitchison, G.: Biological Sequence Analysis: Probabilistic models of Proteins and Nucleic Acids. Cambridge U. Press (1999)Google Scholar
  3. 3.
    Esparza, J., Gaiser, A., Kiefer, S.: Computing least fixed points of probabilistic systems of polynomials. In: Proc. 27th STACS, pp. 359–370 (2010)Google Scholar
  4. 4.
    Esparza, J., Kiefer, S., Luttenberger, M.: Computing the least fixed point of positive polynomial systems. SIAM J. on Computing 39(6), 2282–2355 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Esparza, J., Kučera, A., Mayr, R.: Model checking probabilistic pushdown automata. Logical Methods in Computer Science 2(1), 1–31 (2006)Google Scholar
  6. 6.
    Etessami, K., Stewart, A., Yannakakis, M.: Polynomial time algorithms for branching Markov decision processes and probabilistic min(max) polynomial Bellman equations. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 314–326. Springer, Heidelberg (2012); See full version at ArXiv:1202.4798CrossRefGoogle Scholar
  7. 7.
    Etessami, K., Stewart, A., Yannakakis, M.: Polynomial-time algorithms for multi-type branching processes and stochastic context-free grammars. In: Proc. 44th ACM STOC, Full version is available at ArXiv:1201.2374 (2012)Google Scholar
  8. 8.
    Etessami, K., Stewart, A., Yannakakis, M.: Stochastic Context-Free Grammars, Regular Languages, and Newton’s method, Full preprint of this paper: ArXiv:1302.6411 (2013)Google Scholar
  9. 9.
    Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. Journal of the ACM 56(1) (2009)Google Scholar
  10. 10.
    Etessami, K., Yannakakis, M.: Model checking of recursive probabilistic systems. ACM Trans. Comput. Log. 13(2), 12 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge U. Press (1985)Google Scholar
  12. 12.
    Jelinek, F., Lafferty, J.D.: Computation of the probability of initial substring generation by stochastic context-free grammars. Computational Linguistics 17(3), 315–323 (1991)Google Scholar
  13. 13.
    Knudsen, B., Hein, J.: Pfold: RNA secondary structure prediction using stochastic context-free grammars. Nucleic Acids Res 31, 3423–3428 (2003)CrossRefGoogle Scholar
  14. 14.
    Manning, C., Schütze, H.: Foundations of Statistical Natural Language Processing. MIT Press (1999)Google Scholar
  15. 15.
    Nederhof, M.-J., Satta, G.: Estimation of consistent probabilistic context-free grammars. In: HLT-NAACL (2006)Google Scholar
  16. 16.
    Nederhof, M.-J., Satta, G.: Computing partition functions of PCFGs. Research on Language and Computation 6(2), 139–162 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Nederhof, M.-J., Satta, G.: Probabilistic parsing. New Developments in Formal Languages and Applications 113, 229–258 (2008)CrossRefGoogle Scholar
  18. 18.
    Nederhof, M.-J., Satta, G.: Computation of infix probabilities for probabilistic context-free grammars. In: EMNLP, pp. 1213–1221 (2011)Google Scholar
  19. 19.
    Sánchez, J., Benedí, J.-M.: Consistency of stochastic context-free grammars from probabilistic estimation based on growth transformations. IEEE Trans. Pattern Anal. Mach. Intell. 19(9), 1052–1055 (1997)CrossRefGoogle Scholar
  20. 20.
    Stewart, A., Etessami, K., Yannakakis, M.: Upper bounds for Newton’s method on monotone polynomial systems, and P-time model checking of probabilistic one-counter automata, Arxiv:1302.3741 (2013) (conference version to appear in CAV 2013)Google Scholar
  21. 21.
    Stolcke, A.: An efficient probabilistic context-free parsing algorithm that computes prefix probabilities. Computational Linguistics 21(2), 167–201 (1995)MathSciNetGoogle Scholar
  22. 22.
    Wojtczak, D., Etessami, K.: Premo: an analyzer for probabilistic recursive models. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 66–71. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kousha Etessami
    • 1
  • Alistair Stewart
    • 1
  • Mihalis Yannakakis
    • 2
  1. 1.School of InformaticsUniversity of EdinburghUK
  2. 2.Department of Computer ScienceColumbia UniversityUSA

Personalised recommendations