Skip to main content
SpringerLink
Log in

Recursive Markov Chains, Stochastic Grammars, and Monotone Systems of Nonlinear Equations

  • Conference paper
STACS 2005 (STACS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3404))

Included in the following conference series:

Abstract

We introduce and study Recursive Markov Chains (RMCs), which extend ordinary finite state Markov chains with the ability to invoke other Markov chains in a potentially recursive manner. They offer a natural abstract model for probabilistic programs with procedures, and are a probabilistic version of Recursive State Machines. RMCs generalize Stochastic Context-Free Grammars (SCFG) and multi-type Branching Processes, and are intimately related to Probabilistic Pushdown Systems. We focus here on termination and reachability analysis for RMCs. We present both positive and negative results for the general class of RMCs, as well as for important subclasses including SCFGs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alur, R., Etessami, K., Yannakakis, M.: Analysis of recursive state machines. In: Berry, G., Comon, H., Finkel, A. (eds.) CAV 2001. LNCS, vol. 2102, pp. 304–313. Springer, Heidelberg (2001)

    Google Scholar 

  2. Abney, S., McAllester, D., Pereira, F.: Relating probabilistic grammars and automata. In: Proc. 37th Ann. Ass. for Comp. Linguistics, pp. 542–549 (1999)

    Google Scholar 

  3. Bouajjani, A., Esparza, J., Maler, O.: Reachability analysis of pushdown automata: App’s to model checking. In: Mazurkiewicz, A., Winkowski, J. (eds.) CONCUR 1997. LNCS, vol. 1243, pp. 135–150. Springer, Heidelberg (1997)

    Google Scholar 

  4. Benedikt, M., Godefroid, P., Reps, T.: Model checking of unrestricted hierarchical state machines. In: Orejas, F., Spirakis, P.G., van Leeuwen, J. (eds.) ICALP 2001. LNCS, vol. 2076, pp. 652–666. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  5. Basu, S., Pollack, R., Roy, M.F.: On the combinatorial and algebraic complexity of quantifier elimination. J. of the ACM 43(6), 1002–1045 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Basu, S., Pollack, F., Roy, M.F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2003)

    MATH  Google Scholar 

  7. Booth, T.L., Thompson, R.A.: Applying probability measures to abstract languages. IEEE Transactions on Computers 22(5), 442–450 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  8. Canny, J.: Some algebraic and geometric computations in PSPACE. In: Prof. of 20th ACM STOC, pp. 460–467 (1988)

    Google Scholar 

  9. Chi, Z., Geman, S.: Estimation of probabilistic context-free grammars. In: Computational Linguistics (1997)

    Google Scholar 

  10. Esparza, J., Hansel, D., Rossmanith, P., Schwoon, S.: Efficient algorithms for model checking pushdown systems. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 232–247. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  11. Esparza, J., Kučera, A.: Personal communication (2004)

    Google Scholar 

  12. Esparza, J., Kučera, A., Mayr, R.: Model checking probabilistic pushdown automata. In: LICS 2004 (2004)

    Google Scholar 

  13. Everett, C.J., Ulam, S.: Multiplicative systems, part i., ii, and iii. Technical Report 683,690,707, Los Alamos Scientific Laboratory (1948)

    Google Scholar 

  14. Etessami, K., Yannakakis, M.: Algorithmic verification of recursive probabilistic systems. Technical report, School of Informatics, U. of Edinburgh (2004) (submitted)

    Google Scholar 

  15. Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of non-linear equations. Technical report, School of Informatics, University of Edinburgh (2004) (Full paper)

    Google Scholar 

  16. Garey, M.R., Graham, R.L., Johnson, D.S.: Some NP-complete geometric problems. In: 8th ACM STOC, pp. 10–22 (1976)

    Google Scholar 

  17. Grenander, U.: Lectures on Pattern Theory, vol. 1. Springer, Heidelberg (1976)

    Google Scholar 

  18. Harris, T.E.: The Theory of Branching Processes. Springer, Heidelberg (1963)

    MATH  Google Scholar 

  19. Jagers, P.: Branching Processes with Biological Applications. Wiley, Chichester (1975)

    MATH  Google Scholar 

  20. Kolmogorov, A.N., Sevastyanov, B.A.: The calculation of final probabilities for branching random processes. Dokaldy 56, 783–786 (1947) (Russian)

    MATH  Google Scholar 

  21. Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn. Academic Press, London (1985)

    MATH  Google Scholar 

  22. Malajovich, G.: An effective version of Kronecker’s theorem on simultaneous diophantine equations. Technical report, City U. of Hong Kong (1996)

    Google Scholar 

  23. Manning, C., Schütze, H.: Foundations of Statistical Natural Language Processing. MIT Press, Cambridge (1999)

    MATH  Google Scholar 

  24. Ortega, J.M., Rheinbolt, W.C.: Iterative solution of nonlinear equations in several variables. Academic Press, London (1970)

    MATH  Google Scholar 

  25. Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. parts i,ii, iii. J. of Symbolic Computation, 255–352 (1992)

    Google Scholar 

  26. Stoer, J., Bulirsch, R.: Introduction to Numerical Analysis. Springer, Heidelberg (1993)

    MATH  Google Scholar 

  27. Sakakibara, Y., Brown, M., Hughey, R., Mian, I.S., Sjolander, K., Underwood, R., Haussler, D.: Stochastic context-free grammars for tRNA modeling. Nucleic Acids Research 22(23), 5112–5120 (1994)

    Article  Google Scholar 

  28. Sevastyanov, B.A.: The theory of branching processes. Uspehi Mathemat. Nauk 6, 47–99 (1951) (in Russian)

    MathSciNet  Google Scholar 

  29. Tiwari, P.: A problem that is easier to solve on the unit-cost algebraic ram. Journal of Complexity, 393–397 (1992)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Informatics, University of Edinburgh,  

    Kousha Etessami

  2. Department of Computer Science, Columbia University,  

    Mihalis Yannakakis

Authors
  1. Kousha Etessami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mihalis Yannakakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Universität Stuttgart, FMI, Germany

    Volker Diekert

  2. LIF, CNRS & Univ. de Provence,, Marseille

    Bruno Durand

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Etessami, K., Yannakakis, M. (2005). Recursive Markov Chains, Stochastic Grammars, and Monotone Systems of Nonlinear Equations. In: Diekert, V., Durand, B. (eds) STACS 2005. STACS 2005. Lecture Notes in Computer Science, vol 3404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31856-9_28

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-31856-9_28

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24998-6

  • Online ISBN: 978-3-540-31856-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation