Skip to main content
SpringerLink
Log in
Book cover

International Colloquium on Automata, Languages, and Programming

ICALP 2014: Automata, Languages, and Programming pp 268–279Cite as

Stability and Complexity of Minimising Probabilistic Automata

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8573))

Abstract

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for “lossy” minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.

For a full version of this paper, see [19].

This is a preview of subscription content, 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   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aggarwal, A., Booth, H., O’Rourke, J., Suri, S., Yap, C.K.: Finding minimal convex nested polygons. Information and Computation 83(1), 98–110 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arnoldi, W.E.: The principle of minimized iteration in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics 9, 17–29 (1951)

    MATH  MathSciNet  Google Scholar 

  3. Beimel, A., Bergadano, F., Bshouty, N.H., Kushilevitz, E., Varricchio, S.: Learning functions represented as multiplicity automata. Journal of the ACM 47(3), 506–530 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berstel, J., Reutenauer, C.: Rational Series and Their Languages. Springer (1988)

    Google Scholar 

  5. Bonchi, F., Bonsangue, M.M., Hansen, H.H., Panangaden, P., Rutten, J.J.M.M., Silva, A.: Algebra-coalgebra duality in Brzozowski’s minimization algorithm. ACM Transactions on Computational Logic (to appear)

    Google Scholar 

  6. Brzozowski, J.A.: Canonical regular expressions and minimal state graphs for definite events. In: Symposium on Mathematical Theory of Automata. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, Polytechnic Institute of Brooklyn (1962)

    Google Scholar 

  7. Bukharaev, R.G.: Probabilistic automata. Journal of Soviet Mathematics 13(3), 359–386 (1980)

    Article  MATH  Google Scholar 

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

    Google Scholar 

  9. Cho, S., Huynh, D.T.: The parallel complexity of finite-state automata problems. Information and Computation 97(1), 1–22 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  10. Das, G., Goodrich, M.T.: On the complexity of approximating and illuminating three-dimensional convex polyhedra. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 74–85. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  11. Das, G., Joseph, D.: Minimum vertex hulls for polyhedral domains. Theoretical Computer Science 103(1), 107–135 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Golub, G.H., van Loan, C.F.: Matrix Computations. John Hopkins University Press (1989)

    Google Scholar 

  13. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM (2002)

    Google Scholar 

  14. Householder, A.S.: Unitary triangularization of a nonsymmetric matrix. Journal of the ACM 5(4), 339–342 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  15. Culik II, K., Kari, J.: Image compression using weighted finite automata. Computers & Graphics 17(3), 305–313 (1993)

    Article  Google Scholar 

  16. Jiang, T., Ravikumar, B.: Minimal NFA problems are hard. SIAM Journal on Computing 22(6), 1117–1141 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kiefer, S., Murawski, A.S., Ouaknine, J., Wachter, B., Worrell, J.: Language equivalence for probabilistic automata. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 526–540. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  18. Kiefer, S., Murawski, A.S., Ouaknine, J., Wachter, B., Worrell, J.: On the complexity of equivalence and minimisation for Q-weighted automata. Logical Methods in Computer Science 9(1:8), 1–22 (2013)

    Google Scholar 

  19. Kiefer, S., Wachter, B.: Stability and complexity of minimising probabilistic automata. Technical report, arxiv.org (2014), http://arxiv.org/abs/1404.6673

  20. Mateus, P., Qiu, D., Li, L.: On the complexity of minimizing probabilistic and quantum automata. Information and Computation 218, 36–53 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mitchell, J.S.B., Suri, S.: Separation and approximation of polyhedral objects. In: Proceedings of SODA, pp. 296–306 (1992)

    Google Scholar 

  22. Paz, A.: Introduction to probabilistic automata. Academic Press (1971)

    Google Scholar 

  23. Rabin, M.O.: Probabilistic automata. Information and Control 6(3), 230–245 (1963)

    Article  MATH  Google Scholar 

  24. Renegar, J.: On the computational complexity and geometry of the first-order theory of the reals. Parts I–III. Journal of Symbolic Computation 13(3), 255–352 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Schützenberger, M.-P.: On the definition of a family of automata. Information and Control 4, 245–270 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  26. Silio, C.B.: An efficient simplex coverability algorithm in E 2 with application to stochastic sequential machines. IEEE Transactions on Computers C-28(2), 109–120 (1979)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Oxford, UK

    Stefan Kiefer & Björn Wachter

Authors
  1. Stefan Kiefer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Björn Wachter
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstrasse 3, 85748, München, Germany

    Javier Esparza

  2. LIAFA, Université Paris Diderot-Paris 7, Case 7014, 75205, Paris Cedex 13, France

    Pierre Fraigniaud

  3. IT University of Copenhagen, Rued Langgaards Vej 7, 2300, Copenhagen, Denmark

    Thore Husfeldt

  4. Department Computer Science, University of Oxford, Wolfson Building, Parks Road, OX1 3QD, Oxford, UK,

    Elias Koutsoupias

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Kiefer, S., Wachter, B. (2014). Stability and Complexity of Minimising Probabilistic Automata. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43951-7_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-43951-7_23

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-43950-0

  • Online ISBN: 978-3-662-43951-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation