Skip to main content
SpringerLink
Log in

Finding Lower Bounds for Nondeterministic State Complexity Is Hard

  • Conference paper
Developments in Language Theory (DLT 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4036))

Included in the following conference series:

Abstract

We investigate the following lower bound methods for regular languages: The fooling set technique, the extended fooling set technique, and the biclique edge cover technique. It is shown that the maximal attainable lower bound for each of the above mentioned techniques can be algorithmically deduced from a canonical finite graph, the so called dependency graph of a regular language. This graph is very helpful when comparing the techniques with each other and with nondeterministic state complexity. In most cases it is shown that for any two techniques the gap between the best bounds can be arbitrarily large. Moreover, we show that deciding whether a certain lower bound w.r.t. one of the investigated techniques can be achieved is in most cases computationally hard, i.e., PSPACE-complete and hence is as hard as minimizing nondeterministic finite automata.

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 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Adorna, H.N.: Some descriptional complexity problems in finite automata theory. In: Saldaña, R.P., Chua, C. (eds.) Proceedings of the 5th Philippine Computing Science Congress, Cebu City, Philippines, March 2005, pp. 27–32. Computing Society of the Philippines (2005)

    Google Scholar 

  2. Bezrukov, S., Fronček, D., Rosenberg, S.J., Kovář, P.: On biclique coverings (preprint, 2005)

    Google Scholar 

  3. Birget, J.-C.: Intersection and union of regular languages and state complexity. Information Processing Letters 43, 185–190 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  4. Brzozowski, J.A.: Mathematical theory of automata. In: Canonical Regular Expressions and Minimal State Graphs for Definite Events. MRI Symposia Series, vol. 12, pp. 529–561. Polytechnic Press, NY (1962)

    Google Scholar 

  5. Cameron, K.: Induced matchings. Discrete Applied Mathematics 24, 97–102 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  6. Glaister, I., Shallit, J.: A lower bound technique for the size of nondeterministic finite automata. Information Processing Letters 59, 75–77 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hopcroft, J.E., Ullman, J.D.: Formal Languages and Their Relation to Automata. Addison-Wesley, Reading (1968)

    Google Scholar 

  8. Hromkovič, J.: Communication Complexity and Parallel Computing. Springer, Heidelberg (1997)

    Book  MATH  Google Scholar 

  9. Hromkovič, J.: Descriptional complexity of finite automata: Concepts and open problems. Journal of Automata, Languages and Combinatorics 7(4), 519–531 (2002)

    MATH  MathSciNet  Google Scholar 

  10. Hromkovič, J., Karhumäki, J., Klauck, H., Schnitger, G., Seibert, S.: Measures of nondeterminism in finite automata. Report TR00-076, Electronic Colloquium on Computational Complexity (ECCC) (2000)

    Google Scholar 

  11. Jiang, T., McDowell, E., Ravikumar, B.: The structure and complexity of minimal NFAs over a unary alphabet. International Journal of Foundations of Computer Science 2(2), 163–182 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Jirásková, G.: Note on minimal automata and uniform communication protocols. In: Martín-Vide, C., Mitrana, V. (eds.) Grammars and Automata for String Processing. Topics in Computer Mathematics, vol. 9, pp. 163–170. Taylor and Francis, Abington (2003)

    Chapter  Google Scholar 

  14. Kameda, T., Weiner, P.: On the state minimization of nondeterministic finite automata. IEEE Transactions on Computers C-19(7), 617–627 (1970)

    Article  MathSciNet  Google Scholar 

  15. Moore, F.R.: On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Transaction on Computing C-20, 1211–1219 (1971)

    Article  Google Scholar 

  16. Rabin, M.O., Scott, D.: Finite automata and their decision problems. IBM Journal of Research and Development 3, 114–125 (1959)

    Article  MathSciNet  Google Scholar 

  17. Salomaa, A., Wood, D., Yu, S.: On the state complexity of reversals of regular languages. Theoretical Computer Science 320(2–3), 315–329 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstraße 3, D-85748, Garching bei München, Germany

    Hermann Gruber & Markus Holzer

Authors
  1. Hermann Gruber
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Markus Holzer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of California, Santa Barbara

    Oscar H. Ibarra

  2. School of Electrical Engineering and Computer Science, Washington State University, WA 99164, Pullman, USA

    Zhe Dang

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Gruber, H., Holzer, M. (2006). Finding Lower Bounds for Nondeterministic State Complexity Is Hard. In: Ibarra, O.H., Dang, Z. (eds) Developments in Language Theory. DLT 2006. Lecture Notes in Computer Science, vol 4036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779148_33

Download citation

  • DOI: https://doi.org/10.1007/11779148_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-35428-4

  • Online ISBN: 978-3-540-35430-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation