Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Analysing Complexity in Classes of Unary Automatic Structures

  • Conference paper
Language and Automata Theory and Applications (LATA 2009)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5457))

Abstract

This paper addresses the time complexity of several queries (including membership and isomorphism) in classes of unary automatic structures and the state complexity of describing these classes. We focus on unary automatic equivalence relations, linear orders, trees, and graphs with finite degree. We prove that in various senses, the complexity of these classes is low: (1) For the isomorphism problem, we either greatly improve known algorithms (reducing highly exponential bounds to small polynomials) or answer open questions about the existence of a decision procedure; (2) for state complexity, we provide polynomial bounds with respect to natural measures of the sizes of the structures.

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. Libkin, L.: Elements of Finite Model Theory. EATCS. Springer, Heidelberg (2004)

    Book  MATH  Google Scholar 

  2. Vardi, M.Y., Wolper, P.: An automata-theoretic approach to automatic program verification (preliminary report). In: Proceedings of the First Annual IEEE Symposium on Logic in Computer Science (LICS 1986), pp. 332–344. IEEE Computer Society Press, Los Alamitos (1986)

    Google Scholar 

  3. Vardi, M.: Model checking for database theoreticians. In: Eiter, T., Libkin, L. (eds.) ICDT 2005. LNCS, vol. 3363, pp. 1–16. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  4. Khoussainov, B., Nies, A., Rubin, S., Stephan, F.: Automatic structures: Richness and limitations. In: Proc. 19th LICS, pp. 44–53. IEEE Computer Society, Los Alamitos (2004)

    Google Scholar 

  5. Rubin, S.: Automatic Structures. PhD thesis, University of Auckland (2004)

    Google Scholar 

  6. Olver, G., Thomas, R.: Automatic presentation for finitely generated groups. In: Proc. 5th DLT. LNCS, pp. 130–144. Springer, Heidelberg (2002)

    Google Scholar 

  7. Vinokurov, N.: Complexity of some natural problems in automatic structures. Siberian Mathematical Journal 46, 56–61 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. Khoussainov, B., Minnes, M.: Model theoretic complexity of automatic structures (Extended abstract). In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 520–531. Springer, Heidelberg (2008)

    Google Scholar 

  9. Blumensath, A., Grädel, E.: Automatic structures. In: Proc. 15th LICS, pp. 51–62. IEEE Computer Society, Los Alamitos (2000)

    Google Scholar 

  10. Blumensath, A., Grädel, E.: Finite presentations of infinite structures: Automata and interpretations. Theory of Computing Systems 37, 641–674 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Campeanu, C., Culik II, K., Salomaa, K., Yu, S.: State complexity of basic operations on finite languages, automata implementation. In: Boldt, O., Jürgensen, H. (eds.) WIA 1999. LNCS, vol. 2214, pp. 60–70. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  12. Yu, S.: Regular languages. Handbook of Formal Languages, ch. 2 (1997)

    Google Scholar 

  13. Yu, S.: State complexity: recent results and open problems. Fundamenta Informaticae 64(1-4), 471–480 (2005)

    MathSciNet  MATH  Google Scholar 

  14. Hopcroft, J.E., Motwani, R., Ullman, J.: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Reading (2001)

    MATH  Google Scholar 

  15. Khoussainov, B., Minnes, M.: Three lectures on automatic structures. In: Proc. LC 2007. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  16. Rubin, S.: Automata presenting structures: A survey of the finite string case. Bulletin of Symbolic Logic 14(2), 169–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hodgson, B.: On direct products of automaton decidable theories. Theoretical Computer Science 19, 331–335 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Khoussainov, B., Nerode, A.: Automatic presentations of structures. In: Leivant, D. (ed.) LCC 1994. LNCS, vol. 960, pp. 367–392. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  19. Blumensath, A.: Automatic Structures. Diploma thesis, RWTH Aachen (1999)

    Google Scholar 

  20. Khoussainov, B., Rubin, S.: Graphs with automatic presentations over a unary alphabet. J. of Automata, Languages and Combinatorics 6(4), 467–480 (2001)

    MathSciNet  MATH  Google Scholar 

  21. Khoussainov, B., Liu, J., Minnes, M.: Unary automatic graphs: An algorithmic perspective. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 548–559. Springer, Heidelberg (2008)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Jiamou Liu

  2. Department of Mathematics, MIT, USA

    Mia Minnes

Authors
  1. Jiamou Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mia Minnes
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Rovira i Virgili University, Av. Catalunya, 35, 43002, Tarragona, Spain

    Adrian Horia Dediu

  2. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Tarragona, Spain

    Armand Mihai Ionescu

  3. European Research Council Executive Agency, 16 Place Rogier, 1210, Brussels, Belgium

    Carlos Martín-Vide

Rights and permissions

Reprints and permissions

Copyright information

© 2009 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Liu, J., Minnes, M. (2009). Analysing Complexity in Classes of Unary Automatic Structures. In: Dediu, A.H., Ionescu, A.M., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2009. Lecture Notes in Computer Science, vol 5457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00982-2_44

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-00982-2_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00981-5

  • Online ISBN: 978-3-642-00982-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation