Skip to main content
SpringerLink
Log in

The Inherent Dimension of Bounded Counting Classes

  • Conference paper
  • First Online:
Computing and Combinatorics (COCOON 1998)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1449))

Included in the following conference series:

Abstract

Cronauer et al. [2] introduced the chain method to separate counting classes by oracles. Among the classes, for which this method is applicable, are NP, coNP, MOD-classes, . . . As these counting classes are defined via subsets of ℕk, it is natural to ask for the minimum value of k, such that a given class can be defined via such a set. We call this value the inherent dimension of the respective class.

The inherent dimension is a very natural concept, but it is quite hard to check the value for a given bounded counting class. Thus, we complement this notion by the notion of type-3 dimension, which is less natural than inherent dimension, but very easy to check. We compare type-3 dimension and inherent dimension, with the result that for classes of inherent dimension less than 3, both notions coincide, and generally the inherent dimension is never greater than the type-3 dimension.

For k ≤ 2 we can completely solve the questions, whether a given class has inherent dimension k, and which are the minimal classes with that dimension. For k ≥ 3 we give a sufficient condition for a class being of dimension at least k. We disprove the conjecture that this is also a necessary condition by a counterexample.

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. Bovet, D. P., Crescenzi, P., Silvestri, R.: A Uniform Approach to Define Complexity Classes. Theoretical Computer Science 104 (1992) 263–283.

    Article  MathSciNet  Google Scholar 

  2. Cronauer, K., Hertrampf, U., Vollmer, H., Wagner, K.W.: The Chain Method to Separate Counting Classes. Theory of Computing Systems 31 (1998) 93–108.

    Article  MathSciNet  Google Scholar 

  3. Fenner, S., Fortnow, L. Kurtz, S.: Gap-definable Counting Classes. Journal of Computer and System Sciences 48 (1994) 116–148.

    Article  MathSciNet  Google Scholar 

  4. Gundermann, T., Nasser, N.A., Wechsung, G.: A Survey on Counting Classes. Proceedings of the 5th Structure in Complexity Theory Conference. IEEE (1990) 140–153.

    Google Scholar 

  5. Hertrampf, U.: Complexity Classes with Finite Acceptance Types. Proceedings of the 11th Symp. on Theoretical Aspects of Computer Science. LNCS 775 (1994) 543–553.

    Google Scholar 

  6. Hertrampf, U.: Classes of Bounded Counting Type and Their Inclusion Relations. Proceedings of the 12th Symp. on Theoretical Aspects of Computer Science. LNCS 900 (1995) 60–70.

    Google Scholar 

  7. Hertrampf, U.: The Inherent Dimension of Bounded Counting Classes. Technical Report. University of Stuttgart (1998).

    Google Scholar 

  8. Hertrampf, U., Lautemann, C., Schwentick, T., Vollmer, H., Wagner, K.W.: On the Power of Polynomial Time Bit-Reductions. Proceedings of the 8th Structure in Complexity Theory Conference. IEEE (1993) 200–207.

    Google Scholar 

  9. Hertrampf, U., Vollmer, H., Wagner, K.W.: On Balanced vs. Unbalanced Computation Trees. Mathematical Systems Theory 29 (1996) 411–421.

    Article  MathSciNet  Google Scholar 

  10. Jenner, B., McKenzie, P., Thérien, D.: Logspace and Logtime Leaf Languages. Proceedings of the 9th Structure in Complexity Theory Conference. IEEE (1994) 242–254.

    Google Scholar 

  11. Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading, Mass. (1994).

    MATH  Google Scholar 

  12. Valiant, L.G.: The Complexity of Computing the Permanent. Theoretical Computer Science 8 (1979) 189–201.

    Article  MathSciNet  Google Scholar 

  13. Vereshchagin, N.K.: Relativizable and Non-relativizable Theorems in the Polynomial Theory of Algorithms. (In Russian.) Izvestija Rossijskoj Akademii Nauk 57 (1993) 51–90.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Universität Stuttgart, Breitwiesenstr. 20-22, D-70565, Stuttgart, Germany

    Ulrich Hertrampf

Authors
  1. Ulrich Hertrampf
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institute of Information Science, Academia Sinica, 11529, Nankang, Taipei, Taiwan

    Wen-Lian Hsu

  2. Department of Computer Science, Yale University, 51 Prospect Street, New Haven, CT, 06520, USA

    Ming-Yang Kao

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Hertrampf, U. (1998). The Inherent Dimension of Bounded Counting Classes. In: Hsu, WL., Kao, MY. (eds) Computing and Combinatorics. COCOON 1998. Lecture Notes in Computer Science, vol 1449. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-68535-9_19

Download citation

  • DOI: https://doi.org/10.1007/3-540-68535-9_19

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64824-6

  • Online ISBN: 978-3-540-68535-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation