Skip to main content
SpringerLink
Log in
Book cover

Annual Symposium on Theoretical Aspects of Computer Science

STACS 1998: STACS 98 pp 607–617Cite as

Uniformly defining complexity classes of functions

  • Complexity IV
  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1373))

Abstract

We introduce a general framework for the definition of function classes. Our model, which is based on polynomial time nondeterministic Turing transducers, allows uniform characterizations of FP, FPNP, counting classes (#·P, #·NP, #·coNP, GapP, GapPNP), optimization classes (max·P, min·P, max·NP, min·NP), promise classes (NPSV, #few·P, c#·P), multivalued classes (FewFP, NPMV) and many more. Each such class is defined in our model by a certain family of functions. We study a reducibility notion between such families, which leads to a necessary and sufficient criterion for relativizable inclusion between function classes. As it turns out, this criterion is easily applicable and we get as a consequence e.g. that there are oracles A, B, such that min.PA \(\nsubseteq\) #·NPA, and max.NPB \(\nsubseteq\) c#·coNPB (note that no structural consequences are known to follow from the corresponding positive inclusions).

Supported by the Deutsche Forschungsgemeinschaft (DFG), grant Wa 847/1-2.

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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. L. Balcázar, J. Diaz, and J. Gabarró. Structural Complexity I. Texts in Theoretical Computer Science. Springer-Verlag, Berlin Heidelberg, 2nd edition, 1995.

    Google Scholar 

  2. R. V. Book, T. Long, and A. Selman. Quantitative relativizations of complexity classes. SIAM Journal on Computing, 13:461–487, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  3. B. Borchert. Predicate classes, promise classes, and the acceptance power of regular languages. PhD thesis, Naturwissenschaftlich-Mathematische Fakultät, Universität Heidelberg, 1994.

    Google Scholar 

  4. D. P. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. International Series in Computer Science. Prentice Hall, London, 1994.

    Google Scholar 

  5. D. P. Bovet, P. Crescenzi, and R. Silvestri. A uniform approach to define complexity classes. Theoretical Computer Science, 104:263–283, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  6. S. Fenner, L. Fortnow, and S. Kurtz. Gap-definable counting classes. Journal of Computer and System Sciences, 48:116–148, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Glaßer and G. Wechsung. Relativizing function classes. Manuscript, 1997.

    Google Scholar 

  8. L. Hemaspaandra and H. Vollmer. The satanic notations: counting classes beyond #P and other definitional adventures. Complexity Theory Column 8, ACM SIGACT-Newsletter, 26(1):2–13, 1995.

    Google Scholar 

  9. H. Hempel and G. Wechsung. The operators min and max on the polynomial hierarchy. In Proceedings 14th Symposium on Theoretical Aspects of Computer Science, volume 1200 of Lecture Notes in Computer Science, pages 93–104. Springer-Verlag, 1997.

    Google Scholar 

  10. U. Hertrampf. Classes of bounded counting type and their inclusion relations. In Proceedings 12th Symposium on Theoretical Aspects of Computer Science, volume 900 of Lecture Notes in Computer Science, pages 60–70. Springer-Verlag, 1995.

    Google Scholar 

  11. U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, and K. W. Wagner. On the power of polynomial time bit-reductions. In Proceedings 8th Structure in Complexity Theory, pages 200–207, 1993.

    Google Scholar 

  12. U. Hertrampf, H. Vollmer, and K. W. Wagner. On the power of number-theoretic operations with respect to counting. In Proceedings 10th Structure in Complexity Theory, pages 299–314, 1995.

    Google Scholar 

  13. B. Jenner, P. McKenzie, and D. Th6rien. Logspate and logtime leaf languages. In 9th Annual Conference Structure in Complexity Theory, pages 242–254, 1994.

    Google Scholar 

  14. B. Jenner and J. Torán. Computing functions with parallel queries to NP. Theoretical Computer Science, 141:175–193, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Köbler. Strukturelle Komplexitdt von Anzahlproblemen. PhD thesis, Universität Stuttgart, Fakultät für Informatik, 1989.

    Google Scholar 

  16. J. Köbler, U. Schöning, and J. Torán. On counting and approximation. Acta Informatica, 26:363–379, 1989.

    MATH  MathSciNet  Google Scholar 

  17. S. Kosub. On cluster machines and function classes. Technical Report 172, Institut für Informatik, Universität Würzburg, 1997.

    Google Scholar 

  18. M. W. Krentel. The complexity of optimization functions. Journal of Computer and System Sciences, 36:490–509, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  19. M. W. Krentel. Generalizations of OptP to the polynomial hierarchy. Theoretical Computer Science, 97:183–198, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  20. C. H. Papadimitriou. Computational Complexity. Addison-Wesley, Reading, MA, 1994.

    MATH  Google Scholar 

  21. H. Schmitz. Nichtdeterministische Polynomialzeit-Berechnung von Funktionen. Master's thesis, Institut für Informatik, Universität Würzburg, 1996.

    Google Scholar 

  22. A. Selman. A taxonomy on complexity classes of functions. Journal of Computer and System Sciences, 48:357–381, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Toda. PP is as hard as the polynomial time hierarchy. SIAM Journal on Computing, 20:865–877, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  24. L. G. Valiant. The complexity of enumeration and reliabilty problems. SIAM Journal of Computing, 8(3):411–421, 1979.

    Article  MathSciNet  Google Scholar 

  25. N. K. Vereshchagin. Relativizable and non-relativizable theorems in the polynomial theory of algorithms. Izvestija Rossijskoj Akademii Nauk, 57:51–90, 1993. In Russian.

    Google Scholar 

  26. H. Vollmer. On different reducibility notions for function classes. In Proceedings 11th Symposium on Theoretical Aspects of Computer Science, volume 775 of Lecture Notes in Computer Science, pages 449–460. Springer-Verlag, 1994.

    Google Scholar 

  27. H. Vollmer and K. W. Wagner. The complexity of finding middle elements. International Journal of Foundations of Computer Science, 4:293–307, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  28. H. Vollmer and K. W. Wagner. Complexity classes of optimization functions. Information and Computation, 120:198–219, 1995.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Theoretische Informatik, Universität Würzburg, Am Exerzierplatz 3, 97072, Würzburg, Germany

    Sven Kosub, Heinz Schmitz & Heribert Vollmer

Authors
  1. Sven Kosub
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Heinz Schmitz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Heribert Vollmer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Michel Morvan Christoph Meinel Daniel Krob

Rights and permissions

Reprints and permissions

Copyright information

© 1998 Springer-Verlag

About this paper

Cite this paper

Kosub, S., Schmitz, H., Vollmer, H. (1998). Uniformly defining complexity classes of functions. In: Morvan, M., Meinel, C., Krob, D. (eds) STACS 98. STACS 1998. Lecture Notes in Computer Science, vol 1373. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028595

Download citation

  • DOI: https://doi.org/10.1007/BFb0028595

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-64230-5

  • Online ISBN: 978-3-540-69705-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation