Skip to main content
Springer Nature Link
Log in

On balanced versus unbalanced computation trees

  • Published:
Mathematical systems theory Aims and scope Submit manuscript

Abstract

A great number of complexity classes between P and PSPACE can be defined via leaf languages for computation trees of nondeterministic polynomial-time machines. Jenner, McKenzie, and Thérien (Proceedings of the 9th Conference on Structure in Complexity Theory, 1994) raised the issue of whether considering balanced or unbalanced trees makes any difference. For a number of leaf-language classes, coincidence of both models was shown, but for the very prominent example of leaf-language classes from the alternating logarithmic-time hierarchy the question was left open. It was only proved that in the balanced case these classes exactly characterize the classes from the polynomial-time hierarchy. Here, we show that balanced trees apparently make a difference: In the unbalanced case, a class from the logarithmic-time hierarchy characterizes the corresponding class from the polynomial-time hierarchy with a PP-oracle. Along the way, we get an interesting normal form for PP computations.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. L. Balcázar, J. Díaz, J. Gabarró,Structural Complexity I; Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  2. J. L. Balcázar, J. Díaz, J. Gabarró,Structural Complexity II; Springer-Verlag, Berlin, 1990.

    MATH  Google Scholar 

  3. D. P. Bovet, P. Crescenzi, R. Silvestri, Complexity classes and sparse oracles;Proceedings of the 6thStructure in Complexity Theory Conference (1991), pp. 102–108.

  4. D. P. Bovet, P. Crescenzi, R. Silvestri, A uniform approach to define complexity classes;Theoretical Computer Science 104 (1992), 263–283.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. K. Chandra, D. C. Kozen, L. J. Stockmeyer, Alternation;Journal of the ACM 28 (1981), 114–133.

    Article  MathSciNet  MATH  Google Scholar 

  6. J. Gill, Computational complexity of probabilistic complexity classes;SIAM Journal on Computing 6 (1977), 675–695.

    Article  MathSciNet  MATH  Google Scholar 

  7. Y. Han, L. Hemachandra, T. Thierauf, Threshold computation and cryptographic security;Proceedings of the 4th International Symposium on Algorithms and Computation (1993), LNCS 762, pp. 230–239.

  8. U. Hertrampf, Locally definable acceptance types-the three valued case;Proceedings of the 1stLatin American Symposium on Theoretical Informatics (1992), LNCS 583, pp. 262–271.

  9. U. Hertrampf, Locally definable acceptance types for polynomial time machines;Proceedings of the 9thSymposium on Theoretical Aspects of Computer Science (1992), LNCS 577, pp. 199–207.

  10. U. Hertrampf, Complexity classes defined via k-valued functions;Proceedings of the 9thStructure in Complexity Theory Conference (1994), pp. 224–234.

  11. U. Hertrampf, Über Komplexitätsklassen, die mit Hilfe k-wertiger Funktionen definiert werden; Habilitationsschrift, Universität Würzburg, 1994.

  12. U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, K. W. Wagner, On the power of polynomial time bit-reductions;Proceedings of the 8thStructure in Complexity Theory Conference (1993), pp. 200–207.

  13. J. E. Hopcroft, J. D. Ullman,Introduction to Automata Theory, Languages, and Computation; Addison-Wesley, Reading, MA, 1979.

    MATH  Google Scholar 

  14. B. Jenner, P. McKenzie, D. Thérien, Logspace and logtime leaf languages;Proceedings of the 9thStructure in Complexity Theory Conference (1994), pp. 242–254.

  15. J. Simon, On some central problems in computational complexity; Ph.D. Thesis, Cornell University, Ithaca, NY, 1975.

    Google Scholar 

  16. M. Sipser, Borel sets and circuit complexity;Proceedings of the 15th Symposium on the Theory of Computing (1983), pp. 61–69.

  17. C. Wrathall, Complete sets and the polynomial-time hierarchy;Theoretical Computer Science 3 (1977), 23–33.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    U. Hertrampf, H. Vollmer & K. W. Wagner

Authors
  1. U. Hertrampf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. Vollmer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. K. W. Wagner
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The first and third authors were supported by Deutsche Forschungsgemeinschaft, Grant No. Wa 847/1-1, “k-wertige Schaltkreise.” The second author was supported in part by an Alexander von Humboldt fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hertrampf, U., Vollmer, H. & Wagner, K.W. On balanced versus unbalanced computation trees. Math. Systems Theory 29, 411–421 (1996). https://doi.org/10.1007/BF01192696

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01192696

Keywords