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A Discrete Approximation and Communication Complexity Approach to the Superposition Problem

  • Farid Ablayev
  • Svetlana Ablayeva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2138)

Abstract

The superposition (or composition) problem is a problem of representation of a function f by a superposition of “simpler” (in a different meanings) set Ω of functions. In terms of circuits theory this means a possibility of computing f by a finite circuit with 1 fan-out gates Ω of functions.

Using a discrete approximation and communication approach to this problem we present an explicit continuous function f from Deny class, that can not be represented by a superposition of a lower degree functions of the same class on the first level of the superposition and arbitrary Lipshitz functions on the rest levels. The construction of the function f is based on particular Pointer function g (which belongs to the uniform AC0) with linear one-way communication complexity.

Keywords

Boolean Function Communication Complexity Discrete Approximation Discrete Function Arbitrary Continuous Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Ab]
    F. Ablayev, Communication method of the analyses of superposition of continuous functions, in Proceedings of the international conference ”Algebra and Analyses part II. Kazan, 1994, 5–7 (in Russian). See also F. Ablayev, Communication complexity of probabilistic computations and some its applications, Thesis of doctor of science dissertation, Moscow State University, 1995, (in Russian).Google Scholar
  2. [Ar]
    V. Arnold, On functions of Three Variables, Dokladi Akademii Nauk, 114,4, (1957), 679–681.Google Scholar
  3. [Hi]
    D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Gottingen (1900) 253–297; Gesammelete Abhandlungen, Bd. 3 (1935), 290–329.Google Scholar
  4. [Hr]
    J. Hromkovic, Communication Complexity and Parallel Computing, EATCS Series, Springer-Verlag, (1997).Google Scholar
  5. [KaRaWi]
    M. Karchmer, R. Raz, and A. Wigderson, Super-logarithmic Depth Lower Bounds Via the Direct Sum in Communication Complexity, Computational Complexity, 5, (1995), 191–204.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [Ko]
    A. Kolmogorov, On Representation of Continuous Functions of Several Variables by a superposition of Continuous Functions of one Variable and Sum Operation. Dokladi Akademii Nauk, 114,5, (1957), 953–956.zbMATHMathSciNetGoogle Scholar
  7. [KuNi]
    E. Kushilevitz and N. Nisan, Communication complexity, Cambridge University Press, (1997).Google Scholar
  8. [Lo]
    G. Lorenz, Metric Entropy, Widths and Superpositions Functions, Amer. Math. Monthly 69,6, (1962), 469–485.CrossRefMathSciNetGoogle Scholar
  9. [Ma]
    S. Marchenkov, On One Method of Analysis of superpositions of Continuous Functions, Problemi Kibernetici, 37, (1980), 5–17.MathSciNetGoogle Scholar
  10. [Vi]
    A. Vitushkin, On Representation of Functions by Means of Superpositions and Related Topics, L’Enseignement mathematique, 23, fasc.3–4, (1977), 255–320.zbMATHMathSciNetGoogle Scholar
  11. [Yao]
    A. C. Yao, Some Complexity Questions Related to Distributive Computing, in Proc. of the 11th Annual ACM Symposium on the Theory of Computing, (1979), 209–213.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Farid Ablayev
    • 1
    • 2
  • Svetlana Ablayeva
    • 1
    • 2
  1. 1.Dept. of Theoretical CyberneticsKazan State UniversityKazanRussia
  2. 2.Department of Differential EquationsKazan State UniversityKazanRussia

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