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.
The research Supported partially Russia Fund for Basic Research under the grant 99-01-00163 and Fund “Russia Universities” under the grant 04.01.52
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
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).
V. Arnold, On functions of Three Variables, Dokladi Akademii Nauk, 114,4, (1957), 679–681.
D. Hilbert, Mathematische Probleme, Nachr. Akad. Wiss. Gottingen (1900) 253–297; Gesammelete Abhandlungen, Bd. 3 (1935), 290–329.
J. Hromkovic, Communication Complexity and Parallel Computing, EATCS Series, Springer-Verlag, (1997).
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.
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.
E. Kushilevitz and N. Nisan, Communication complexity, Cambridge University Press, (1997).
G. Lorenz, Metric Entropy, Widths and Superpositions Functions, Amer. Math. Monthly 69,6, (1962), 469–485.
S. Marchenkov, On One Method of Analysis of superpositions of Continuous Functions, Problemi Kibernetici, 37, (1980), 5–17.
A. Vitushkin, On Representation of Functions by Means of Superpositions and Related Topics, L’Enseignement mathematique, 23, fasc.3–4, (1977), 255–320.
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.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ablayev, F., Ablayeva, S. (2001). A Discrete Approximation and Communication Complexity Approach to the Superposition Problem. In: Freivalds, R. (eds) Fundamentals of Computation Theory. FCT 2001. Lecture Notes in Computer Science, vol 2138. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44669-9_7
Download citation
DOI: https://doi.org/10.1007/3-540-44669-9_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-42487-1
Online ISBN: 978-3-540-44669-9
eBook Packages: Springer Book Archive