On the asymptotic behavior of Shannon-type functions characterizing the computing complexity of systems of monomials

S.A. Korneev

Lomonosov Moscow State University, Moscow, 119991 Russia Keldysh Institute of Applied Mathematics,

Russian Academy of Sciences, Moscow, 125047 Russia

E-mail: korneev.sa.42@gmail.com

Received July 15, 2020

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DOI: 10.26907/2541-7746.2020.3.300-310

For citation: Korneev S.A. On the asymptotic behavior of Shannon-type functions characterizing the computing complexity of systems of monomials. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 3, pp. 300–310. doi: 10.26907/2541-7746.2020.3.300-310. (In Russian)

 

Abstract

In this paper, we examined the computational complexity of systems of monomials for some models that allow multiple use of intermediate results, such as composition circuits and multiplication circuits.

For these models, we studied Shannon-type functions that characterize the maximum computational complexity of systems of monomials with exponents not exceeding the corresponding elements of a given matrix A. We found that for composition circuits, under the condition of unlimited growth of the maximum of matrix elements, this function grows asymptotically as the binary logarithm of the maximum absolute value (without regard to the sign) of the term from the determinant of the matrix A. Using generalized circuits as an auxiliary model, we transferred this result (under some restrictions) to the model of multiplication circuits.

Keywords: set of monomials, computation complexity, circuit complexity, Shannon function

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