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 Prikl. Diskr. Mat.: Year: Volume: Issue: Page: Find

 Prikl. Diskr. Mat., 2019, Number 45, Pages 13–25 (Mi pdm667)

Theoretical Backgrounds of Applied Discrete Mathematics

On the degree of restrictions of $q$-valued logic functions to linear manifolds

V. G. Ryabov

NP “GST”, Moscow, Russia

Abstract: In case of a finite field $\mathbb{F}_q$, the degree of restricting a $q$-valued logic function in $n$ variables to a $r$-dimensional linear manifold of the vector space $\mathbb{F}_q^n$ is defined as the degree of a polynomial in $r$ variables that represents this restriction. For manifolds of a fixed dimension, the probability of occurrence of restrictions with a degree not higher than the given one is estimated, and the asymptotics of the number of manifolds on which the restrictions are affine is obtained. It is shown that if $n \to \infty$, for almost all $q$-valued logic functions in $n$ variables, the value of the maximum dimension of a linear manifold on which the restriction is affine belongs to the segment $[\lfloor \log_q n+\log_q \log_q n \rfloor, \lceil \log_q n+\log_q \log_q n \rceil]$, while the analogous parameter for the case of fixing variables is in the range $[\lfloor \log_q n \rfloor, \lceil \log_q n \rceil]$.

Keywords: many-valued logic, Boolean function, restriction, linear manifold, degree.

DOI: https://doi.org/10.17223/20710410/45/2

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UDC: 519.1,519.7

Citation: V. G. Ryabov, “On the degree of restrictions of $q$-valued logic functions to linear manifolds”, Prikl. Diskr. Mat., 2019, no. 45, 13–25

Citation in format AMSBIB
\Bibitem{Rya19} \by V.~G.~Ryabov \paper On the degree of restrictions of $q$-valued logic functions to linear manifolds \jour Prikl. Diskr. Mat. \yr 2019 \issue 45 \pages 13--25 \mathnet{http://mi.mathnet.ru/pdm667} \crossref{https://doi.org/10.17223/20710410/45/2}