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Diskr. Mat., 1993, Volume 5, Issue 3, Pages 40–43 (Mi dm689)  

This article is cited in 2 scientific papers (total in 2 papers)

On the number of threshold functions

A. A. Irmatov


Abstract: A Boolean function is called a threshold function if its truth domain is a part of the $n$-cube cut off by some hyperplane. The number of threshold functions of $n$ variables $P(2,n)$ was estimated in [1, 2, 3]. Obtaining the lower bounds is a problem of special difficulty. Using a result of [4], Yu. A. Zuev showed [3] that for sufficiently large $n$
$$ P(2,n)>2^{n^2(1-10/\ln n)}. $$
In the present paper a new proof which gives a more precise lower bound of $P(2,n)$ is proposed, namely, it is proved that for sufficiently large $n$
$$ P(2,n)>2^{n^2(1-7/\ln n)}P(2,[\frac{7(n-1)\ln 2}{\ln(n-1)}]). $$


Full text: PDF file (443 kB)

English version:
Discrete Mathematics and Applications, 1993, 3:4, 429–432

Bibliographic databases:

Received: 02.07.1992

Citation: A. A. Irmatov, “On the number of threshold functions”, Diskr. Mat., 5:3 (1993), 40–43; Discrete Math. Appl., 3:4 (1993), 429–432

Citation in format AMSBIB
\Bibitem{Irm93}
\by A.~A.~Irmatov
\paper On the number of threshold functions
\jour Diskr. Mat.
\yr 1993
\vol 5
\issue 3
\pages 40--43
\mathnet{http://mi.mathnet.ru/dm689}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1266256}
\zmath{https://zbmath.org/?q=an:0796.94016}
\transl
\jour Discrete Math. Appl.
\yr 1993
\vol 3
\issue 4
\pages 429--432


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Irmatov A.A., “Arrangements of hyperplanes and the number of threshold functions”, Acta Applicandae Mathematicae, 68:1–3 (2001), 211–226  crossref  mathscinet  zmath  isi
    2. A. D. Korshunov, “Some unsolved problems in discrete mathematics and mathematical cybernetics”, Russian Math. Surveys, 64:5 (2009), 787–803  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
  • Дискретная математика
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