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 Sib. Èlektron. Mat. Izv., 2017, Volume 14, Pages 1100–1107 (Mi semr850)

Discrete mathematics and mathematical cybernetics

Asymptotics of growth for non-monotone complexity of multi-valued logic function systems

V. V. Kochergina, A. V. Mikhailovichb

a Lomonosov Moscow State University, GSP-1, Leninskie Gory, 119991, Moscow, Russia
b Higher School of Economics, str. Myasnitskaya, 20, 101000, Moscow, Russia

Abstract: The problem of the complexity of multi-valued logic functions realization by circuits in a special basis is investigated. This kind of basis consists of elements of two types. The first type of elements are monotone functions with zero weight. The second type of elements are non-monotone elements with unit weight. The non-empty set of elements of this type is finite.
In the paper the minimum number of non-monotone elements for an arbitrary multi-valued logic function system $F$ is established. It equals $\lceil\log_{u}(d(F)+1)\rceil - O(1)$. Here $d(F)$ is the maximum number of the value decrease over all increasing chains of tuples of variable values for at least one function from system $F$; $u$ is the maximum (over all non-monotone basis functions and all increasing chains of tuples of variable values) length of subsequence such that the values of the function decrease over these subsequences.

Keywords: combinational machine (logic circuits), circuits complexity, bases with zero weight elements, $k$-valued logic functions, inversion complexity, Markov's theorem, Shannon function.

DOI: https://doi.org/10.17377/semi.2017.14.093

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UDC: 519.714
MSC: 94C10
Received June 1, 2017, published November 9, 2017
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Citation: V. V. Kochergin, A. V. Mikhailovich, “Asymptotics of growth for non-monotone complexity of multi-valued logic function systems”, Sib. Èlektron. Mat. Izv., 14 (2017), 1100–1107

Citation in format AMSBIB
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