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 Diskr. Mat., 2016, Volume 28, Issue 4, Pages 80–90 (Mi dm1394)

The minimum number of negations in circuits for systems of multi-valued functions

V. V. Kocherginab, A. V. Mikhailovichc

a Lomonosov Moscow State University
b Lomonosov Moscow State University, Bogoliubov Institute for Theoretical Problems of Microphysics
c National Research University "Higher School of Economics" (HSE), Moscow

Abstract: The paper is concerned with the complexity of realization of $k$-valued logic functions by logic circuits over an infinite complete bases containing all monotone functions; the weight of monotone functions (the cost of use) is assumed to be $0$. The complexity problem of realizations of Boolean functions over a basis having negation as the only nonmonotone element was completely solved by A. A. Markov. In 1957 he showed that the minimum number of NOT gates sufficient for realization of any Boolean function $f$ (the inversion complexity of the function $f$) is $\lceil\log_2(d(f)+1)\rceil$. Here $d(f)$ is the maximum number of the changes of the function $f$ from larger to smaller values over all increasing chains of tuples of variables values. In the present paper Markov's result is extended to the case of realization of $k$-valued logic functions. We show that the minimum number of Post negations (that is, functions of the form $x+1\pmod{k}$) that is sufficient to realize an arbitrary function of $k$-valued logic is $\lceil\log_2(d(f)+1)\rceil$ and the minimum number of Łukasiewicz negation (that is, functions of the form $k-1-x$) that is sufficient to realize an arbitrary $k$-valued logic function is $\lceil\log_k(d(f)+1)\rceil$. In addition, another classical Markov's result on the inversion complexity of systems of Boolean functions is extended to the setting of systems of functions of $k$-valued logic.

Keywords: multi-valued logic functions, logic circuits, circuit complexity, nonmonotone complexity, inversion complexity, Markov's theorem.

 Funding Agency Grant Number National Research University Higher School of Economics 14-01-0144 Russian Foundation for Basic Research 14-01-00598_à This study was supported by the Academic Fund Programme of the National Research University Higher School of Economics in 2014/2015 (research grant no. 14-01-0144). The first author was supported by the Russian Foundation for Basic Research (project no. 14–01–00598).

DOI: https://doi.org/10.4213/dm1394

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English version:
Discrete Mathematics and Applications, 2017, 27:5, 295–302

Bibliographic databases:

UDC: 519.714

Citation: V. V. Kochergin, A. V. Mikhailovich, “The minimum number of negations in circuits for systems of multi-valued functions”, Diskr. Mat., 28:4 (2016), 80–90; Discrete Math. Appl., 27:5 (2017), 295–302

Citation in format AMSBIB
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\paper The minimum number of negations in circuits for systems of multi-valued functions
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\pages 80--90
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\transl
\jour Discrete Math. Appl.
\yr 2017
\vol 27
\issue 5
\pages 295--302
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• http://mi.mathnet.ru/eng/dm1394
• https://doi.org/10.4213/dm1394
• http://mi.mathnet.ru/eng/dm/v28/i4/p80

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This publication is cited in the following articles:
1. V. V. Kochergin, A. V. Mikhailovich, “Asymptotics of growth for non-monotone complexity of multi-valued logic function systems”, Sib. elektron. matem. izv., 14 (2017), 1100–1107
2. V. V. Kochergin, A. V. Mikhailovich, “On the complexity of multivalued logic functions over some infinite basis”, J. Appl. Industr. Math., 12:1 (2018), 40–58
3. V. V. Kochergin, A. V. Mikhailovich, “Exact Value of the Nonmonotone Complexity of Boolean Functions”, Math. Notes, 105:1 (2019), 28–35
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