We investigate the realization complexity of k -valued logic functions k 2 by combinational circuits in an infinite basis that includes the negation of the Lukasiewicz function, i.e., the function k−1−x, and all monotone functions. Complexity is understood as the total number of circuit elements. For an arbitrary function f, we establish lower and upper complexity bounds that differ by at most by 2 and have the form 2 log (d(f) + 1) + o(1), where d(f) is the maximum number of times the function f switches from larger to smaller value (the maximum is taken over all increasing chains of variable tuples). For all sufficiently large n, we find the exact value of the Shannon function for the realization complexity of the most complex function of n variables.
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V. V. Kochergin and A. V. Mikhailovich, “On the complexity of multi-valued logic functions in one infinite basis,” Diskretn. Anal. Issled. Operatsii, 25, No. 1, 42–74 (2018).
V. V. Kochergin and A. V. Mikhailovich, “On the minimal number of negations for the realization of systems of k-valued logic functions,” Diskr. Matem., 28, No. 4, 80–90 (2016).
A. A. Markov, “On inversion complexity of systems of functions,” Dokl. Akad. Nauk SSSR, 116, No. 6, 917–919 (1957).
A. A. Markov, “On inversion complexity of systems of Boolean functions,” Dokl. Akad. Nauk SSSR, 150, No. 3, 477–479 (1963).
O. B. Lupanov, Asymptotic Complexity Bounds for Control Systems [in Russian], Izd. MGU, Moscow (1984).
V. V. Kochergin and A. V. Mikhailovich, “On the complexity of circuits in bases containing monotone elements with zero weights,” Prikl. Diskr. Matem., No. 4(30), 24–31 (2015).
V. V. Kochergin and A. V. Mikhailovich, “Asymptotics of growth for the non-monotone complexity of multi-valued logic function systems,” Siberian Electronic Mathematical Reports (http://semr.math.msc.ru), 14, 1100–1107 (2017).
V. V. Kochergin and A. V. Mikhailovich, “Exact value of non-monotone complexity of Boolean functions,” Matem. Zametki (in print).
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Translated from Prikladnaya Matematika i Informatika, No. 58, 2018, pp. 21–34.
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Kochergin, V.V., Mikhailovich, A.V. Circuit Complexity of k-Valued Logic Functions in One Infinite Basis. Comput Math Model 30, 13–25 (2019). https://doi.org/10.1007/s10598-019-09430-5
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DOI: https://doi.org/10.1007/s10598-019-09430-5