Bounds of non-monotone complexity for the multi-valued logic functions

V.V. Kochergina,b∗ , A.V. Mikhailovichb∗∗

aLomonosov Moscow State University, Moscow, 119991 Russia

bNational Research University – Higher School of Economics, Moscow, 101000 Russia

E-mail: ∗vvkoch@yandex.ru, ∗∗avmikhailovich@gmail.com

Received August 17, 2020

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DOI: 10.26907/2541-7746.2020.3.311-321

For citation: Kochergin V.V., Mikhailovich A.V. Bounds of non-monotone complexity for the multi-valued logic functions. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2020, vol. 162, no. 3, pp. 311–321. doi: 10.26907/2541-7746.2020.3.311-321. (In Russian)

Abstract

The non-monotone complexity of realization of k-valued logic functions by circuits in a special basis was investigated. The basis consists of elements of two types: the first type comprises all monotone functions (with respect to the order 0 < 1 < 2 <···< k−1 ) with zero weight; the second type includes non-monotone elements with unit weight, the non-empty set of which is finite. The upper and lower bounds of non-monotone complexity (the minimum number of non-monotone elements) for an arbitrary k-valued logic function were established. The difference between the upper and lower bounds does not exceed a universal constant. The difference between the best upper and lower bounds known before is a constant that depends on the basis. The range of values for these constants is infinite.

Keywords: logic circuits, circuit complexity, k-valued logic functions, bases with zero weight elements, inversion complexity, non-monotone complexity

Acknowledgments. This study was supported in part by the Russian Foundation for Basic Research (project no. 18-01-00337).

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