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

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.