Inversion complexity of functions of multi-valued logic

The minimum number of NOT gates in a logic circuit computing a Boolean function is called the inversion complexity of the function. In 1957, A. A. Markov determined the inversion complexity of every Boolean function and proved that ⌈log 2 (d(f)+ 1)⌉ NOT gates are necessary and sufficient to compute any Boolean function f (where d(f) is the maximum number of value changes from greater to smaller one over all increasing chains of tuples of variables values). This result is extended on k-valued functions computing in the paper. Thereupon one can use monotone functions " for free " like in Boolean case. It is shown that the minimal sufficient for a realization of the arbitrary k-valued logic function f number of non-monotone gates is equal to ⌈log 2 (d(f) + 1)⌉ if Post negation (function x + 1 (mod k)) is used in NOT nodes and is also equal to ⌈log k (d(f) + 1)⌉, if Lukasiewicz negation (function k − 1 − x) is used in NOT nodes. Similar extension for another classical result of A. A. Markov for the inversion complexity of system of Boolean functions to k-valued logic functions has been obtained. Let P k be the set of all functions of k-valued logic and M be the set of all functions that are monotone relative to order 0 < 1 <. .. < k − 1. We will