The paper is concerned with the complexity of realization of k -valuedlogicfunctions by logic circuits over an infinite complete bases containing all monotone functions;the weight of monotone functions (the cost of ...
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The paper is concerned with the complexity of realization of k -valuedlogicfunctions 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 [loge (d( f) + 1)1. 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 -valuedlogicfunctions. We show that the minimum number of Post negations (that is, functions of the form x + 1 (mod k)) that is sufficient to realize an arbitrary function of k -valuedlogic is ilog2 (d( f) + 1)1 and the minimum number of Lukasiewicz negation (that is, functions of the form k 1 x) that is sufficient to realize an arbitrary k -valuedlogic function is [logk (d( f) + 1)1. 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
We investigate the realization complexity of k -valuedlogicfunctions 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 monot...
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