On the Complexity of Multivalued Logic Functions over Some Infinite Basis

Abstract

Under study is the complexity of the realization of k-valued logic functions (k ≥ 3) by logic circuits in the infinite basis consisting of the Post negation (i.e., the function (x + 1) mod k) and all monotone functions. The complexity of the circuit is the total number of elements of this circuit. For an arbitrary function f, we find the lower and upper bounds of complexity, which differ from one another at most by 1 and have the form 3 log₃(d(f)+ 1)+O(1), where d(f) is the maximal number of the decrease of the value of f taken over all increasing chains of tuples of values of the variables. We find the exact value of the corresponding Shannon function which characterizes the complexity of the most complex function of a given number of variables.