Circuit Complexity of k-Valued Logic Functions in One Infinite Basis

Abstract

We investigate the realization complexity of k -valued logic functions 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 monotone functions. Complexity is understood as the total number of circuit elements. For an arbitrary function f, we establish lower and upper complexity bounds that differ by at most by 2 and have the form 2 log (d(f) + 1) + o(1), where d(f) is the maximum number of times the function f switches from larger to smaller value (the maximum is taken over all increasing chains of variable tuples). For all sufficiently large n, we find the exact value of the Shannon function for the realization complexity of the most complex function of n variables.