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

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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.

## Keywords

multi-valued logic functions logic circuits circuit complexity infinite basis non-monotone complexity inversion complexity Markov theorem## Preview

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