Complexity of implementing functions of k-valued logic by circuits and formulas in functionally complete bases

Abstract

Algorithmic problems are considered that are related to implementing bounded-deterministic functions by circuits and formulas of the minimum size in automaton bases. The problem of finding the asymptotics of the Shannon function is known to be algorithmically undecidable in the case of complete bases, but the coefficient in the formula for the Shannon function can be found with arbitrary accuracy. In the paper the so called strong algorithmic undecidability of the problem of finding the asymptotics of the Shannon function in the case of functionally complete bases is proved. A basis is called cf-equivalent if the constants in the asymptotic formulas for the Shannon function in the classes of circuits and formulas coincide. The existence of bases that are not cf-equivalent is proved in the case of functionally complete bases. It is proved that the recognition problem for the cf-equivalence of a basis is algorithmically undecidable in the strong sense.