The complexity of the descriptiveness of Boolean circuits over different sets of gates

Abstract. Any Boolean function can be defined by a Boolean circuit, provided we may use sufficiently strong functions in its gates. On the other hand, it depends on these gate functions, what Boolean functions can be defined: Each set B of gate functions defines the class of Boolean functions that can be defined by circuits over B. Although these classes are known since the 1920s, their computational complexity was never investigated. In this paper we will study how difficult it is to decide for a Boolean function f and a class B, whether f is in B. Moreover we will provide such a decision algorithm with additional information: How difficult is it to decide whether or not f is in B, provided we already know a circuit for f, but with gates from another class A? Given such a circuit, we know that f is in A. Is the problem harder if we do not have a concrete representation for f, but still know that it is from A? For nearly all possible combinations, we show that this is not the case,and that the problem is either in P or coNP-complete.