On ways of characterizing complete sets

The traditional method for constructing criteria for completeness with respect to reducibility is by describing the property of (in general, weakened) productiveness satisfied by the complement of a set which is complete with respect to the given reducibility. Originally this property was tied to the reducibility of a creative set to the complete set. Such a method appeals directly to the universality of the creative set in the class of all recursively enumerable sets.
However, for several reducibilities it is possible to determine the completeness of a recursively enumerable set from the fact that a certain set of degree below the degree of the creative sets is reducible to the given set. This second, “test” set is, of course, not recursively enumerable. In addition, in place of the property of effective nonrecursive enumerability which productive sets have, it is possible to substitute variants of the property of diagonal nonrecursiveness, although not for all reducibilities.
In this paper we examine the connection between these two approaches – specifically, between different weakenings of the property of productiveness on the one hand, and diagonal nonrecursiveness on the other – and we present results obtained by these methods for Turing and truth-table reducibilities.