Resource-bounded genericity

Abstract

Resource-bounded genericity concepts have been introduced by Ambos-Spies, Fleischhack and Huwig (1984, 1988), Lutz (1990), and Fenner (1991). Though it was known that these concepts are incompatible, the relations among these notions were not fully understood. We survey these notions and clarify the relations among them by specifying the types of diagonalizations captured by the individual concepts. Moreover, we introduce new, stronger resource-bounded genericity concepts corresponding to the fundamental diagonalization concepts in complexity theory. In particular we introduce general genericity, which generalizes the previous concepts and captures both standard finite extension arguments and slow diagonalizations. As we also point out, however, there is no strongest resource-bounded genericity concept. This is shown by giving a strict hierarchy of genericity notions corresponding to delayed diagonalizations. Finally we study some properties of the Baire category notions on E induced by the genericity concepts and we point out the relations between resource-bounded genericity and resource-bounded randomness.