Notions of resource-bounded category and genericity

Abstract

The author investigates the strength of resource-bounded generic sets for deciding results in relativized complexity. He makes technical improvements to J.H. Lutz's notion of resource-bounded Baire category (1987, 1989) to show that almost every exponential-time set (in the author's sense of category) separate P from NP. It is shown that the author's improved notion of category, while strictly more powerful, still has all the other desirable properties of Lutz's characterization of resource-bounded category in terms of Banach-Mazur games. He then considers the amount of genericity needed to prove result of M. Blum and R. Impagliazzo (1987) regarding NP intersection co-NP and one-way functions. It is found that although these results hold for 1-generic sets, they cannot be guaranteed even by extremely powerful but slightly weaker generics. A crucial difference between 1-genericity and weaker notions is thus isolated. The author studies this weaker notion of genericity and shows that it has recursion-theoretic properties radically different from 1-genericity.<>