Genericity and measure for exponential time

Abstract

Recently Lutz [14,15] introduced a polynomial time bounded version of Lebesgue measure. He and others (see e.g. [11,13,14,15,16,17,18,20]) used this concept to investigate the quantitative structure of Exponential Time (E=DTIME(2lin)). Previously, Ambos-Spies, Fleischhack and Huwig [2,3] introduced polynomial time bounded genericity concepts and used them for the investigation of structural properties of NP (under appropriate assumptions) and E. Here we relate these concepts to each other. We show that, for any c≥1 the class of nc-generic sets has p-measure 1. This allows us to simplify and extend certain p-measure 1-results. To illustrate the power of generic sets we take the Small Span Theorem of Juedes and Lutz [11] as an example and prove a generalization for bounded query reductions.