Resource Bounded Randomness and Weakly Complete Problems 1

We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ((7, 8]). We concentrate on n c-randomness (c 2) which corresponds to the polynomial time bounded (p-) measure of Lutz, and which is adequate for studying the internal and quantitative structure of E = DTIME(2 lin). However we will also comment on E2 = DTIME(2 pol) and its corresponding (p2-) measure. First we show that the class of n c-random sets has p-measure 1. This provides a new, simpliied approach to p-measure 1-results. Next we compare ran-domness with genericity (in the sense of 2, 3]) and we show that n c+1-random sets are n c-generic, whereas the converse fails. From the former we conclude that n c-random sets are not p-btt-complete for E. Our technical main results describe the distribution of the n c-random sets under p-m-reducibility. We show that every n c-random set in E has n k-random predecessors in E for any k 1, whereas the amount of randomness of the successors is bounded. We apply this result to answer a question raised by Lutz 10]: We show that the class of weakly complete sets has measure 1 in E and that there are weakly complete problems which are not p-btt-complete for E.