Resource bounded randomness and weakly complete problems - Ambos-Spies, Klaus; Terwijn, Sebastiaan A.; Zheng, Xizhong (Theoretical computer science, 1997)
We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure [7, 8]. We concentrate on nc-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(2lin). However, we will also comment on E2 = DTIME(2pol) and its corresponding (p2-) measure. First we show that the class of nc-random sets has p-measure 1. This provides a new, simplified approach to p-measure 1-results. Next we compare randomness with genericity (in the sense of [2, 3]) and we show that nc + 1-random sets are nc-generic, whereas the converse fails. From the former we conclude that nc-random sets are not p-btt-complete for E. Our technical main results describe the distribution of the nc-random sets under p-m-reducibility. We show that every nc-random set in E has nκ-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.