Resource bounded randomness and weakly complete problems

Abstract

We introduce and study resource bounded random sets based on Lutz's concept of resource bounded measure ([5, 6]). 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 quantative structure of E = DTIME(2^lin). First we show that the class of n ^c-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 [1, 2]) and we show that n ^c+1-random sets are n ^c-generic, whereas the converse fails. From the former we conclude thatn ^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 [8]: 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.

This research was done while the second and third author visited the University of Heidelberg in 1993/94.

The first author was supported in part by the Human Capital and Mobility program of the European Community under grant CHRX-CT93-0415.

The second author was supported by the Dutch VSB foundation.

The third author was supported by the Chinese State Education Commission.