Resource Bounded Randomness and Weakly Complete Problems
Klaus Ambos-Spies, Sebastiaan A. Terwijn, Zheng Xizhong

Abstract:
We introduce and study resource bounded random sets based on Lutz's
concept of resource bounded measure. We concentrate on n^c­randomness 
(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 E_2 = DTIME(2^pol) and its corresponding (p_2­)measure. First
we show that the class of n^c­random sets has p­measure 1. This provides
a new, simplified approach to p­measure 1­results. Next we compare 
randomness with genericity, and we show that n^(c+1)­random sets are 
n^c­generic, whereas the converse fails. From the former
we conclude that n^c­random sets are not p­btt­complete for E. Our technical 
main results describe the distribution of the n^c­random sets under
p­m­reducibility. We show that every n^c­random set in E has n^k­random
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: We show that the class of weakly complete sets has measure 1 in E and 
that there are weakly complete problems which are not p­btt­complete for E.