CT-1994-09: Resource Bounded Randomness and Weakly Complete Problems

CT-1994-09: Ambos-Spies, Klaus and Terwijn, Sebastiaan A. and Xizhong, Zheng (1994) Resource Bounded Randomness and Weakly Complete Problems. [Report]

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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.

Item Type: Report
Report Nr: CT-1994-09
Series Name: Computation and Complexity Theory (CT)
Year: 1994
Date Deposited: 12 Oct 2016 14:39
Last Modified: 12 Oct 2016 14:39
URI: https://eprints.illc.uva.nl/id/eprint/1057

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