Resource bounded randomness and weakly complete problems

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Abstract

We introduce and study resource bounded random sets based on Lutz’s concept of resource bounded measure ([5, 6]). 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 quantative structure of E = DTIME(21in). 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 [1, 2]) 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 n%random sets under pm-reducibility. We show that every nO-random set in E has nk-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.

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Ambos-Spies, K., Terwijn, S. A., & Xizhong, Z. (1994). Resource bounded randomness and weakly complete problems. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 834 LNCS, pp. 369–377). Springer Verlag. https://doi.org/10.1007/3-540-58325-4_201

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