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Resource bounded randomness and weakly complete problems

  • Klaus Ambos-Spies
  • Sebastiaan A. Terwijn
  • Zheng Xizhong
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 834)

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(2lin). 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 thatnc-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 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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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Klaus Ambos-Spies
    • 1
  • Sebastiaan A. Terwijn
    • 2
  • Zheng Xizhong
    • 3
  1. 1.Mathematisches InstitutUniversität HeidelbergHeidelbergGermany
  2. 2.FWIUniversiteit van AmsterdamTV AmsterdamThe Netherlands
  3. 3.Department of MathematicsNanjing UniversityNanjingChina

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