9 research outputs found
Downward Collapse from a Weaker Hypothesis
Hemaspaandra et al. proved that, for and : if
\Sigma_i^p \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed under complementation,
then . This sharply asymmetric
result fails to apply to the case in which the hypothesis is weakened by
allowing the to be replaced by any class in its difference
hierarchy. We so extend the result by proving that, for and : if DIFF_s(\Sigma_i^p) \BoldfaceDelta DIFF_m(\Sigma_k^p) is closed
under complementation, then
Extending Cooper’s theorem to Δ30 Turing degrees
© 2018 - IOS Press and the authors. All rights reserved. In 1971 B. Cooper proved that there exists a 2-c.e. Turing degree which doesn't contain a c.e. set. Thus, he showed that the second level of the Ershov hierarchy is proper. In this paper we investigate proper levels of some extensions of the Ershov hierarchy to higher levels of the arithmetical hierarchy. Thus we contribute to the theory of ' " 3 0 -degrees by extending Cooper's theorem to some levels of the fine hierarchy within ' " 3 0 -sets