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Upper bounds on ideals in the computably enumerable Turing degrees

George Barmpalias & André Nies

Annals of Pure and Applied Logic 162 (6):465-473 (2011)

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(1 other version)Algorithmic Randomness and Measures of Complexity.George Barmpalias - 2013 - Bulletin of Symbolic Logic 19 (3):318-350.details We survey recent advances on the interface between computability theory and algorithmic randomness, with special attention on measures of relative complexity. We focus on reducibilities that measure the initial segment complexity of reals and the power of reals to compress strings, when they are used as oracles. The results are put into context and several connections are made with various central issues in modern algorithmic randomness and computability. Download Export citation Bookmark 1 citation
On the C.E. Degrees realizable in $\pi ^0_1$ classes.Barbara F. Csima, Rod Downey & N. G. Keng Meng - 2024 - Journal of Symbolic Logic 89 (3):1370-1395.details We study for each computably bounded $\Pi ^0_1$ class P the set of degrees of c.e. paths in P. We show, amongst other results, that for every c.e. degree a there is a perfect $\Pi ^0_1$ class where all c.e. members have degree a. We also show that every $\Pi ^0_1$ set of c.e. indices is realized in some perfect $\Pi ^0_1$ class, and classify the sets of c.e. degrees which can be realized in some $\Pi ^0_1$ class as exactly (...) Download Export citation Bookmark

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