Kolmogorov Complexity, Strong Reducibilities, and Computably Enumerable Sets
We also study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded if there is an m-incomplete computably enumerable set A such that every set in S is m-reducible to A. For example, we show that the class of effectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded if and only if B is low2. For r = bwtt, tt, wtt, and T, there is a bounded class intersecting every computably enumerable r-degree; for r = c, d and p, no such class exists.
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign