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arXiv · 1506.06869

Partial functions and domination

C. T. Chong·Gordon Hoi·Frank Stephan·Daniel Turetsky
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Abstract

The current work introduces the notion of pdominant sets and studies their recursion-theoretic properties. Here a set A is called pdominant iff there is a partial A-recursive function ψ such that for every partial recursive function ϕ and almost every x in the domain of ϕ there is a y in the domain of ψ with y<= x and ψ(y) > ϕ(x). While there is a full π01-class of nonrecursive sets where no set is pdominant, there is no π01-class containing only pdominant sets. No weakly 2-generic set is pdominant while there are pdominant 1-generic sets below K. The halves of Chaitin's Ω are pdominant. No set which is low for Martin-Löf random is pdominant. There is a low r.e. set which is pdominant and a high r.e. set which is not pdominant.

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C. T. Chong, Gordon Hoi, Frank Stephan, Daniel Turetsky. 2016-01-05. Partial functions and domination. https://doi.org/10.2168/lmcs-11(3%3A16)2015

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