Partial functions and domination

TL;DR

This paper introduces pdominant sets, a new class in recursion theory, and explores their properties, showing their relationships with various well-known classes and randomness notions.

Contribution

It defines pdominant sets and analyzes their recursion-theoretic properties, revealing their nonexistence in certain classes and their presence in others.

Findings

No pdominant sets in certain nonrecursive classes

Weakly 2-generic sets are not pdominant

Halves of Chaitin's Omega are pdominant

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 {\psi} such that for every partial recursive function {\phi} and almost every x in the domain of {\phi} there is a y in the domain of {\psi} with y<= x and {\psi}(y) > {\phi}(x). While there is a full {\pi}01-class of nonrecursive sets where no set is pdominant, there is no {\pi}01-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 {\Omega} are pdominant. No set which is low for Martin-L\"of random is pdominant. There is a low r.e. set which is pdominant and a high r.e. set which is not pdominant.