Degrees of randomized computability: decomposition into atoms

TL;DR

This paper explores the structural decomposition of LV-degrees of collections of sequences, introducing atoms and infinitely divisible elements, and applies these concepts to hyperimmune sequences, advancing the understanding of probabilistic computability.

Contribution

It constructs atoms and divisible elements within LV-degrees, providing a new decomposition framework and applying it to hyperimmune sequences.

Findings

Decomposition of the maximal LV-degree into atoms and an infinitely divisible part.

Construction of atoms from hyperimmune sequences.

Representation of LV-degree of all hyperimmune sequences as a union of atoms and an infinitely divisible element.

Abstract

In this paper we study structural properties of LV-degrees of the algebra of collections of sequences that are non-negligible in the sense that they can be computed by a probabilistic algorithm with positive probability. We construct atoms and infinitely divisible elements of this algebra generated by sequences, which cannot be Martin-L\"of random and, moreover, these sequences cannot be Turing equivalent to random sequences. The constructions are based on the corresponding templates which can be used for defining the special LV-degrees. In particular, we present the template for defining atoms of the algebra of LV-degrees and obtain the decomposition of the maximal LV-degree into a countable sequence of atoms and their non-zero complement -- infinitely divisible LV-degree. We apply the templates to establish new facts about specific LV-degrees, such as the LV-degree of the collection of sequences of hyperimmune degree. We construct atoms defined by collections of hyperimmune sequences, moreover, a representation of LV-degree of the collection of all hyperimmune sequences will be obtained in the form of a union of an infinite sequence of atoms and an infinitely divisible element.