Arithmetic complexity via effective names for random sequences

TL;DR

This paper explores the complexity and enumeration properties of various classes of left-recursively enumerable sets and their relation to randomness and the arithmetic hierarchy, revealing fundamental differences in their numberings.

Contribution

It characterizes the complexity of random sets and classes via effective names, and establishes equivalences between arithmetic complexity and numberings of left-r.e. classes.

Findings

Characterization of the complexity of random sets like Martin-Löf and Kurtz.

Existence of equivalence between arithmetic hierarchy levels and numberings of left-r.e. classes.

Fundamental differences between left-r.e. numberings for sets and reals.

Abstract

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals.