Effective forcing with Cantor manifolds

TL;DR

This paper constructs a set of integers with specific computability properties using an effective forcing method involving infinite dimensional Cantor manifolds, extending the understanding of totality variants in computability theory.

Contribution

It introduces a novel effective forcing construction with Cantor manifolds to produce a set with unique enumeration degree properties, expanding the methods in computability theory.

Findings

Constructed a set with cototal and almost total properties

The set is neither cylinder-cototal nor telograph-cototal

Uses an effectivization of Zapletal's half-Cohen forcing

Abstract

A set $A$ of integers is called total if there is an algorithm which, given an enumeration of $A$, enumerates the complement of $A$, and called cototal if there is an algorithm which, given an enumeration of the complement of $A$, enumerates $A$. Many variants of totality and cototality have been studied in computability theory. In this note, by an effective forcing construction with strongly infinite dimensional Cantor manifolds, which can be viewed as an effectivization of Zapletal's "half-Cohen" forcing (i.e., the forcing with Henderson compacta), we construct a set of integers whose enumeration degree is cototal, almost total, but neither cylinder-cototal nor telograph-cototal.