Universality and Almost Decidability

TL;DR

This paper introduces new concepts of universality and almost decidability for unary functions, exploring the properties of halting sets and solving an open problem regarding their decidability status.

Contribution

It defines almost decidability and demonstrates the existence of universal functions with halting sets that are generic or negligible, addressing an open problem in the field.

Findings

Existence of infinitely many universal functions with generic halting sets

Existence of universal functions with negligible halting sets that are not almost decidable

Resolution of an open problem about the decidability of halting sets

Abstract

We present and study new definitions of universal and programmable universal unary functions and consider a new simplicity criterion: almost decidability of the halting set. A set of positive integers S is almost decidable if there exists a decidable and generic (i.e. a set of natural density one) set whose intersection with S is decidable. Every decidable set is almost decidable, but the converse implication is false. We prove the existence of infinitely many universal functions whose halting sets are generic (negligible, i.e. have density zero) and (not) almost decidable. One result - namely, the existence of infinitely many universal functions whose halting sets are generic (negligible) and not almost decidable - solves an open problem in [9]. We conclude with some open problems.