-Generic Computability, Turing Reducibility and Asymptotic Density

TL;DR

This paper explores the concept of generic computability in classical computability theory, establishing new relationships between generic and coarse computability, and embedding Turing degrees into generic degrees.

Contribution

It introduces a formal framework for generic reducibility, demonstrates the existence of sets with specific computability properties, and embeds Turing degrees into generic degrees.

Findings

Existence of c.e. sets that are generically but not coarsely computable.

Every nonzero Turing degree contains a set not coarsely computable.

A natural embedding of Turing degrees into generic degrees is established.

Abstract

Generic computability has been studied in group theory and we now study it in the context of classical computability theory. A set A of natural numbers is generically computable if there is a partial computable function f whose domain has density 1 and which agrees with the characteristic function of A on its domain. A set A is coarsely computable if there is a computable set C such that the symmetric difference of A and C has density 0. We prove that there is a c.e. set which is generically computable but not coarsely computable and vice versa. We show that every nonzero Turing degree contains a set which is not coarsely computable. We prove that there is a c.e. set of density 1 which has no computable subset of density 1. As a corollary, there is a generically computable set A such that no generic algorithm for A has computable domain. We define a general notion of generic reducibility in the spirt of Turing reducibility and show that there is a natural order-preserving embedding of the Turing degrees into the generic degrees which is not surjective.