Generic computability, Turing degrees, and asymptotic density

Abstract

Generic decidability has been extensively studied in group theory, and we now study it in the context of classical computability theory. A set A of natural numbers is called generically computable if there is a partial computable function that agrees with the characteristic function of A on its domain D, and furthermore D has density 1, that is, lim n→∞ |{k<n: k∈D} |/n=1. A set A is called coarsely computable if there is a computable set R such that the symmetric difference of A and R has density 0. We prove that there is a computably enumerable (c.e.) set that is generically computable but not coarsely computable and vice versa. We show that every nonzero Turing degree contains a set that is neither generically computable nor coarsely computable. We prove that there is a c.e. set of density 1 that has no computable subset of density 1. Finally, we define and study generic reducibility.