-Generic Computability, Turing Reducibility and Asymptotic Density
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.
Cite
@article{arxiv.1010.5212,
title = {-Generic Computability, Turing Reducibility and Asymptotic Density},
author = {Carl G. Jockusch and Paul E. Schupp},
journal= {arXiv preprint arXiv:1010.5212},
year = {2014}
}