A Generic m-Reducibility

Abstract

Kapovich, Myasnikov, Schupp and Shpilrain in 2003 developed generic approach to algorithmic problems, which considers an algorithmic problem on “most” of the inputs (i.e., on a generic set) instead of the entire domain and ignores it on the rest of inputs (a negligible set). Jockusch and Schupp in 2012 began the study of generic computability in the context of classical computability theory. In particular, they defined a generic analog of Turing reducibility. In this paper we introduce a generic analog of m-reducibility as m-reducibility by computable functions, which preserve the non-negligibility of sets. We study generic m-reducibility of computable and c.e. sets. We prove the existence of generically m-complete c.e. sets, incomparable c.e. sets and m-degrees, which contain more than one generic m-degree.