Quasiminimal pairs for c.e. degrees of generic and coarse reducibilities

Abstract

Abstract Generic approach to algorithmic problems in combinatorial group theory was suggested in 2003 by Kapovich, Myasnikov, Schupp and Shpilrain. This approach deals with algorithms, which solves algorithmic problems on ”most” of the inputs (i.e., on a generic set) instead of the entire domain and output undefined answer (do not halt) on the rest of inputs (a negligible set). Generic analog of Turing reducibility was introduced by Jockusch and Schupp in 2012. Later Hirschfeldt, Jockusch, Kuyper and Schupp defined coarse reducibility. In this paper we prove that there exist quasiminimal pairs for generic and coarse reducibilities of computably enumerable (c.e.) degrees. The work was supported by Russian Science Foundation, grant number 18-71-10028.