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Quasi-Minimal Enumeration Degrees and Minimal Turing Degrees

By Theodore Slaman and Andrea Sorbi


We show that there exists a set A such that A has quasi--minimal enumeration degree, and there are uncountably many sets B such that A is enumeration reducible to B and B has minimal Turing degree. Answering a related question raised by Solon, we also show that there exists a nontotal enumeration degree which is not e-hyperimmune. 1 Introduction We adopt the formalization of enumeration reducibility given by Friedberg and Rogers (1959), and our exposition follows (Rogers, 1967). In the Friedberg-- Rogers formalization, an enumeration operator # z : 2 # # 2 # is derived from a recursively enumerable set W z by the equation # z (B) = {x : (#u)[#x, u# # W z and D u # B]}, where D u is the finite set with canonical index u. Henceforth, we may write #x, D# instead of #x, u#, when D is equal to D u . Given an e#ective listing {W z : z # #} of the recursively enumerable sets, we get a corresponding indexing {# z : z # #} of the e-operators. # During the preparation o..

Year: 2007
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