Defining totality in the enumeration degrees

Abstract

We show that if A and B form a nontrivial K-pair, then there is a semi-computable set C such that A ≤e C and B ≤e C. As a consequence, we obtain a definition of the total enumeration degrees: a nonzero enumeration degree is total if and only if it is the join of a nontrivial maximal K-pair. This answers a longstanding question of Hartley Rogers, Jr. We also obtain a definition of the “c.e. in” relation for total degrees in the enumeration degrees. Two of the most fundamental concepts in computability theory are effective computation and effective enumeration, i.e., computation and enumeration of sets of natural numbers as achieved by computer programs. Each notion induces a natural measure of the complexity of sets of natural numbers. Turing reducibility was defined by Turing in the late 1930’s to capture the relative complexity of computing sets of natural numbers: A ≤T B means that a set A is computable from another set B. We use the word “reducibility” because the problem of computing A is being reduced to the problem of computing B. In the same way, Enumeration reducibility was defined by Friedberg and Rogers in the late 1950’s to capture the relative complexity of enumerating sets: A ≤e B means that there is a way to effectively produce an enumeration of a set A from any enumeration of a set B. Turing reducibility has been studied extensively, but enumeration reducibility also arises naturally in various contexts. Enumeration reducibility restricted to partial functions is equivalent to Kleene’s [12] notion of relative effective computability of partial functions. Scott [20] used enumeration reducibility to give a countable graph model of λ-calculus. C. F. Miller (unpublished manuscript) and M. Ziegler [28] applied enumeration reducibility in group theory, to state and prove an extension of Higman’s embedding theorem for finitely generated groups. J. Miller [15] used enumeration reducibility in his work on computable analysis; he answered a question of Pour El and Lempp by showing that Turing reducibility is not sufficient to measure the complexity of continuous functions on R (but enumeration reducibility is). Similarly, Richter [16] constructed a wide variety of countable structures for which enumeration reducibility, but not Turing reducibility, is sufficient to measure the complexity of their isomorphism type. There are several other occasions in 2010 Mathematics Subject Classification. 03D30.