Partial degrees and the density problem

Abstract

A notion of relative reducibility for partial functions, which coincides with Turing reducibility on the total functions, was first given by S.C. Kleene in Introduction to metamathematics [4]. Following Myhill [7], this was made more explicit in Hartley Rogers, Jr., Theory of recursive functions and effective computability [8, pp. 146, 279], where some basic properties of the partial degrees or (equivalent, but notationally more convenient) the enumeration degrees, were derived. The question of density of this proper extension of the degrees of unsolvability was left open, although Medvedev's result [6] that there are quasi-minimal partial degrees (that is, nonrecursive partial degrees with no nonrecursive total predecessors) is proved.In 1971, Sasso [9] introduced a finer notion of partial degree, which also contained the Turing degrees as a proper substructure (intuitively, Sasso's notion of reducibility between partial functions differed from Rogers' in that computations terminated when the oracle was asked for an undefined value, whereas a Rogers computation could be thought of as proceeding simultaneously along a number of different branches of a ‘consistent’ computation tree—cf. Sasso [10]). His construction of minimal ‘partial degrees’ [11], while of interest in itself, left open the analogous problem for the more standard partial degree structure.