Recursively enumerable language

About: Recursively enumerable language is a research topic. Over the lifetime, 1508 publications have been published within this topic receiving 32382 citations.

1-Genericity in the Enumeration Degrees

Abstract: The structure of the Turing degrees of generic and n -generic sets has been studied fairly extensively, especially for n = 1 and n = 2. The original formulation of 1-generic set in terms of recursively enumerable sets of strings is due to D. Posner [11], and much work has since been done, particularly by C. G. Jockusch and C. T. Chong (see [5] and [6]). In the enumeration degrees (see definition below), attention has previously been restricted to generic sets and functions. J. Case used genericity for many of the results in his thesis [1]. In this paper we develop a notion of 1-generic partial function, and study the structure and characteristics of such functions in the enumeration degrees. We find that the e-degree of a 1-generic function is quasi-minimal. However, there are no e-degrees minimal in the 1-generic e-degrees, since if a 1-generic function is recursively split into finitely or infinitely many parts the resulting functions are e-independent (in the sense defined by K. McEvoy [8]) and 1-generic. This result also shows that any recursively enumerable partial ordering can be embedded below any 1-generic degree. Many results in the Turing degrees have direct parallels in the enumeration degrees. Applying the minimal Turing degree construction to the partial degrees (the e-degrees of partial functions) produces a total partial degree a e which is minimal-like ; that is, all functions in degrees below a e have partial recursive extensions.