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Title:Some results on e-genericity and recursively enumerable weak truth table degrees
Author(s):Blaylock, Richard Warren
Doctoral Committee Chair(s):Jockusch, Carl G., Jr.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:In this manuscript we explore two topics in recursion theory and their interaction.
The first topic is e-genericity, a notion of genericity for recursively enumerable (r.e.) sets introduced by C. G. Jockusch, Jr. The second is weak truth table reducibility (w-reducibility), a strong reducibility (i.e., stronger than the most general Turing reducibility) first introduced by Friedberg and Rogers. In Chapter 1 we give a brief introduction to these topics and establish the relevant terminology and notation.
In Chapter 2 we give some closure and non-closure properties for the classes of e-generic sets and degrees, which are predicted by analogous results for previous notions of genericity. For example, the e-generic sets are not closed under union, intersection, or join, but on the other hand if the join $A \oplus B$ of two sets is e-generic, then so are $A,B, A \cup B$, and $A \cap B$.
In Chapter 3 we investigate the structure of the weak truth table degrees (w-degrees) inside an e-generic Turing degree. Here we show that e-generic Turing degrees are highly noncontiguous in the sense that they contain no greatest and no least r.e. w-degree.
Finally in Chapter 4 we obtain some results on the ordering of the r.e. w-degrees in general. The main result is the existence of a nontrivial r.e. w-degree a which has a greatest lower bound with every r.e. w-degree b. We also show that these nontrivial completely cappable degrees can neither be low nor promptly simple.
Issue Date:1991
Rights Information:Copyright 1991 Blaylock, Richard Warren
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9210749
OCLC Identifier:(UMI)AAI9210749

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