Reducibilities in L Rafr 1 Recursive Function Theory

REDUCIBILITIES IN RECURSIVE FUNCTION THEORY Carl Groos Jockusch, Jr. Submitted to the Department of Mathematics on May 13, 1966 in partial fulfillment of the requirements for the degree of Doctor of Philosophy. In this dissertation some reducibilities of recursive function theory are analyzed, with particular emphasis on the relationships between many-one reducibility and various kinds of truth-table reducibility. In the first section, the theory of cylinders as developed by Rogers is given. Then the notion of "R-cylinder,, is defined for any reducibility R, and the properties of R-cylinders are studied. In the second section, the R-cylinders are characterized for many kinds of truth-table reducibilities. The characterizations are employed to prove that not every btt-degree has a maximum m-degree and several similar theorems. It is also shown that there are r.e., nonrecursive, noncreative sets A such that AxA 4m A. In the third section, it is pointed out that the reducibilities mentioned in the second section differ in general on the r.e. sets, but theorems are proved to show that they occasionally coincide under special hypotheses. In the fourth section, the notion of "semirecursive set" is introduced and studied. It is shown that there are semirecursive sets in every tt-degree and hyperimmune semirecursive sets in every r.e. nonrecursive T-degree. It is proved that the p-degree of a semirecursive set consists of a single m-degree, where p-reducibility is as defined in section two. Priority constructions are used to prove that it is possible to have r.e. semirecursive sets A and B such that A join B is not semirecursive and r.e. sets A and B such that B is semirecursive, A is not semirecursive, and A ettB. Finally it is shown that immune semirecursive sets are hyperimmune, not hyperhyperimmune and in Et in the arithmetical hierarchy and that retraceable or effectively immune semi-recursive sets are co-r.e. In the final section it is shown that the m-degreesof A,AxA,... are all distinct for sets A such that A is simple but not hypersimple or I is immune, non-hyperimmune and retraced by a total function. From this it follows that every nonrecursive tt-degree has infinitely many m-degrees and every r.e. nonrecursive T-degree has infinitely many r.e m-degrees. It is also proved that every r.e. T-degree has an r.e. m-degree consisting of a single 1-degree. The theorems on r.e. nonrecursive T-degrees depend on a construction of Yates for simple but not hypersimple sets and use the propositional calculus as a tool. These theorems seem to be bound together by the fact that if A is simple but not hypersimple, then ix I Dx C. A) acts in many ways like a creative set. Finally, the notion of "inverse R-cylinder," is defined and shown to be relevant only for m-reducibility.