On the strongly bounded turing degrees of simple sets

Abstract

We study the r-degrees of simple sets under the strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) which are de ned in terms of Turing functionals where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We call a c.e. r-degree a simple if it contains a simple set and we call a nonsimple otherwise. As we show, the ibTdegree of a c.e. set A is simple if and only if the cl-degree of A is simple, and there are nonsimple c.e. r-degrees > 0. Moreover, we analyze the distribution of the simple and nonsimple r-degrees in the partial ordering of the c.e. r-degrees. Among the resultswe obtain are the following. (i) For any c.e. r-degree a > 0, there are simple r-degrees which are below a, above a and incomparable with a. (ii) For any c.e. r-degree a > 0, there are nonzero nonsimple c.e. r-degrees which are below a and incomparable with a; and there is a nonsimple c.e. r-degree above a if and only if a is not contained in the complete wtt-degree. (iii) There are in nite intervals of c.e. r-degrees entirely consisting of nonsimple c.e. r-degrees respectively simple rdegrees. (iv) Any c.e. r-degree is the join of two nonsimple c.e. r-degrees whereas the class of the nonzero c.e. r-degrees is not generated by the simple r-degrees under join though any simple r-degree is the join of two lesser simple r-degrees. Moreover, neither the class of the nonsimple c.e. r-degrees nor the class of the simple r-degrees generates the class of c.e. r-degrees under meet.