Graver degrees are not polynomially bounded by true circuit degrees

Degrees That Are Not Degrees of Categoricity

A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B ∼= A there is an isomorphism f : B → A with f ≤T x. A degree x is a degree of categoricity if there is a computable structure A such that A is x-computably categorical, and for all y, if A is y-computably categorical then x ≤T y. We construct a Σ2 set whose degree is not a degree o...

The d.r.e. Degrees are Not Dense

We indicate how to fix an error in the proof of the main theorem of our original paper, pointed out to us by Yong Liu and Keng Meng Ng. The main result of our paper [1] read as follows: Main Theorem. There is an incomplete d.c.e. degree maximal in the n-c.e. degrees (for all n ≥ 2) as well as in the ω-c.e. degrees. Thus the n-c.e. degrees (for n ≥ 2) and the ω-c.e. degrees are not dense. The er...

The 02 Enumeration Degrees Are Not

We show that the 0 2 enumeration degrees are not dense. This answers a question posed by Cooper. 1 Background Cooper 3] showed that the enumeration degrees (e-degrees) are not dense. He also showed that the e-degrees of the 0 2 sets are dense 2]. Since Cooper's nondensity proof constructs a 0 7 set, he posed the question: What is the least n (2 n 6) such that the e-degrees below 0 n are not den...

Degrees which do not bound minimal degrees

The high/low hierarchy respects the ordering of degrees, and so we can expect to find properties of order which are possessed by all degrees in a given hierarchy class. Such properties can also be found for classes of the generalized high/low hierarchy even though that hierarchy does not respect the ordering of the degrees. For example, Jockusch and Posner [4], extending a result of Cooper [l],...

Turing Degrees and Muchnik Degrees of Recursively Bounded DNR Functions