A class of recursively enumerable sets may be classified either as an object in itself — the range of a two-place function in the obvious way — or by means of the corresponding set of indices. The latter approach is not only more precise but also, as we show below, provides an alternative method for solving certain problems on recursively enumerable sets and their degrees of unsolvability. The ...

We show that the structure of the enumeration degrees De has a finite automorphism base consisting of finitely many total elements below the first jump of its least element. As a consequence we obtain that the rigidity of the structure of the enumeration degrees is implied by the rigidity of the local structures of the Σ2 enumeration degrees, the ∆ 0 2 Turing degrees and the computably enumerab...

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a∪ c)∩ (b∪ c), a∪ c ∣∣b∪ c, and c < a ∪ b.

We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a...

Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with ≤T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The conditions in the forcing partial specify finite initial segments of the generic reals and impose a infinite cons...

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [33], [34], we give syntactical conditions necessary and sufficient for a relation to be intrinsically Π1 on a structure. We consider some examples of computable structures A and intrinsically Π1 relations R. We also consider a general family o...

It is shown that for any computably enumerable degree a 6= 0, any degree c 6= 0, and any Turing degree s, if s ≥ 0, and c.e. in a, then there exists a c.e. degree x with the following properties, (1) x < a, c 6≤ x, (2) a is splittable over x, and (3) x = s. This implies that the Sacks’ splitting theorem and the Sacks’ jump theorem can be uniformly combined. A corollary is that there is no atomi...

This thesis is concerned with the interaction between computability and randomness. In the first part, we study the notion of traceability. This combinatorial notion has an increasing influence in the study of algorithmic randomness. We prove a separation result about the bounds on jump traceability, and show that the index set of the strongly jump traceable computably enumerable (c.e.) sets is...

An irreversible k-threshold process (also a k-neighbor bootstrap percolation) is a dynamic process on a graph where vertices change color from white to black if they have at least k black neighbors. An irreversible k-conversion set of a graph G is a subset S of vertices of G such that the irreversible k-threshold process starting with S black eventually changes all vertices of G to black. We sh...

Toward establishing the decidability of the two quantifier theory of the ∆ 2 Turing degrees with join, we study extensions of embeddings of upper-semi-lattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular the degree of the halting set 0. We obtain a good deal of sufficient and necessary conditions.