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.

A real is computable if its left cut, L( ); is computable. If (qi)i is a computable sequence of rationals computably converging to ; then fqig; the corresponding set, is always computable. A computably enumerable (c.e.) real is a real which is the limit of an increasing computable sequence of rationals, and has a left cut which is c.e. We study the Turing degrees of representations of c.e. real...

We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...

We prove that the ∆2 Turing degrees have a finite automorphism base. We apply this result to show that the automorphism group of DT (≤ 0′) is countable and that all its members have arithmetic presentations. We prove that every relation on DT (≤ 0′) induced by an arithmetically definable degree invariant relation is definable with finitely many ∆2 parameters and show that rigidity for DT (≤ 0′)...

We introduce a Π1-uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω1) = ω1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π1 maximal chains and thin maximal antichains in the Turing degrees. We also use the Π1-uniformization principle to study Martin’s conjectures on cones of Turing d...

We study some generalized notions of cohesiveness which arise naturally in connection with effective versions of Ram-sey's Theorem. An infinite set A of natural numbers is n–cohesive (respectively, n–r–cohesive) if A is almost homogeneous for every computably enumerable (respectively, computable) 2–coloring of the n–element sets of natural numbers. (Thus the 1–cohesive and 1–r–cohesive sets coi...

The strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) are defined in terms of Turing reductions where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We show that, for r = ibT, cl, every computably enumerable (c.e.) r-degree a > 0 has the an...

The minimum independent generalized t-degree of a graph G = (V,E) is ut = min{ IN(H H is an independent set of t vertices of G}, with N(H) = UxtH N(x). In a KI,~+I -free graph, we give an upper bound on u! in terms of r and the independence number CI of G. This generalizes already known results on u2 in KI,,+I-free graphs and on U, in KI,x-free graphs.