Daniel Turetsky

LIMITWISE MONOTONIC FUNCTIONS AND THEIR APPLICATIONS

We survey what is known about limitwise monotonic functions and sets and discuss their applications in effective algebra and computable model theory. Additionally, we characterize the computably enumerable degrees that are totally limitwise monotonic, show the support strictly increasing 0′-limitwise monotonic sets on Q do not capture the sets with computable strong η-representations, and study the limitwise monotonic spectra of a set.

Characterizing lowness for Demuth randomness

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [35]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

Linear orders realized by C.e. Equivalence relations

Let E be a computably enumerable (c.e.) equivalence relation on the set ω of natural numbers. We say that the quotient set ω/E (or equivalently, the relation E) realizes a linearly ordered set L if there exists a c.e. relation E respecting E such that the induced structure (ω/E;E) is isomorphic to L. Thus, one can consider the class of all linearly ordered sets that are realized by ω/E; formally, K(E) = {L | the order-type L is realized by E}. In this paper we study the relationship between computability-theoretic properties of E and algebraic properties of linearly ordered sets realized by E. One can also define the following pre-order 6lo on the class of all c.e. equivalence relations: E1 6lo E2 if every linear order realized by E1 is also realized by E2. Following the tradition of computability theory, the lo-degrees are the classes of equivalence relations induced by the pre-order 6lo. We study the partially ordered set of lo-degrees. For instance, we construct various chains and antichains and show the existence of a maximal element among the lo-degrees.

INHERENT ENUMERABILITY OF STRONG JUMP-TRACEABILITY

We show that every strongly jump-traceable set obeys every be- nign cost function. Moreover, we show that every strongly jump-traceable set is computable from a computably enumerable strongly jump-traceable set. This allows us to generalise properties of c.e. strongly jump-traceable sets to all such sets. For example, the strongly jump-traceable sets induce an ideal in the Turing degrees; the strongly jump-traceable sets are precisely those that are computable from all superlow Martin-Lof random sets; the strongly jump- traceable sets are precisely those that are a base for DemuthBLR-randomness; and strong jump-traceability is equivalent to strong superlowness.

Decidability and Computability of Certain Torsion-Free Abelian Groups

We study completely decomposable torsion-free abelian groups of the form GS := ⊕n∈SQpn for sets S ⊆ ω. We show that GS has a decidable copy if and only if S is Σ2 and has a computable copy if and only if S is Σ 0 3.

Natural Large Degree Spectra

We show that the collection of array non-recursive degrees, the collection of non-jump-traceable degrees, and the collection of degrees which compute a function not dominated by any ω-computably approximable function, are all degree spectra of countable structures.

Connectedness properties of dimension level sets

We prove that the set of all points of effective Hausdorff dimension 1 in R^n(n>=2) is connected, and simultaneously that the complement of this set is not path-connected when n=2.

UNIFORM PROCEDURES IN UNCOUNTABLE STRUCTURES

This paper contributes to the general program of extending techniques and ideas of effective algebra to computable metric space theory. It is well-known that relative computable categoricity (to be defined) of a computable algebraic structure is equivalent to having a c.e. Scott family with finitely many parameters (e.g., [1]). The first main result of the paper extends this characterisation to computable Polish metric spaces. The second main result illustrates that just a slight change of the definitions will give us a new notion of categoricity unseen in the countable case (to be stated formally). The second result also shows that the characterisation of computably categorical closed subspaces of Rn contained in [17] cannot be improved. The third main result extends the characterisation to not necessarily separable structures of cardinality κ using κ-computability.

Computability and uncountable linear orders I: Computable categoricity

We study the computable structure theory of linear orders of size א1 within the framework of admissible computability theory. In particular, we characterize which of these linear orders are computably categorical.

Computability and uncountable linear orders II: Degree spectra

We study the computable structure theory of linear orders of size