Failure modes for structural highness notions

Computability theoretic aspects of Polish metric spaces are studied by adapting notions and methods of computable structure theory. In this dissertation, we mainly investigate index sets and classification problems for computably presentable Polish metric spaces. We find the complexity of a number of index sets, isomorphism problems, and embedding problems for computably presentable metric spaces. We also provide several computable structure theory results related to some classical Polish metric spaces such as the Urysohn space $\mathbb {U}$ , the Cantor space $2^{\mathbb {N}}$ , the Baire space $\mathbb {N}^{\mathbb {N}}$ , and spaces of continuous functions.Abstract prepared by Teerawat Thewmorakot.E-mail: teerawat.thew@hotmail.com

In computability theory, the standard tool to classify preorders is provided by the computable reducibility. If [Formula: see text] and [Formula: see text] are preorders with domain [Formula: see text], then [Formula: see text] is computably reducible to [Formula: see text] if and only if there is a computable function [Formula: see text] such that for all [Formula: see text] and [Formula: see text], [Formula: see text] [Formula: see text][Formula: see text]. We study the complexity of preorders which arise in a natural way in computable structure theory. We prove that the relation of computable isomorphic embeddability among computable torsion abelian groups is a [Formula: see text] complete preorder. A similar result is obtained for computable distributive lattices. We show that the relation of primitive recursive embeddability among punctual structures (in the setting of Kalimullin et al.) is a [Formula: see text] complete preorder.

The relationship between enumeration degrees and abstract models of computability inspires a new direction in the field of computable structure theory. Computable structure theory uses the notions and methods of computability theory in order to find the effective contents of some mathematical problems and constructions. The paper is a survey on the computable structure theory from the point of view of enumeration reducibility.

Using methods from computable analysis, we establish a new connection between two seemingly distant areas of logic: computable structure theory and invariant descriptive set theory. We extend several fundamental results of computable structure theory to the more general setting of topological group actions. As we will see, the usual action of ${S_\infty }$ on the space of structures in a given language is effective in a certain algorithmic sense that we need, and ${S_\infty }$ itself carries a natural computability structure (to be defined). Among other results, we give a sufficient condition for an orbit under effective ${\cal G}$-action of a computable Polish ${\cal G}$ to split into infinitely many disjoint effective orbits. Our results are not only more general than the respective results in computable structure theory, but they also tend to have proofs different from (and sometimes simpler than) the previously known proofs of the respective prototype results.

We investigate the effects of adding uniformity requirements to concepts in computable structure theory such as computable categoricity (of a structure) and intrinsic computability (of a relation on a computable structure). We consider and compare two different notions of uniformity, previously studied by Kudinov and by Ventsov. We discuss some of their results and establish new ones, while also exploring the connections with the relative computable structure theory of Ash, Knight, Manasse, and Slaman and Chisholm and with previous work of Ash, Knight, and Slaman on uniformity in a general computable structure-theoretical setting.

The class of abelian $p$-groups are an example of some very interesting phenomena in computable structure theory. We will give an elementary first-order theory $T_p$ whose models are each bi-interpretable with the disjoint union of an abelian $p$-group and a pure set (and so that every abelian $p$-group is bi-interpretable with a model of $T_p$) using computable infinitary formulas. This answers a question of Knight by giving an example of an elementary first-order theory of Ulm type: Any two models, low for $\omega_1^{CK}$, and with the same computable infinitary theory, are isomorphic. It also gives a new example of an elementary first-order theory whose isomorphism problem is $\mathbf{\Sigma}^1_1$-complete but not Borel complete.

Enumerations in computable structure theory

An old classical result in computable structure theory is Ash’s theorem stating that for every computable ordinal α ≥ 2, under some additional conditions, a computable structure is Δ 0 -categorical iff it has a computable Σ α Scott family. We construct a counterexample revealing that the proof of this theorem has a serious error. Moreover, we show how the error can be corrected by revising the proof. In addition, we formulate a sufficient condition under which the Δ 0 -dimension of a computable structure is infinite.

This paper contributes to the general program which aims to eliminate an unbounded search from proofs and procedures in computable structure theory. A countable structure in a finite language is punctual if its domain is ω \omega and its operations and relations are primitive recursive. A function f f is punctual if both f f and f − 1 f^{-1} are primitive recursive. We prove that there exists a countable rigid algebraic structure which has exactly two punctual presentations, up to punctual isomorphism.

We introduce a topology on the space of all isomorphism types represented in a given class of countable models, and use this topology as an aid in classifying the isomorphism types. This mixes ideas from effective descriptive set theory and computable structure theory, extending concepts from the latter beyond computable structures to examine the isomorphism problem on arbitrary countable structures. We give examples using specific classes of fields and of trees, illustrating how the new concepts can yield classifications that reveal differences between seemingly similar classes. Finally, we use a computable homeomorphism to define a measure on the space of isomorphism types of algebraic fields, and examine the prevalence of relative computable categoricity under this measure.

We survey the current results about degrees of categoricity and the degrees that are low for isomorphism as well as the proof techniques used in the constructions of elements of each of these classes. We conclude with an analysis of these classes, what we may deduce about them given the sorts of proof techniques used in each case, and a discussion of future lines of inquiry.

Klaimullin, Melnikov and Ng [KMNa] have recently suggested a new systematic approach to algorithms in algebra which is intermediate between computationally feasible algebra [CR91, KNRS07] and abstract computable structure theory [AK00, EG00]. In this short survey we discuss some of the key results and ideas of this new topic [KMNa, KMNc, KMNb]. We also suggest several open problems.

Limitwise monotonic sets and functions constitute an important tool in computable structure theory. We investigate limitwise monotonic numberings. A numbering ν of a family is limitwise monotonic (l.m.) if every set is the range of a limitwise monotonic function, uniformly in k. The set of all l.m. numberings of S induces the Rogers semilattice . The semilattices exhibit a peculiar behavior, which puts them in‐between the classical Rogers semilattices (for computable families) and Rogers semilattices of ‐computable families. We show that every Rogers semilattice of a ‐computable family is isomorphic to some semilattice . On the other hand, there are infinitely many isomorphism types of classical Rogers semilattices which can be realized as semilattices . In particular, there is an l.m. family S such that is isomorphic to the upper semilattice of c.e. m‐degrees. We prove that if an l.m. family S contains more than one element, then the poset is infinite, and it is not a lattice. The l.m. numberings form an ideal (w.r.t. reducibility between numberings) inside the class of all ‐computable numberings. We prove that inside this class, the index set of l.m. numberings is ‐complete.

We continue work from [1] on computable structure theory in the setting of ω1, where the countable ordinals play the role of natural numbers, and countable sets play the role of finite sets. In the present paper, we define the arithmetical hierarchy through all countable levels (not just the finite levels). We consider two different ways of doing this—one based on the standard definition of the hyperarithmetical hierarchy, and the other based on the standard definition of the effective Borel hierarchy. For each definition, we define computable infinitary formulas through all countable levels, and we obtain analogues of the well-known results from [2] and [4] saying that a relation is relatively intrinsically Σα just in case it is definable by a computable Σα formula. Although we obtain the same results for the two definitions of the arithmetical hierarchy, we conclude that the definition resembling the standard definition of the hyperarithmetical hierarchy seems preferable.

These are the lecture notes from a 10-hour course that the author gave at the University of Notre Dame in September 2010. The objective of the course was to introduce some basic concepts in computable structure theory and develop the background needed to understand the author’s research on back-and-forth relations.