Classification from a computable viewpoint

Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence, in terms of relatively simple invariants. Where this is impossible, it is useful to have concrete results saying so. In model theory and descriptive set theory, there is a large body of work showing that certain classes of mathematical structures admit classification while others do not. In the present paper, we describe some recent work on classification in computable structure theory. Section 1 gives some background from model theory and descriptive set theory. From model theory, we give sample structure and non-structure theorems for classes that include structures of arbitrary cardinality. We also describe the notion of Scott rank, which is useful in the more restricted setting of countable structures. From descriptive set theory, we describe the basic Polish space of structures for a fixed countable language with fixed countable universe. We give sample structure and non-structure theorems based on the complexity of the isomorphism relation, and on Borel embeddings. Section 2 gives some background on computable structures. We describe three approaches to classification for these structures. The approaches are all equivalent. However, one approach, which involves calculating the complexity of the isomorphism relation, has turned out to be more productive than the others. Section 3 describes results on the isomorphism relation for a number of mathematically interesting classes—various kinds of groups and fields. In Section 4, we consider a setting similar to that in descriptive set theory. We describe an effective analogue of Borel embedding which allows us to make distinctions even among classes of finite structures. Section 5 gives results on computable structures of high Scott rank. Some of these results make use of computable embeddings. Finally, in Section 6, we mention some open problems and possible directions for future work

Degrees of Categoricity and the Isomorphism Problem

In this thesis, we study notions of complexity related to computable structures. We first study degrees of categoricity for computable tree structures. We show that, for any computable ordinal $\alpha$, there exists a computable tree of rank $\alpha+1$ with strong degree of categoricity ${\bf 0}^{(2\alpha)}$ if $\alpha$ is finite, and with strong degree of categoricity ${\bf 0}^{(2\alpha+1)}$ if $\alpha$ is infinite. For a computable limit ordinal $\alpha$, we show that there is a computable tree of rank $\alpha$ with strong degree of categoricity ${\bf 0}^{(\alpha)}$ (which equals ${\bf 0}^{(2\alpha)}$). In general, it is not the case that every Turing degree is the degree of categoricity of some structure. However, it is known that every degree that is of a computably enumerable (c.e.) set\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a successor ordinal, is a degree of categoricity. In this thesis, we include joint work with Csima, Deveau and Harrison-Trainor which shows that every degree c.e.\ in and above $\mathbf{0}^{(\alpha)}$, for $\alpha$ a limit ordinal, is a degree of categoricity. We also show that every degree c.e.\ in and above $\mathbf{0}^{(\omega)}$ is the degree of categoricity of a prime model, making progress towards a question of Bazhenov and Marchuk. After that, we study the isomorphism problem for tree structures. It follows from our proofs regarding the degrees of categoricity for these structures that, for every computable ordinal $\alpha>0$, the isomorphism problem for trees of rank $\alpha$ is $\Pi_{2\alpha}$-complete. We also discuss the isomorphism problem for pregeometries in which dependent elements are dense and the closure operator is relatively intrinsically computably enumerable. We show that, if $K$ is a class of such pregeometries, then the isomorphism problem for the class $K$ is $\Pi_3$-hard. Finally, we study the Turing ordinal. We observed that the definition of the Turing ordinal has two parts each of which alone can define a specific ordinal which we call the upper and lower Turing ordinals. The Turing ordinal exists if and only if these two ordinals exist and are equal. We give examples of classes of computable structures such that the upper Turing ordinal is $\beta$ and the lower Turing ordinal is $\alpha$ for all computable ordinals $\alpha<\beta$

Computability Theory and Some Applications

We explore various areas of computability theory, ranging from applications in computable structure theory primarily focused on problems about computing isomorphisms, to a number of new results regarding the degree-theoretic notion of the bounded Turing hierarchy. In Chapter 2 (joint with Csima, Harrison-Trainor, Mahmoud), the set of degrees that are computably enumerable in and above $\mathbf{0}^{(\alpha)}$ are shown to be degrees of categoricity of a structure, where $\alpha$ is a computable limit ordinal. We construct such structures in a particularly useful way: by restricting the construction to a particular case (the limit ordinal $\omega$) and proving some additional facts about the widgets that make up the structure, we are able to produce a computable prime model with a degree of categoricity as high as is possible. This then shows that a particular upper bound on such degrees is exact. In Chapter 3 (joint with Csima and Stephenson), a common trick in computable structure theory as it relates to degrees of categoricity is explored. In this trick, the degree of an isomorphism between computable copies of a rigid structure is often able to be witnessed by the clever choice of a computable set whose image or preimage through the isomorphism actually attains the degree of the isomorphism itself. We construct a pair of computable copies of $(\omega, <)$ where this trick will not work, examine some problems with decidability of the structures and work with $(\omega^2, <)$ to resolve them by proving a similar result. In Chapter 4, the effectivization of Walker's Cancellation Theorem in group theory is discussed in the context of uniformity. That is, if we have an indexed collection of instances of sums of finitely generated abelian groups A_i \join G_i \cong A_i \join H_i and the code for the isomorphism between them, then we wish to know to what extent we can give a single procedure that, given an index $i$, produces an isomorphism between $G_i$ and $H_i$. Finally, in Chapter 5, several results pertaining to the bounded Turing degrees (also known as the weak truth-table degrees) and the bounded jump are investigated, with an eye toward jump inversion. We first resolve a potential ambiguity in the definition of sets used to characterize degrees in the bounded Turing hierarchy. Then we investigate some open problems related to lowness and highness as it appears in this realm, and then generalize a characterization about reductions to iterated bounded jumps of arbitrary sets. We use this result to prove the non-triviality of the hierarchy of successive applications of the bounded jump above any set, showing that the problem of jump inversion must be non-trivial if it is true in any relativized generality

Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science