Degrees of categoricity and spectral dimension

© 2018 The Association for Symbolic Logic. A Turing degree d is the degree of categoricity of a computable structure S if d is the least degree capable of computing isomorphisms among arbitrary computable copies of S. A degree d is the strong degree of categoricity of S if d is the degree of categoricity of S, and there are computable copies A and B of S such that every isomorphism from A onto B computes d. In this paper, we build a c.e. degree d and a computable rigid structure Msuch that d is the degree of categoricity of M, but d is not the strong degree of categoricity of M. This solves the open problem of Fokina, Kalimullin, andMiller [13]. For a computable structure S, we introduce the notion of the spectral dimension of S, which gives a quantitative characteristic of the degree of categoricity of S. We prove that for a nonzero natural number N, there is a computable rigid structureMsuch that 0 is the degree of categoricity ofM, and the spectral dimension ofMis equal to N

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 $\alpha<\beta$