Notes on degrees of relative computable categoricity

We are studying the degrees in which a computable structure is relatively computably categoricity, i.e., computably categorcial among all non-computable copies of the structure. Unlike the degrees of computable categoricity we can bound the possible degrees of relative computable categoricity by the oracle 0". In the case of rigid structures the bound is in fact 0'. These estimations are precise, in particular we can build a computable structure which is relatively computably categorical only in the degrees above 0"

Categoricity Spectra for Rigid Structures

© 2016 by University of Notre Dame. For a computable structure M, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of M. If the spectrum has a least degree, this degree is called the degree of categoricity of M. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity

Degrees of categoricity of computable structures

Defining the degree of categoricity of a computable structure M to be the least degree d for which M is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d. c. e. in and above 0(n) can be so realized, as can the degree 0(ω). © 2009 Springer-Verlag

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 $\alpha<\beta$