Computable categoricity of trees of finite height

Computable categoricity of trees of finite height

We characterize the structure of computably categorical trees of finite height, and prove that our criterion is both necessary and sufficient. Intuitively, the characterization is easiest to express in terms of isomorphisms of (possibly infinite) trees, but in fact it is equivalent to a Σ3-condition. We show that all trees which are not computably categorical have computable dimension ω. Finall...

Computable Categoricity of Graphs with Finite Components

A computable graph is computably categorical if any two computable presentations of the graph are computably isomorphic. In this paper we investigate the class of computably categorical graphs. We restrict ourselves to strongly locally finite graphs; these are the graphs all of whose components are finite. We present a necessary and sufficient condition for certain classes of strongly locally f...

Finite Computable Dimension and Degrees of Categoricity

We first give an example of a rigid structure of computable dimension 2 such that the unique isomorphism between two non-computably isomorphic computable copies has Turing degree strictly below 0′′, and not above 0′. This gives a first example of a computable structure with a degree of categoricity that does not belong to an interval of the form [0(α),0(α+ 1)] for any computable ordinal α. We t...

Computable Categoricity versus Relative Computable Categoricity

We study the notion of computable categoricity of computable structures, comparing it especially to the notion of relative computable categoricity and its relativizations. We show that every 1-decidable computably categorical structure is relatively ∆2-categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also...

Categoricity of computable infinitary theories