Degree Spectra of Relations on Computable Structures

Abstract

There has been increasing interest over the last few decades in the study of the effective content of Mathematics. One field whose effective content has been the subject of a large body of work, dating back at least to the early 1960s, is model theory. (A valuable reference is the handbook [7]. In particular, the introduction and the articles by Ershov and Goncharov and by Harizanov give useful overviews, while the articles by Ash and by Goncharov cover material related to the topic of this communication.)Several different notions of effectiveness of model-theoretic structures have been investigated. This communication is concerned withcomputablestructures, that is, structures with computable domains whose constants, functions, and relations are uniformly computable.In model theory, we identify isomorphic structures. From the point of view of computable model theory, however, two isomorphic structures might be very different. For example, under the standard ordering of ω the success or relation is computable, but it is not hard to construct a computable linear ordering of type ω in which the successor relation is not computable. In fact, for every computably enumerable (c. e.) degree a, we can construct a computable linear ordering of type ω in which the successor relation has degree a. It is also possible to build two isomorphic computable groups, only one of which has a computable center, or two isomorphic Boolean algebras, only one of which has a computable set of atoms. Thus, for the purposes of computable model theory, studying structures up to isomorphism is not enough.

Π10 classes and strong degree spectra of relations

“…For a computable structure B such that B ∼ = A, the Turing degree spectrum of R on A with respect to B, in symbols DgSp A,B (R), is the set of all Turing degrees of the images f (R) ⊆ |B| (where |B| is the domain of B) under all isomorphisms f from A to B. The study of Turing degree spectra has been continued in many papers by Ash, Cholak, Downey, Goncharov, Harizanov, Hirschfeldt, Khoussainov, Knight, Shore, White, and others (for example, see [1,2,11,15,17,18,19,20,21,22,23,28]). …”

Spaces of orders and their Turing degree spectra

“…For a computable group G, we define the notion of a Turing degree spectrum of LO(G). (For another notion of a degree spectrum of a relation on a computable structure see [16], [18]. )…”

On the complexity of categoricity in computable structures

“…The case where Γ is the collection of hyperarithmetic sets has been studied by Ash [1], while the case where Γ is the collection of all computable sets was originally studied by Goncharov [5]. Further work on computable categoricity includes that of Kudinov [12], Cholak, Goncharov, Khoussainov, and Shore [3], Hirschfeldt, Khoussainov, Slinko, and Shore [9], and Hirschfeldt [8].…”