Autostability spectra for decidable structures

Abstract

We study autostability spectra relative to strong constructivizations (SC-autostability spectra). For a decidable structure $\mathcal{S}$, the SC-autostability spectrum of $\mathcal{S}$ is the set of all Turing degrees capable of computing isomorphisms among arbitrary decidable copies of $\mathcal{S}$. The degree of SC-autostability for $\mathcal{S}$ is the least degree in the spectrum (if such a degree exists).We prove that for a computable successor ordinal α, every Turing degree c.e. in and above 0(α) is the degree of SC-autostability for some decidable structure. We show that for an infinite computable ordinal β, every Turing degree c.e. in and above 0(2β+1) is the degree of SC-autostability for some discrete linear order. We prove that the set of all PA-degrees is an SC-autostability spectrum. We also obtain similar results for autostability spectra relative to n-constructivizations.

Computable embeddings for pairs of linear orders

“…First, these pairs play an important role in computable structure theory. The technique of pairs of computable structures, which was developed by Ash and Knight [1,2], found many applications in studying various computability-theoretic properties of structures (in particular, their degree spectra and effective categoricity, see, e.g., [2,3,10]).…”

Categoricity Spectra of Computable Structures

“…In a subsequent work, this result was refined in the following way. Theorem 2.13 (see [19]). There exists a decidable prime model such that its decidable categoricity spectrum contains precisely the P A-degrees.…”

“…In particular, it was proved in [54] that for any c.e. Turing degree d, there is a decidable prime model with strong degree of decidable categoricity d. Theorem 2.12 (see [12,19]). Assume that α is a computable ordinal, and d is a Turing degree such that d ≥ 0 (α+1) and d is c.e.…”