Computable categoricity versus relative computable categoricity

Abstract

Abstract. 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 ∆ 0 2 -categorical. We study the complexity of various index sets associated with computable categoricity and relative computable categoricity. We also introduce and study a variation of relative computable categoricity, comparing it to both computable categoricity and relative computable categoricity and its relativizations.

Index Sets for n-Decidable Structures Categorical Relative to m-Decidable Presentations

“…Note that property ( * ) is a weakening of computable categoricity. Downey, Kach, Lempp, and Turetsky showed that if a structure is computably categorical and 1-decidable, then it is relatively Δ 0 2 -categorical [20]. Inspection of that proof reveals that use was not made of the full power of computable categoricity; instead, advantage was taken of property ( * ) only.…”

“…In [24], it was proved that a 2-decidable computably categorical structure is relatively computably categorical. In [20], it was stated that the index set of relatively computably categorical structures is Σ 0 3 -complete. In fact, it was shown that the index set of 2-decidable computably categorical structures is Σ 0 3 -complete.…”

Categoricity Spectra of Computable Structures

“…In [71] it was shown that this result cannot be extended to 1-decidable structures: there is a computably categorical, 1-decidable structure with no formally Σ 0 1 Scott family. Nevertheless, in [34] it was proved that any computably categorical, 1-decidable structure is relatively Δ 0 2 -categorical. There are a lot of works that study relative Δ 0 1+α categoricity and 0 (α) -computable categoricity for structures from familiar algebraic classes: Boolean algebras [3,10,11,78,79], linear orders [4,46,78,79], abelian groups [35][36][37]81], fields [67,87,88], etc.…”