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We study the algorithmic complexity of embeddings between bi-embeddable equivalence structures. We define the notions of computable bi-embeddable categoricity, (relative) $Delta^0_alpha$ bi-embeddable categoricity, and degrees of bi-embeddable categoricity. These notions mirror the classical notions used to study the complexity of isomorphisms between structures. We show that the notions of $Delta^0_alpha$ bi-embeddable categoricity and relative $Delta^0_alpha$ bi-embeddable categoricity coincide for equivalence structures for $alpha=1,2,3$. We also prove that computable equivalence structures have degree of bi-embeddable categoricity $mathbf{0},mathbf{0}$, or $mathbf{0}$. We obtain results on index sets of computable equivalence structure with respect to bi-embeddability.
We investigate the complexity of embeddings between bi-embeddable structures. In analogy with categoricity spectra, we define the bi-embeddable categoricity spectrum of a structure $mathcal A$ as the family of Turing degrees that compute embeddings b
In several classes of countable structures it is known that every hyperarithmetic structure has a computable presentation up to bi-embeddability. In this article we investigate the complexity of embeddings between bi-embeddable structures in two such
A computable structure A is x-computably categorical for some Turing degree x, if for every computable structure B isomorphic to A there is an isomorphism f:B -> A with f computable in x. A degree x is a degree of categoricity if there is a computabl
A computable structure $mathcal{A}$ has degree of categoricity $mathbf{d}$ if $mathbf{d}$ is exactly the degree of difficulty of computing isomorphisms between isomorphic computable copies of $mathcal{A}$. Fokina, Kalimullin, and Miller showed that e
While most research in Gold-style learning focuses on learning formal languages, we consider the identification of computable structures, specifically equivalence structures. In our core model the learner gets more and more information about which pa