Abstract
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 furthermore obtain results on the index set complexity of computable equivalence structure with respect to bi-embeddability.
Highlights
The systematic study of the complexity of isomorphisms between computable copies of structures was initiated in the 1950s by Fröhlich and Shepherdson [1] and independently by Maltsev [2]
We prove that computable equivalence structures have degree of bi-embeddable categoricity 0, 0, or 0
In this article we study the algorithmic complexity of embeddings between biembeddable equivalence structures
Summary
The systematic study of the complexity of isomorphisms between computable copies of structures was initiated in the 1950s by Fröhlich and Shepherdson [1] and independently by Maltsev [2]. Fokina et al [7] studied degree spectra with respect to the bi-embeddability relation and noticed that any countable equivalence structure is bi-embeddable with a computable one. For this reason, the study of the algorithmic complexity of embeddings is interesting for this class of structures. In addition, A has two computable biembeddable copies A0, A1 such that for all embeddings μ: A0 → A1, ν: A1 → A0, μ ⊕ ν ≥T d, d is the strong degree of bi-embeddable categoricity of A. 4 we obtain results on the complexity of the index sets of equivalence structures with degrees of bi-embeddable categoricity 0, 0 , and 0
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
