Generically and coarsely computable isomorphisms

Author:

Calvert Wesley1,Cenzer Douglas2,Harizanov Valentina3

Affiliation:

1. School of Mathematical and Statistical Sciences, Mail Code 4408, Southern Illinois University, 1245 Lincoln Drive, Carbondale, Illinois 62901, USA

2. Department of Mathematics, University of Florida, Gainesville, FL 32611, USA

3. Department of Mathematics, George Washington University, Washington, DC 20052, USA

Abstract

Inspired by the study of generic and coarse computability in computability theory, we extend such investigation to the context of computable model theory. In this paper, we continue our study initiated in the previous paper (Journal of Logic and Computation 32 (2022) 581–607) , where we introduced and studied the notions of generically and coarsely computable structures and their generalizations. In this paper, we introduce the notions of generically and coarsely computable isomorphisms, and their weaker variants. We sometimes also require that the isomorphisms preserve the density structure. For example, for any coarsely computable structure A, there is a density preserving coarsely computable isomorphism from A to a computable structure. We demonstrate that each notion of generically and coarsely computable isomorphisms, density preserving or not, gives interesting insights into the structures we consider, focusing on various equivalence structures and injection structures.

Publisher

IOS Press

Subject

Artificial Intelligence,Computational Theory and Mathematics,Computer Science Applications,Theoretical Computer Science

Reference11 articles.

1. Densely computable structures;Calvert;Journal of Logic and Computation,2022

2. Effective categoricity of equivalence structures;Calvert;Annals of Pure and Applied Logic,2006

3. Σ 1 0 and Π 1 0 structures;Cenzer;Annals of Pure and Applied Logic,2011

4. Computability-theoretic properties of injection structures;Cenzer;Algebra and Logic,2014

5. Equivalence structures and isomorphisms in the difference hierarchy;Cenzer;Journal of Symbolic Logic,2009

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1. Generically Computable Abelian Groups;Unconventional Computation and Natural Computation;2023

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