Quasi-Isometric Reductions Between Infinite Strings

TL;DR

This paper explores the geometric relationships between infinite strings through quasi-isometric reductions, revealing that some reductions are inherently non-recursive, thus advancing the understanding of large-scale geometric properties in recursion theory.

Contribution

It constructs specific infinite recursive strings demonstrating that certain quasi-isometric reductions cannot be achieved recursively, solving an open problem in the field.

Findings

Existence of infinite recursive strings with non-recursive quasi-isometric reductions

Characterization of equivalence classes of such reductions

Resolution of an open problem by Khoussainov and Takisaka

Abstract

This paper studies the recursion-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We investigate several notions of quasi-isometric reductions between recursive infinite strings and prove various results on the equivalence classes of such reductions. The main result is the construction of two infinite recursive strings $\alpha$ and $\beta$ such that $\alpha$ is strictly quasi-isometrically reducible to $\beta$, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.