Quasi-Isometric Reductions Between Infinite Strings

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 $\hat{I}\pm$ and ${\hat{I}}^2$ such that $\hat{I}\pm$ is strictly quasi-isometrically reducible to ${\hat{I}}^2$, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka.