Quasi-isometric reductions between infinite strings

Abstract

This paper studies the recursion- and automata-theoretic aspects of large-scale geometries of infinite strings, a subject initiated by Khoussainov and Takisaka (2017). We first 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 α and β such that α is strictly quasi-isometrically reducible to β, but the reduction cannot be made recursive. This answers an open problem posed by Khoussainov and Takisaka. Furthermore, we also study automatic quasi-isometric reductions between automatic structures, and show that automatic quasi-isometry may be separable from general quasi-isometry depending on the growth of the automatic domain.