Large Scale Geometries of Infinite Strings
Bakhadyr Khoussainov, Toru Takisaka
TL;DR
The paper pioneers the use of large-scale geometry, via colour-preserving quasi-isometries, to study infinite strings in formal language theory, treating strings as coloured metric spaces. It develops a rich poset of large-scale geometries, proves the existence of a greatest element, and shows infinite chains, anti-chains, and infinitely many minimal elements, while refining the notion of quasi-isometry through eventually periodic cases and componentwise reducibility. It connects these geometries to Büchi automata with a linear-time atlas-equality algorithm, analyzes the quasi-isometry problem as a $oldsymbol{ ext{Σ}}_3^0$-complete task, and investigates asymptotic cones, including interactions with algorithmic randomness (e.g., Martin-Löf randomness). Collectively, the results fuse formal languages, geometric group theory, automata theory, and computability, yielding new invariants and efficient decision procedures for global string patterns.
Abstract
We introduce geometric consideration into the theory of formal languages. We aim to shed light on our understanding of global patterns that occur on infinite strings. We utilise methods of geometric group theory. Our emphasis is on large scale geometries. Two infinite strings have the same large scale geometry if there are colour preserving bi-Lipschitz maps with distortions between the strings. Call these maps quasi-isometries. Introduction of large scale geometries poses several questions. The first question asks to study the partial order induced by quasi-isometries. This partial order compares large scale geometries; as such it presents an algebraic tool for classification of global patterns. We prove there is a greatest large scale geometry and infinitely many minimal large scale geometries. The second question is related to understanding the quasi-isometric maps on various classes of strings. The third question investigates the sets of large scale geometries of strings accepted by computational models, e.g. Büchi automata. We provide an algorithm that describes large scale geometries of strings accepted by Büchi automata. This links large scale geometries with automata theory. The fourth question studies the complexity of the quasi-isometry problem. We show the problem is $Σ_3^0$-complete thus providing a bridge with computability theory. Finally, the fifth question asks to build algebraic structures that are invariants of large scale geometries. We invoke asymptotic cones, a key concept in geometric group theory, defined via model-theoretic notion of ultra-product. Partly, we study asymptotic cones of algorithmically random strings thus connecting the topic with algorithmic randomness.