无穷串的大尺度几何
Large Scale Geometries of Infinite Strings
AI总结：
本文将几何群论引入形式语言研究无穷串的大尺度几何，通过拟等距映射分类全局模式，并探讨了与Büchi automata、可计算性及算法随机性相关的五个核心问题。
摘要：
我们将几何考量引入形式语言理论。旨在阐明无穷串上出现的全局模式。我们利用几何群论的方法，重点关注大尺度几何。若两个无穷串之间存在保持颜色的双Lipschitz映射且具有畸变，则它们具有相同的大尺度几何。称这些映射为拟等距。引入大尺度几何引出了若干问题。第一个问题研究由拟等距诱导的偏序。该偏序比较了大尺度几何，从而为全局模式的分类提供了代数工具。我们证明了存在一个最大大尺度几何和无穷多个极小大尺度几何。第二个问题与理解各类串上的拟等距映射有关。第三个问题考察由计算模型（如Büchi automata）接受的串的大尺度几何集合。我们提供了一个描述Büchi automata接受的串的大尺度几何的算法，将大尺度几何与automata theory联系起来。第四个问题研究拟等距问题的复杂度。我们证明该问题是$Σ_3^0$-完全的，从而搭建了与可计算性理论的桥梁。第五个问题要求构建作为大尺度几何不变量的代数结构。我们引入了asymptotic cones，这是几何群论中的核心概念，通过model-theoretic notion of ultra-product定义。我们部分研究了algorithmically random strings的asymptotic cones，从而将该主题与algorithmic randomness联系起来。
英文摘要：
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.
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