Three lectures on automatic structures

Abstract

PrefaceThis paper grew out of three tutorial lectures on automatic structures given at the Logic Colloquium 2007 in Wroc law (Poland). The paper will follow the outline of the tutorial lectures, supplementing some material along the way. We discuss variants of automatic structures related to several models of computation: word automata, tree automata, Büchi automata, and Rabin automata. Word automata process finite strings, tree automata process finite labeled trees, Büchi automata process infinite strings, and Rabin automata process infinite binary labeled trees. Finite automata are the most commonly known in the general computer science community. Such an automaton reads finite input strings from left to right, making state transitions along the way. Depending on its last state after processing a given string, the automaton either accepts or rejects the input string. Automatic structures are mathematical objects which can be represented by (word, tree, Büchi, or Rabin) automata. The study of properties of automatic structures is a relatively new and very active area of research.We begin with some motivation and history for studying automatic structures. We introduce definitions of automatic structures, present examples, and discuss decidability and definability theorems. Next, we concentrate on finding natural isomorphism invariants for classes of automatic structures. These classes include well-founded partial orders, Boolean algebras, linear orders, trees, and finitely generated groups. Finally, we address the issue of complexity for automatic structures. In order to measure complexity of automatic structures we involve topology (via the Borel hierarchy), model theory (Scott ranks), computability theory (Σ 1 1 -completeness), and computational complexity (the class P).This paper consists of three sections based on the tutorial lectures. The first lecture provides motivating questions and historical context, formal definitions of the different types of automata involved, examples of automatic structures, and decidability and definability results about automatic structures. The second lecture considers tree and Rabin automatic structures, and outlines techniques for proving non-automaticity. We then turn our attention to the study of algorithmic and structural properties of automatic trees, Boolean algebras, and finitely generated groups. The final lecture presents a framework for reducing certain questions about computable structures to questions about automatic structures. These reductions have been used to show that, in some cases, the sharp bound on complexity of automatic structures is as high as the bounds for computable structures. We conclude by looking at Borel structures from descriptive set theory and connecting them to Büchi and Rabin automatic structures.1 Basics Motivating questionsThe study of structures has played a central role in the development of logic.In the course of this study, several themes have been pursued. We will see how questions related to each of these themes is addressed in the ...

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