Uniting dozens of seemingly disparate results from different fields, this book combines concepts from mathematics and computer science to present the first integrated treatment of sequences generated by 'finite automata'. The authors apply the theory to the study of automatic sequences and their generalizations, such as Sturmian words and k-regular sequences. And further, they provide applications to number theory (particularly to formal power series and transcendence in finite characteristic), physics, computer graphics, and music. Starting from first principles wherever feasible, basic results from combinatorics on words, numeration systems, and models of computation are discussed. Thus this book is suitable for graduate students or advanced undergraduates, as well as for mature researchers wishing to know more about this fascinating subject. Results are presented from first principles wherever feasible, and the book is supplemented by a collection of 460 exercises, 85 open problems, and over 1600 citations to the literature.

‘… the very first book that gathers together results from various fields of mathematics and computer science that appeared in a broad variety of journals. It presents the results in a unified manner, making use of a consistent notation and introduces some good unifying terminology, and including most of the material needed for a comprehensive exposition … useful also as a handbook for self study … extremely valuable as a reference text … I strongly recommend this excellent book to anybody interested in interaction between theoretical computer science and mathematics.’

Source: ACM SIGACT News

‘The book is very well written, and contains a tremendous amount of information … Advanced students and researchers will delight in reading Automatic Sequences.’

Source: Bulletin of the London Mathematical Society

'The material presented in this book is well balanced, and is presented in a consistent framework … I highly recommend this book to interested readers, from advanced undergraduates to experts in the field of formal languages and number theory.'

Source: Computing Reviews

‘The book will be of use for beginners as well as for advanced students and researchers.‘

Source: EMS Newsletter

‘This book has been written with the greatest care, in a clear and concise style. The authors have included a number of well-chosen examples which illustrate the general theory. Specialists would particularly appreciate the notes at the end of the chapter, recalling the main contributions and the history of the concepts that are discussed. They also provide a useful link with the impressive bibliography.‘

Source: Semigroup Forum

'Allouche and Shallit's books presents an introduction to the fascinating subject of automatic sequences. This is the first book that systematically develops the theory of these sequences.'

Source: Zentralblatt MATH