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Reviewer: Charles William Bash

Allouche and Shallit present the theory and generalization of automatic sequences by combining work encompassing pure mathematics and theoretical computer science. They have succeeded in presenting the material in an integrated framework. The presentation of each topic is systematic, and the authors present results from first principles in order to maintain the suitability to a large audience of scientists wishing to explore this subject. They also discuss applications relating to mathematics, physics, computer graphics, and music. The authors have demonstrated great insight into the subject. The book is divided into 17 chapters. Each chapter ends with exercises, open problems (where unsolved research problems are given), and notes (given for further study and research). Additionally, an appendix is provided, with hints and solutions to selected exercises given in each of the chapters. Finally, the bibliography section consists of over 1600 references related to the subject. In chapter 1, "Stringology," the authors discuss the basic definitions and properties of finite and infinite words, as well as the definition of a language, which is a set of words. Then they define morphism, which is a way of transforming words. Additionally, they describe the mathematical foundation for repetitions in a word and the overlap-free binary words. Chapter 2 introduces the basic mathematical results on algebra and number theory, such as divisibility and valuations, rational and irrational numbers, algebraic and transcendental numbers, continued fractions, Diophantine approximation, algebraic structures, vector spaces, fields, polynomials rational functions and formal power series, p-adic numbers, and asymptotic notation. Chapter 3 presents the numeration systems, and, in particular, how numbers can be represented by strings over a finite alphabet with the classical k-base representation, Fibonacci representation, and the complex base representation. In chapter 4, the authors discuss models of computations, in particular, finite automata and its variants and Turing machines. In chapter 5, the authors introduce the basic concepts of automatic sequences. Chapter 6 covers the concepts of fixed points of uniform morphisms and the closure properties of automatic sequences. In chapter 7, the authors present the uniform morphism, as well as the general case where the morphism is not uniform. In chapter 8, the authors prove the theorems about the frequency of letters in automatic and morphic sequences. Chapter 9 covers the definitions and basic properties of a special class of infinite words, called the characteristic words. In chapter 10, the authors introduce the basic properties of subword complexity of automatic sequence, as well as morphic words. They also study recurrence and appearance. In chapter 11, the proof of Cobham's Theorem is presented. In chapter 12, the authors study the algebraicity and transcendence of formal power series and their relation with finite automata. Chapter 13 introduces the concepts of automatic real numbers and investigates their properties. In chapter 14, the authors present the generalization of automatic sequences to the multidimensional case. Chapter 15 covers the automaticity of languages and their generalization, namely the automaticity of sequences. Chapter 16 deals with the generalization of a k-automatic sequence by the k-regular sequence. In chapter 17, the authors show the applicability of automatic and morphic sequences in physics. The material presented in this book is well balanced, and is presented in a consistent framework. The authors refer to automatic sequences that are generated by finite automata; they have discussed in detail the close relation of automatic sequences with number theory. The presentation of the material and the proofs of the theorems stated are thought provoking and of high quality. Fortunately, a list of errata is available on one of the author's Web sites. Whenever there is a need for an algorithmic description, it is given in a pseudocode notation similar to a C language, which is familiar to most scientists these days. Since the subject incorporates results from mathematics and computer science, the authors have presented the material in a self-contained way, using consistent notation. They have also gathered material that was spread among different sources, and sometimes not accessible at large. I highly recommend this book to interested readers, from advanced undergraduates to experts in the field of formal languages and number theory. The reader must have a certain level of mathematical knowledge to understand the concepts presented in the book.