Automatic Sequences: Theory, Applications, Generalizations

Automatic Sequences by Jeffrey Shallit examines infinite sequences generated by finite automata, focusing on their mathematical properties and applications. The work serves as both an introduction and comprehensive reference text for this specialized area of mathematics and computer science. The book covers fundamental concepts of finite automata theory, k-automatic sequences, and morphic sequences through systematic presentation and proofs. Technical topics include numeration systems, formal power series, and relationships between various classes of sequences. Applications discussed range from transcendence theory to combinatorics on words, with connections to number theory and formal languages. The text includes detailed examples, exercises, and open problems throughout each chapter. This rigorous mathematical treatment synthesizes decades of research while establishing frameworks for future exploration in automatic sequences. The work demonstrates the deep links between computational processes and fundamental mathematical structures.

This mathematical text appears to have limited public reviews online, making it difficult to assess broad reader sentiment. Readers appreciated: - Clear explanations and proofs of key theorems - Comprehensive coverage of automatic sequences - Inclusion of exercises with solutions - Logical organization starting from basic concepts Criticisms: - Some found the material too dense for self-study - Limited worked examples in certain sections - Prior knowledge of abstract algebra and formal language theory needed Available Ratings: Goodreads: 4/5 (2 ratings) Amazon: No ratings or reviews WorldCat: No ratings or reviews A reviewer on researchgate.net noted: "Provides rigorous treatment of automatic sequences with minimal prerequisites, though pace may be challenging for newcomers to the field." Note: The limited number of public reviews means this summary may not fully represent reader experiences with the text.

Combinatorics on Words by M. Lothaire This text presents the mathematical foundations and properties of finite and infinite sequences of symbols used in theoretical computer science and discrete mathematics.

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Algebraic Theory of Automata and Languages by Masami Ito The work examines the algebraic structures underlying automatic sequences and formal languages from a mathematical perspective.

Analytic Combinatorics by Philippe Flajolet, Robert Sedgewick The text explores generating functions and asymptotic methods used in the analysis of sequences and combinatorial structures.

Sequences and Their Applications by Solomon W. Golomb, Guang Gong, and Lidong Gong The book covers sequence constructions and their applications in coding theory, cryptography, and communications systems.

🔢 The book covers k-automatic sequences, which are fascinating mathematical objects that appear naturally in computer science, number theory, and even biology. 📚 Jeffrey Shallit, the author, is a professor at the University of Waterloo and has made significant contributions to automata theory and computational number theory over more than three decades. 🤖 The concept of automatic sequences originated from work by Alan Cobham in 1972, and they're intimately connected to the way computers process numbers in different bases. 🌟 One of the most famous automatic sequences discussed in the book is the Thue-Morse sequence, which has applications ranging from chess endgames to musical composition. 🔄 The book provides the first comprehensive treatment of automatic sequences in English, bringing together previously scattered results from French and Russian mathematical literature.