The Logical Approach to Automatic Sequences
Exploring Combinatorics on Words with Walnut
Published on 29. September 2022
Book
Paperback/Softback
374 pages
978-1-108-74524-6 (ISBN)
Description
Automatic sequences are sequences over a finite alphabet generated by a finite-state machine. This book presents a novel viewpoint on automatic sequences, and more generally on combinatorics on words, by introducing a decision method through which many new results in combinatorics and number theory can be automatically proved or disproved with little or no human intervention. This approach to proving theorems is extremely powerful, allowing long and error-prone case-based arguments to be replaced by simple computations. Readers will learn how to phrase their desired results in first-order logic, using free software to automate the computation process. Results that normally require multipage proofs can emerge in milliseconds, allowing users to engage with mathematical questions that would otherwise be difficult to solve. With more than 150 exercises included, this text is an ideal resource for researchers, graduate students, and advanced undergraduates studying combinatorics, sequences, and number theory.
Reviews / Votes
'In this book, Jeffrey Shallit gives combinatorics on words enthusiasts access to new and exciting tools to compute examples and test conjectures. Far from a mere user's manual, the text fully introduces the reader to the interactions of logic and words, proving basic theorems like the decidability of Presburger's arithmetic. It will be of great use to students and researchers, as well as the source of many future developments.' Dominique Perrin, Universite Gustave Eiffel 'This book focuses on a decision procedure, which is rather easy to implement as a computer program and allows one to prove many results, classical and new, in combinatorics on words. It addresses decision problems and enumeration problems on sequences that are expressible in first-order logic. The reader will appreciate the style, which is relaxed and pleasant to read, and the numerous examples and exercises. This book is a useful complement to the previous monograph, Automatic Sequences, co-authored by Shallit and Allouche.' Yann Bugeaud, University of Strasbourg 'This is a marvelous book with a very fresh approach to the decidability and structural analysis of combinatorics on words. It combines three different mathematical research topics: first-order logic, automatic sequences, and combinatorics on words. More precisely, it interprets infinite morphic words as automatic sequences via k-automata and expresses properties (of words) in first-order logic. Due to the decidability of such logic, decision results and structural properties of combinatorics on words are established. A crucial role in this approach is to employ a powerful software package called Walnut. The author illustrates the power of his approach by giving a huge number of results obtained by this method. Not only are old and new results proved, but even some errors in previous ones are corrected. Anybody interested in, or curious about, this topic should be enthusiastic about this masterpiece.' Juhani Karhumaeki, University of Turku (Emeritus) 'You should buy this book for the following reasons. 1. For your own enlightenment. 2. For examples of concepts you can teach in a course in automata theory. That is, even if you do not plan to use or teach Walnut, there is much of interest in the book.' William Gasarch, SIGACT NewsMore details
Series
Language
English
Place of publication
Cambridge
United Kingdom
Product notice
Paperback (trade)
Illustrations
Worked examples or Exercises
Dimensions
Height: 226 mm
Width: 150 mm
Thickness: 23 mm
Weight
544 gr
ISBN-13
978-1-108-74524-6 (9781108745246)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
Other editions
Additional editions
Jeffrey Shallit
The Logical Approach to Automatic Sequences
Exploring Combinatorics on Words with Walnut
E-Book
09/2022
Cambridge University Press
€86.49
Available for download
Person
Jeffrey Shallit is Professor of Computer Science in the Faculty of Mathematics at the University of Waterloo. His research areas include formal languages, finite automata, combinatorics on words, algorithmic number theory, algebra, and the history of mathematics. He has published approximately 300 articles on these topics since 1975. He is also the author or co-author of four books. He is a foreign member of the Finnish Academy of Science and Letters.
Content
1. Introduction; 2. Words and sequences; 3. Number representations and numeration systems; 4. Automata; 5. Automatic sequences; 6. First-order logic and automatic sequences; 7. Using Walnut; 8. First-order formulas for fundamental sequence properties; 9. Regular sequences and enumeration problems; 10. Synchronized sequences; 11. Additive number theory; 12. Paperfolding sequences; 13. A final word; References; Index.