The Logical Approach to Automatic Sequences : Exploring Combinatorics on Words with Walnut (London Mathematical Society Lecture Note Series)

Shallit, Jeffrey

Cambridge University Press（2022/09発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 374 p.
• 言語 ENG
• 商品コード 9781108745246
• DDC分類 511.6

Full 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.

Contents

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.