The Logical Approach to Automatic Sequences

Exploring Combinatorics on Words with Walnut

Jeffrey Shallit

• Publisher: Cambridge University Press
• Year: 2022
• Print ISBN:
  9781108745246
• eBook ISBN:
  9781108786973
• Topic: Computer Science
• Subtopic: Computer Science General
• Index: Computer Science

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.

Table of contents

• Cover
• Reviews
• Title page
• Copyright page
• Contents
• Preface
• Permissions
• 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