To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure firstname.lastname@example.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
In this chapter we will examine about 80 different fundamental properties of automatic sequences, and show how each one can be encoded by first-order logical formulas. We then use Walnut to re-derive new proofs of known results, or prove new results, concerning some famous automatic sequences. You can use these examples to learn what Walnut is capable of, but also as a ‘catalogue’ of first-order statements of fundamental properties of sequences.
In this chapter we give yet another fundamental way to think about automatic sequences, based on first-order logic. This revolutionary approach is originally due to Büuchi, with elaborations and additions by Bruyére, Hansel, Michaux, and Villemaire, and their ideas form the basis for this book. A good reference for the material in this section is the wonderful survey paper by these last four authors .
Walnut is free software originally designed and written in Java by Hamoon Mousavi , and recently modified by Aseem Raj Baranwal, Laindon C. Burnett, Kai Hsiang Yang, and Anatoly Zavyalov. This book is based on the most recent version of Walnut, called Walnut 3.7, which is available for free download at
In this chapter we introduce the main subject of the book, which is automatic sequences. Roughly speaking, an automatic sequence is a sequence over a finite alphabet whose nth term can be computed by a finite automaton reading a representation for n in a regular numeration system (Section 6.4).
Although the automatic sequences form a large and interesting class, one drawback is that they need to take their values in a finite set. But many interesting sequences, such as (s2(n))n.0 (counting the sum of the bits in the base-2 representation of n), take their values in N (or Z, or any semiring). We would like to find a generalization that allows this.
Up to now our logical formulas have allowed us to state formulas concerning a single automatic sequence, or perhaps two at the same time (as in Section 8.9.3). In some cases, however, it’s possible to use our approach to prove results about infinitely many sequences (even uncountably many sequences) at once!
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.