In this chapter we discuss words and infinite words (sequences) in more detail, giving complete definitions. A small amount of this material is repeated from the previous chapter.

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 [56].

Walnut is free software originally designed and written in Java by Hamoon Mousavi [278], 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 discuss a class of sequences that is ‘halfway between’ the automatic sequences and the regular sequences.1

Sequences abound in the mathematical literature! To name just a few, there are the Fibonacci numbers [376]

Additive number theory is the study of the additive properties of integers.1 For example, Lagrange proved (1770) that every natural number is the sum of four squares [188].

In this book we have presented and studied the logical approach to automatic sequences, a powerful tool for doing combinatorics on words, using the Walnut system.

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).

We will need finite automata, both with and without output. For more information about automata, see [203, 16, 346].

Let k 2 be an integer. The usual base-k representation uses the digits in to represent natural numbers. We recall the following familiar result

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!