Decision Algorithms for Fibonacci-Automatic Words, with Applications to Pattern Avoidance

Abstract

We implement a decision procedure for answering questions about a class of infinite words that might be called (for lack of a better name) "Fibonacci-automatic". This class includes, for example, the famous Fibonacci word f = 01001010..., the fixed point of the morphism 0 -> 01 and 1 -> 0. We then recover many results about the Fibonacci word from the literature (and improve some of them), such as assertions about the occurrences in f of squares, cubes, palindromes, and so forth. As an application of our method we prove a new result: there exists an aperiodic infinite binary word avoiding the pattern x x x^R. This is the first avoidability result concerning a nonuniform morphism proven purely mechanically.

inserted new section 9 on abelian properties

References in corpus (1)

• Pattern Avoidability with Involution

Cited by in corpus (6)

• Ostrowski numeration systems, addition and finite automata
• Abelian-Square-Rich Words
• A note on the avoidability of binary patterns with variables and reversals
• The structure of palindromes in the Fibonacci sequence and some applications
• The Number of Fractional Powers in the Fibonacci Word
• The Numbers of Distinct Squares and Cubes in the Tribonacci Sequence