Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences

TL;DR

This paper demonstrates that various combinatorial properties of factors in automatic sequences, like being palindromic or balanced, are first-order definable, making their characteristic functions automatic, with specific results for well-known sequences.

Contribution

It introduces first-order logic definability for properties of automatic sequences and computes their characteristic functions for several famous sequences.

Findings

Characteristic functions are automatic for properties like palindromic and balanced factors.

Privileged words require a new characterization for their property.

Counting palindromic factors is not k-synchronized in automatic sequences.

Abstract

We prove that the property of being closed (resp., palindromic, rich, privileged trapezoidal, balanced) is expressible in first-order logic for automatic (and some related) sequences. It therefore follows that the characteristic function of those n for which an automatic sequence x has a closed (resp., palindromic, privileged, rich, trape- zoidal, balanced) factor of length n is automatic. For privileged words this requires a new characterization of the privileged property. We compute the corresponding characteristic functions for various famous sequences, such as the Thue-Morse sequence, the Rudin-Shapiro sequence, the ordinary paperfolding sequence, the period-doubling sequence, and the Fibonacci sequence. Finally, we also show that the function counting the total number of palindromic factors in a prefix of length n of a k-automatic sequence is not k-synchronized.