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 $\bf x$ has a closed (resp., palindromic, privileged, rich, trapezoidal, balanced) factor of length $n$ is itself 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 the prefix of length $n$ of a $k$-automatic sequence is not $k$-synchronized.
Highlights
We prove that the property of being closed is expressible in first-order logic for automatic sequences
A wide variety of different kinds of words have been studied in the combinatorics on words literature, including the six flavors of the title: closed, palindromic, rich, privileged, trapezoidal, and balanced words
In this paper we show that, for k-automatic sequences x, the property of a factor belonging to each class is expressible in first-order logic; more precisely, in the theory Th(N, +, n → x[n])
Summary
A wide variety of different kinds of words have been studied in the combinatorics on words literature, including the six flavors of the title: closed, palindromic, rich, privileged, trapezoidal, and balanced words. In this paper we show that, for k-automatic sequences x (and some analogs, such as the so-called “Fibonacci-automatic” sequences [19]), the property of a factor belonging to each class is expressible in first-order logic; more precisely, in the theory Th(N, +, n → x[n]). We did this for unbordered factors [22]. For some of the properties, these expressions are surprisingly complicated
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