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 …

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 …

Combinatorics on words aims at finding deep connections between properties of
sequences. The resulting theoretical findings are often used in the design of efficient …

A closed word (aka periodic-like word or complete first return) is a word whose longest
border does not have internal occurrences, or, equivalently, whose longest repeated prefix …

A word w is said to be closed if it has a proper factor x which occurs exactly twice in w, as a
prefix and as a suffix of w. Based on the concept of Ziv–Lempel factorization, we define the …

A string is closed if it has length 1 or has a nonempty border without internal occurrences. In
this paper we introduce the definition of a maximal closed substring (MCS), which is an …

A word is closed if it contains a proper factor that occurs both as a prefix and as a suffix but
does not have internal occurrences, otherwise it is open. We deal with the sequence of open …

A balanced word is one in which any two factors of the same length contain the same
number of each letter of the alphabet up to one. Finite binary balanced words are called …

We say a finite word $ x $ is a palindromic periodicity if there exist two palindromes $ p $
and $ s $ such that $| x|\geq| ps| $ and $ x $ is a prefix of the word $(ps)^\omega …

Given a finite non-empty set A, let A+ denote the free semigroup generated by A consisting
of all finite words u 1 u 2⋯ un with ui∈ A. A word u∈ A+ is said to be closed if either u∈ A or …