Related papers: On prefix palindromic length of automatic words
We illustrate a general technique for enumerating factors of k-automatic sequences by proving a conjecture on the number f(n) of unbordered factors of the Thue-Morse sequence. We show that f(n) <= n for n >= 4 and that f(n) = n infinitely…
The length $a(n)$ of the longest common subsequence of the $n$'th Thue-Morse word and its bitwise complement is studied. An open problem suggested by Jean Berstel in 2006 is to find a formula for $a(n)$. In this paper we prove new lower…
Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The…
This paper studies the complexity of languages of finite words using automata theory. To go beyond the class of regular languages, we consider infinite automata and the notion of state complexity defined by Karp. Motivated by the seminal…
The computational complexity of the isomorphism problem for regular trees, regular linear orders, and regular words is analyzed. A tree is regular if it is isomorphic to the prefix order on a regular language. In case regular languages are…
We revisit the topic of power-free morphisms, focusing on the properties of the class of complementary morphisms. Such morphisms are defined over a $2$-letter alphabet, and map the letters 0 and 1 to complementary words. We prove that every…
A generalized lexicographic order on words is a lexicographic order where the total order of the alphabet depends on the position of the comparison. A generalized Lyndon word is a finite word which is strictly smallest among its class of…
We study a standard operator on classes of languages: unambiguous polynomial closure. We prove that for every class C of regular languages satisfying mild properties, the membership problem for its unambiguous polynomial closure UPol(C)…
We present a new class of binary words: the prefix normal words. They are defined by the property that for any given length $k$, no factor of length $k$ has more $a$'s than the prefix of the same length. These words arise in the context of…
We consider two natural problems about nondeterministic finite automata. First, given such an automaton M of n states, and a length l, does M accept a word of length l? We show that the classic problem of triangle-free graph recognition…
A double occurrence word $w$ over a finite alphabet $\Sigma$ is a word in which each alphabet letter appears exactly twice. Such words arise naturally in the study of topology, graph theory, and combinatorics. Recently, double occurrence…
A word is said to be \emph{bordered} if it contains a non-empty proper prefix that is also a suffix. We can naturally extend this definition to pairs of non-empty words. A pair of words $(u,v)$ is said to be \emph{mutually bordered} if…
A universal word (u-word) for $d$-dimensional permutations of length $n$ is a 2-dimensional word with $d-1$ rows, any size $n$ window of which is order-isomorphic to exactly one permutation of length $n$, and all permutations of length $n$…
We prove that, paying a polynomial increase in size only, every unrestricted two-way nondeterministic finite automaton (2NFA) can be complemented by a 1-limited automaton (1-LA), a nondeterministic extension of 2NFAs still characterizing…
We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general…
Operator precedence languages (OPL) enjoy the local parsability property, which essentially means that a code fragment enclosed within a pair of markers -- playing the role of parentheses -- can be compiled with no knowledge of its external…
By replacing the letters to polynomials in F_2[t], an infinite word, over a finite alphabet, can be seen as the sequence of partial quotients of a continued fraction in F_2((1/t)). Here is described a family of such infinite words,…
We show that for every $n \geq 1$ and over any finite alphabet, there is a word whose circular factors of length $n$ have a one-to-one correspondence with the set of primitive words. In particular, we prove that such a word can be obtained…
For $\alpha\geq 1$, an $\alpha$-gapped repeat in a word $w$ is a factor $uvu$ of $w$ such that $|uv|\leq \alpha |u|$; the two factors $u$ in such a repeat are called arms, while the factor $v$ is called gap. Such a repeat is called maximal…
A gapped repeat is a factor of the form $uvu$ where $u$ and $v$ are nonempty words. The period of the gapped repeat is defined as $|u|+|v|$. The gapped repeat is maximal if it cannot be extended to the left or to the right by at least one…