Reconstructing Words from Right-Bounded-Block Words

关于在存在不要关心词情况下确定性ω-自动机的最小化与学习

On Minimization and Learning of Deterministic $ω$-Automata in the Presence of Don't Care Words

研究在存在不要关心词情况下确定性ω-自动机的最小化问题，证明了在任意不要关心词集合下确定性奇偶自动机优先级可高效最小化，并指出对于具有信息性右同余的确定性奇偶自动机，最小化问题为NP难。

Lie complexity of words

Given a finite alphabet $Σ$ and a right-infinite word $\bf w$ over $Σ$, we define the Lie complexity function $L_{\bf w}:\mathbb{N}\to \mathbb{N}$, whose value at $n$ is the number of conjugacy classes (under cyclic shift) of length-$n$ factors $x$ of $\bf w$ with the property that every element of the conjugacy class appears in $\bf w$. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity, and as a result we show that words of linear factor complexity have at most finitely many primitive factors $y$ with the property that $y^n$ is again a facto…

Automatic Sequences of Rank Two

Given a right-infinite word $\bf x$ over a finite alphabet $A$, the rank of $\bf x$ is the size of the smallest set $S$ of words over $A$ such that $\bf x$ can be realized as an infinite concatenation of words in $S$. We show that the property of having rank two is decidable for the class of $k$-automatic words for each integer $k\ge 2$.

Topological invariants for words of linear factor complexity

Given a finite alphabet $Σ$ and a right-infinite word $w$ over the alphabet $Σ$, we construct a topological space ${\rm Rec}(w)$ consisting of all right-infinite recurrent words whose factors are all factors of $w$, where we work up to an equivalence in which two words are equivalent if they have the exact same set of factors (finite contiguous subwords). We show that ${\rm Rec}(w)$ can be endowed with a natural topology and we show that if $w$ is word of linear factor complexity then ${\rm Rec}(w)$ is a finite topological space. In addition, we note that there are examples which show that if …

On the Structure of Bispecial Sturmian Words

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 Sturmian words. A Sturmian word is bispecial if it can be extended to the left and to the right with both letters remaining a Sturmian word. There is a deep relation between bispecial Sturmian words and Christoffel words, that are the digital approximations of Euclidean segments in the plane. In 1997, J. Berstel and A. de Luca proved that \emph{palindromic} bispecial Sturmian words are precisely the maximal internal factors…

Finite test sets for morphisms which are square-free on some of Thue's square-free ternary words

Let $S$ be one of $\{aba,bcb\}$ and $\{aba, aca\}$, and let $w$ be an infinite square-free word over $Σ=\{a,b,c\}$ with no factor in $S$. Suppose that $f:Σ\rightarrow T^*$ is a non-erasing morphism. Word $f(w)$ is square-free if and only if $f$ is square-free on factors of $w$ of length 8 or less.

A model for a Lindenmayer reconstruction algorithm

Given an input string s and a specific Lindenmayer system (the so-called Fibonacci grammar), we define an automaton which is capable of (i) determining whether s belongs to the set of strings that the Fibonacci grammar can generate (in other words, if s corresponds to a generation of the grammar) and, if so, (ii) reconstructing the previous generation.

Constructing Words with High Distinct Square Densities

Fraenkel and Simpson showed that the number of distinct squares in a word of length n is bounded from above by 2n, since at most two distinct squares have their rightmost, or last, occurrence begin at each position. Improvements by Ilie to $2n-Θ(\log n)$ and by Deza et al. to 11n/6 rely on the study of combinatorics of FS-double-squares, when the maximum number of two last occurrences of squares begin. In this paper, we first study how to maximize runs of FS-double-squares in the prefix of a word. We show that for a given positive integer m, the minimum length of a word beginning with m FS-dou…

Left Lyndon tree construction

We extend the left-to-right Lyndon factorisation of a word to the left Lyndon tree construction of a Lyndon word. It yields an algorithm to sort the prefixes of a Lyndon word according to the infinite ordering defined by Dolce et al. (2019). A straightforward variant computes the left Lyndon forest of a word. All algorithms run in linear time on a general alphabet, that is, in the letter-comparison model.

Almost optimal searching of maximal subrepetitions in a word

For $0<δ<1$ a $δ$-subrepetition in a word is a factor which exponent is less than~2 but is not less than $1+δ$ (the exponent of the factor is the ratio of the factor length to its minimal period). The $δ$-subrepetition is maximal if it cannot be extended to the left or to the right by at least one letter with preserving its minimal period. In the paper we propose an algorithm for searching all maximal $δ$-subrepetitions in a word of length~$n$ in $O(\frac{n}δ\log\frac{1}δ)$ time (the lower bound for this time is $Ω(\frac{n}δ)$).