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Mathematics

Open and closed complexity of infinite words

Profile image of Olga ParshinaOlga Parshina

2020, arXiv: Combinatorics

Abstract

In this paper we study the asymptotic behaviour of two relatively new complexity functions defined on infinite words and their relationship to periodicity. Given a factor $w$ of an infinite word $x=x_1x_2x_3\cdots$ with each $x_i$ belonging to a fixed finite set $\mathbb{A},$ we say $w$ is closed if either $w\in \mathbb{A}$ or if $w$ is a complete first return to some factor $v$ of $x.$ Otherwise $w$ is said to be open. We show that for an aperiodic word $x\in \mathbb{A}^\mathbb{N},$ the complexity functions $Cl_x$ (resp. $Op_x)$ that count the number of closed (resp. open) factors of $x$ of each given length are both unbounded. More precisely, we show that if $x$ is aperiodic then $\liminf_{n\in \mathbb{N}} Op_x(n)=+\infty$ and $\limsup_{n\in S} Cl_x(n)=+\infty $ for any syndetic subset $S$ of $\mathbb{N}.$ However, there exist aperiodic infinite words $x$ verifying $\liminf_{n\in \mathbb{N}}Cl_x(n)<+\infty.$ Keywords: word complexity, periodicity, return words.

FAQs

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What are closed and open complexity functions in symbolic dynamics?add

The study defines two complexity functions, Cl_x and Op_x, which count closed and open factors of infinite words, respectively. These functions offer insight into the periodicity and structure of infinite sequences.

How do open and closed complexities relate to periodicity in words?add

The findings suggest that if the closed complexity is bounded, the word is ultimately periodic. Conversely, unbounded complexity indicates aperiodicity, as demonstrated by established examples in the paper.

What distinct examples illustrate variations in complexity for aperiodic words?add

L. Schaeffer and J. Shallit's work shows that the regular paperfolding word has lim inf Cl_x(n) = 0, highlighting variance in complexity classes. Furthermore, it's noted that the Cantor word exhibits lim inf Cl_c(n) = 1 for specific n values.

Which newly introduced terms are essential for analyzing complexity functions?add

The paper introduces key terms such as 'first return words', which are critical to studying the behavior of symbolic dynamical systems. Additionally, concepts like 'syndetic sets' and 'Rauzy graphs' enhance the understanding of complexity structures.

What is the significance of the findings related to return words?add

Return words provide crucial insights into the classification of infinite words generated by substitutions, impacting diverse fields from combinatorics to dynamical systems. Their properties, as shown in past studies, elucidate connections between complexity and recurrence in symbolic dynamics.

Figures (4)
Proof. The situation is as illustrated on the Figure 1. Since wg is closed, ug cannot be a factor of ui. Hence 7+ |u2| — |u| > 0. Since wy is closed, uy cannot be a factor of ug. Hence i + |u| — |u2| > 0. The result follows. C Corollary 1. Let w , and w2 be as in Proposition 1. If there exists a path between w, and we in the Rauzy graph consisting of only closed factors, then |u| = |u2|. Thus, if there exists a path between w, and wz with n distinct open factors, ||u | — |ug|| <n.
Proof. The situation is as illustrated on the Figure 1. Since wg is closed, ug cannot be a factor of ui. Hence 7+ |u2| — |u| > 0. Since wy is closed, uy cannot be a factor of ug. Hence i + |u| — |u2| > 0. The result follows. C Corollary 1. Let w , and w2 be as in Proposition 1. If there exists a path between w, and we in the Rauzy graph consisting of only closed factors, then |u| = |u2|. Thus, if there exists a path between w, and wz with n distinct open factors, ||u | — |ug|| <n.
Figure 3: The sequence of open and closed factors in the Rauzy graph of order N. w = w}---wy of x (which exists since x is aperiodic). By Proposition 2, there exists 1 < k such that Wa = Wisi: WNaY1 ++ Yjs—1 and Wy) = wWyi1-+: wNbzZ +++ Z%_1 with a 4 b € A are both closed factors of x. See Figure 3: at each step before the rightmost one, either on top, bottom, or both paths, there must be an open factor, and each open factor can only appear once.
Figure 3: The sequence of open and closed factors in the Rauzy graph of order N. w = w}---wy of x (which exists since x is aperiodic). By Proposition 2, there exists 1 < k such that Wa = Wisi: WNaY1 ++ Yjs—1 and Wy) = wWyi1-+: wNbzZ +++ Z%_1 with a 4 b € A are both closed factors of x. See Figure 3: at each step before the rightmost one, either on top, bottom, or both paths, there must be an open factor, and each open factor can only appear once.
Figure 4: v’ overlaps itself with difference smaller than k. Let us denote the frontiers of wg and wy, by u and v respectively. For the illustration of the following reasoning see Figure 4. Applying Lemma 2, we get ||u|— |v|| < k. Since both u and v are longer than k and a ¥ b, they cannot be equal. This implies |u| 4 |v| since wa and wy have a long common prefix. Suppose, without loss of generality, that |u| < |v|. Lemma 3 gives m < |u|. Let u’ and v’ be prefixes of u and v such that u = u/ay,--- yj, and v = v'bz1 +++ z1. Then, |v'| > m—k and wu’ is a prefix and a suffix of v’. Hence v’ overlaps itself with a difference less than k, what contradicts Lemma 1.
Figure 4: v’ overlaps itself with difference smaller than k. Let us denote the frontiers of wg and wy, by u and v respectively. For the illustration of the following reasoning see Figure 4. Applying Lemma 2, we get ||u|— |v|| < k. Since both u and v are longer than k and a ¥ b, they cannot be equal. This implies |u| 4 |v| since wa and wy have a long common prefix. Suppose, without loss of generality, that |u| < |v|. Lemma 3 gives m < |u|. Let u’ and v’ be prefixes of u and v such that u = u/ay,--- yj, and v = v'bz1 +++ z1. Then, |v'| > m—k and wu’ is a prefix and a suffix of v’. Hence v’ overlaps itself with a difference less than k, what contradicts Lemma 1.
Figure 6: u = v would produce an overlap of wu.
Figure 6: u = v would produce an overlap of wu.

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