Abelian Repetition Threshold Revisited

Two words are Abelian equivalent if they have the same multiset of letters (in other terms, if they are anagrams of each other); Abelian repetition is a pair of Abelian equivalent factors in a word. The study of Abelian repetitions originated from the question of Erdös [6]: does there exist an infinite finitary word having no consecutive pair of Abelian equivalent factors? The factors of the form $u\bar{u}$, where $u$ and $\bar{u}$ are Abelian equivalent, are now called Abelian squares. In modern terms, Erdös’s question looks like “are Abelian squares avoidable over some finite alphabet?” This question was answered in the affirmative by Evdokimov [7]; the smallest possible alphabet has cardinality 4, as was proved by Keränen [8]. In a similar way, Abelian $k$th powers are defined for arbitrary $k\geq 2$. Dekking [5] constructed infinite ternary words without Abelian cubes and infinite binary words without Abelian 4th powers. The results by Dekking and Keränen form an Abelian analog of the seminal result by Thue [14]: there exist an infinite ternary word containing no squares (factors of the form $uu$) and an infinite binary word containing no cubes (factors of the form $uuu$).

Integral powers of words were later generalized to rational (fractional) powers: given a word $u$ of length $n$, take a length-$m$ prefix $v$ of the infinite word $uuu\cdots$; then $v$ is the $(\frac{m}{n})$th power of $u$ ($m>n$ is assumed). Usually, $v$ is referred to as a $(\frac{m}{n})$-power. A word is said to be $\alpha$-power-free if it contains no $(\frac{m}{n})$-powers with $\frac{m}{n}\geq\alpha$. Fractional powers gave rise to the notion of repetition threshold which is the function ${\sf RT}(k)=\inf\{\alpha:\text{there exists an infinite $k$-ary $\alpha$-power-free word}\}$. The value ${\sf RT}(2)=2$ is known since Thue [15]. Dejean [4] showed that ${\sf RT}(3)=7/4$ and conjectured the remaining values ${\sf RT}(4)=7/5$ (proved by Pansiot [10]) and ${\sf RT}(k)=\frac{k}{k-1}$ for $k\geq 5$ (proved by efforts of many authors [9, 1, 3, 12]). An extension of the notions of fractional power and repetition threshold to the Abelian case was proposed by Samsonov and Shur [13]. Integral Abelian powers can be generalized to fractional ones in several ways; however, for the case $m\leq 2n$, one definition of an Abelian $(\frac{m}{n})$-power is preferable due to its symmetric nature. According to this definition, a word $vu\bar{v}$ is an Abelian $(\frac{m}{n})$th power of the word $vu$ if $|vu|=n$, $|vu\bar{v}|=m$, and $\bar{v}$ is Abelian equivalent to $v$ (in [13], such Abelian powers were called strong). Note that the reversal of an Abelian $(\frac{m}{n})$-power is also an Abelian $(\frac{m}{n})$-power. In this paper, we consider only strong Abelian powers; see Section 2 for the definition in the case $m>2n$. Given the definition of fractional Abelian powers, one naturally defines Abelian $\alpha$-power-free words and Abelian repetition threshold ${\sf ART}(k)=\inf\{\alpha:\text{there exists an infinite $k$-ary Abelian $\alpha$-power-free word}\}$. Cassaigne and Currie [2] showed that for any $\varepsilon>0$ their exists an Abelian $(1+\varepsilon)$-power-free word over a finite alphabet of size $2^{\mathrm{poly}(\varepsilon^{-1})}$. Surely, this bound is very loose but it proves that $\lim_{k\to\infty}{\sf ART}(k)=1$. In [13], the lower bounds ${\sf ART}(4)\geq 9/5$, ${\sf ART}(k)\geq\frac{k-2}{k-3}$ for $k\geq 5$ were proved and conjectured to be tight; in full, this conjecture is as follows.

Up to now, no exact values of ${\sf ART}(k)$ are known. One reason for the lack of progress in estimating ${\sf ART}(k)$ is the fact that the number of $k$-ary Abelian $\alpha$-power-free words can be finite but so huge that these words cannot be enumerated by exhaustive search.

In the present study, we approach Abelian $\alpha$-power-free words using randomized depth-first search. The language ${\sf AF}(k,\alpha)$ of all $k$-ary Abelian $\alpha$-power-free words is viewed as a prefix tree ${\cal T}_{k,\alpha}$: the elements of ${\sf AF}(k,\alpha)$ are nodes of the tree and $u$ is an ancestor of $v$ in the tree iff $u$ is a prefix of $w$. The search starts at the root (empty word) and is organized as follows. Reaching a node $u$ for the first time, we choose a random letter $a$, check ad hoc whether $ua\in{\sf AF}(k,\alpha)$ and descend to $ua$ if this node exists; visiting $u$ next times, we choose a random letter among the letters not chosen at $u$ before, and proceed the same way. If there is no choice, we return to the parent of $u$ (and thus will never reach $u$ in the future). We repeated the search multiple times and analysed the maximum level of a node reached in the tree and the change of level during the search. Based on this analysis, we state

As a first step in proving this conjecture, we prove ${\sf ART}(k)>\frac{k-2}{k-3}$ for $k=6,7,8,9,10$ by some exhaustive search (Theorem 4.1 in Section 4).

The paper is organized as follows. After preliminary Section 2, we describe the algorithms used in our study and the results obtained through experiments in Sections 3 and 4 respectively. Section 5 contains some final remarks and prospects for future studies.

We study finite words over finite alphabets, using standard notation $\Sigma$ for a (linearly ordered) alphabet, $\sigma$ for its size, $\Sigma^{*}$ for the set of all finite words over $\Sigma$, including the empty word $\lambda$. For a length-$n$ word $u\in\Sigma^{*}$ we write $u=u[1..n]$; the elements of the range $[1..n]$ are positions in $u$, the length of $u$ is denoted by $|u|$. A word $w$ is a factor of $u$ if $u=vwz$ for some (possibly empty) words $v$ and $z$; the condition $v=\lambda$ (resp., $z=\lambda$) means that $w$ is a prefix (resp., suffix) of $u$. Any factor $w$ of $u$ can be represented as $w=u[i..j]$ for some $i$ and $j$ ($j<i$ means $w=\lambda$). A factor $w$ of $u$ can have several such representations; we say that $su[i..j]$ specifies the occurrence of $w$ at position $i$.

A $k$-power of a word $u$ is the concatenation of $k$ copies of $u$, denoted by $u^{k}$. This notion can be extended to $\alpha$-powers for an arbitrary rational $\alpha>1$. The $\alpha$-power of $u$ is the word $u^{\alpha}=u\cdots uu^{\prime}$ such that $|u^{\alpha}|=\alpha|u|$ and $u^{\prime}$ is a prefix of $u$. A word is $\alpha$-free (resp., $\alpha^{+}\!$-free) if no one of its factors is a $\beta$-power with $\beta\geq\alpha$ (resp., $\beta>\alpha$).

The Parikh vector $\Psi(u)$ of a word $u\in\Sigma^{*}$ is an integer vector of length $\sigma$ whose coordinates are the numbers of occurrences of the letters from $\Sigma$ in $u$. Thus, the word $acabac$ over the alphabet $\Sigma=\{a<b<c<d\}$ has the Parikh vector $(3,1,2,0)$. Two words $u$ and $v$ are Abelian equivalent (denoted by $u\sim v$) if $\Psi(u)=\Psi(v)$. The reversal of a word $u=u[1..n]$ is the word $u^{R}=u[n]u[n{-}1]\cdots u[1]$. Clearly, $u\sim u^{R}$. A nonempty word $u$ is an Abelian $k$th power ($k$-A-power) if $u=w_{1}\cdots w_{k}$, where $w_{i}\sim w_{j}$ for all indices $i,j$. A 2-A-power is an Abelian square, and a 3-A-power is an Abelian cube. Thus, $k$-A-powers generalize $k$-powers by relaxing the equality of factors to their Abelian equivalence. However, there are many ways to generalize the notion of an $\alpha$-power to the Abelian case, and all of them have drawbacks. The reason is that $u\sim v$ does not imply $u[i..j]\sim v[i..j]$ for any pair of factors of $u$ and $v$. If $1<\alpha\leq 2$, we define an $\alpha$-A-power as a word $vuv^{\prime}$ such that $\frac{|vuv^{\prime}|}{|vu|}=\alpha$ and $v\sim v^{\prime}$. The advantage of this definition is that the reversal of an $\alpha$-A-power is an $\alpha$-A-power as well. For $\alpha>2$ the situation is worse: all natural definitions compatible with the definition of $k$-A-power are not symmetric with respect to reversals (see [13] for more details). So we give the definition which is compatible with the case $\alpha\leq 2$: an $\alpha$-A-power is a word $u_{1}\cdots u_{k}u^{\prime}$ such that $\frac{|u_{1}\cdots u_{k}u^{\prime}|}{|u_{1}|}=\alpha$, $k=\lfloor\alpha\rfloor$, $u_{1}\sim\cdots\sim u_{k}$, and $u^{\prime}$ is Abelian equivalent to a prefix of $u_{1}$. In [13], such words are called strong Abelian $\alpha$-powers. For a given $\alpha$, $\alpha$-A-free and $\alpha^{+}\!$-A-free words are defined in the same way as $\alpha$-free ($\alpha^{+}\!$-free) words. It is convenient to extend rational numbers with “numbers” of the form $\alpha^{+}$, postulating the equivalence of the inequalities $\beta>\alpha$ and $\beta\geq\alpha^{+}$ (resp., $\beta\leq\alpha$ and $\beta<\alpha^{+}$).

A language is any subset of $\Sigma^{*}$. The reversal $L^{R}$ of a language $L$ consists of the reversals of all words in $L$. The $\alpha$-A-free language over $\Sigma$ (where $\alpha$ belongs to extended rationals) consists of all $\alpha$-A-free words over $\Sigma$ and is denoted by ${\sf AF}(\sigma,\alpha)$. These languages are the main objects of our study aimed at finding the Abelian repetition threshold, which is the function ${\sf ART}(k)=\inf\{\alpha:{\sf AF}(k,\alpha)\text{ is infinite}\}$. The languages ${\sf AF}(\sigma,\alpha)$ are closed under permutations: if $\pi$ is a permutation of the alphabet, then the words $u$ and $\pi(u)$ are $\alpha$-A-free for exactly the same values of $\alpha$. This makes possible the enumeration of the words in languages ${\sf AF}(\sigma,\alpha)$ by considering only lexmin words: a word $u\in\Sigma^{*}$ is lexmin if $u<\pi(u)$ for any permutation $\pi$ of $\Sigma$.

Suppose that a language $L$ is factorial (i.e., closed under taking factors of its words); for example, all languages ${\sf AF}(\sigma,\alpha)$ are factorial. Then $L$ can be represented by its prefix tree ${\cal T}_{L}$, which is a rooted labeled tree whose nodes are elements of $L$ and edges have the form $u\xrightarrow{a}ua$, where $a$ is a letter. Thus $u$ is an ancestor of $v$ iff $u$ is a prefix of $v$. We study the languages ${\sf AF}(\sigma,\alpha)$ through different types of search in their prefix trees.

In this section we present the algorithms we develop for the use in experiments. First we describe the random depth-first search in the prefix tree ${\cal T}={\cal T}_{L}$ of an arbitrary factorial language $L$. Given a number $N$, the algorithm visits $N$ distinct nodes of ${\cal T}$ following the depth-first order and returns the maximum level of a visited node. The search can be easily augmented to return the word corresponding to the node of maximum level, or to log the sequence of levels of visited nodes. Algorithm 1 below describes one iteration of the search. In the algorithm, $u=u[1..n]$ is the word corresponding to the current node; ${\sf Set}[u]$ is the set of all letters $a$ such that the search has not tried the node $ua$ yet; ${\sf ml}$ is the maximum level reached so far; $\mathsf{count}$ is the number of visited nodes; ${\cal L}(u)$ is the predicate returning $\mathsf{true}$ if $u\in L$ and $\mathsf{false}$ otherwise. The lines 3 and 8 refer to the updates of data structures used to compute ${\cal L}(u)$. The search starts with $u=\lambda$, ${\sf ml}=0$, $\mathsf{count}=1$. A variant of this search algorithm was used in [11] to numerically estimate the entropy of some $\alpha$-free and $\alpha$-A-free languages.

The key to an efficient search is a fast algorithm computing the predicate ${\cal L}(u)$. The following fact is very useful: if $u^{\prime}$ is a proper prefix of $u$, then the node for $u^{\prime}$ exists and hence ${\cal L}(u^{\prime})=\mathsf{true}$. Below we present four algorithms checking, for a given $\alpha$, whether a word has a $\beta$-A-power with $\beta\geq\alpha$ as a suffix. The cases $\alpha<2$ and $\alpha>2$ are considered in Sections 3.1 and 3.2 respectively.