Ins-Robust Primitive Words

Abstract: Let Q be the set of primitive words over a finite alphabet with at least two symbols. We characterize a class of primitive words, Q_I, referred to as ins-robust primitive words, which remain primitive on insertion of any letter from the alphabet and present some properties that characterizes words in the set Q_I. It is shown that the language Q_I is dense. We prove that the language of primitive words that are not ins-robust is not context-free. We also present a linear time algorithm to recognize ins-robust primitive words and give a lower bound on the number of n-length ins-robust primitive words.

• The paper characterizes ins-robust primitive words ($Q_I$), defining them as primitive words that remain primitive after inserting any letter from the alphabet.
• The authors develop a linear-time algorithm to recognize these words and establish properties, including their density and a lower bound on their number.
• The study demonstrates that the language of non-ins-robust primitive words is not context-free, showing the complexity and relationship of $Q_I$ to formal language classes.

The paper characterizes a subclass of primitive words referred to as ins-robust primitive words, denoted as $Q_I$, which maintain primitivity upon insertion of any letter from the underlying alphabet $V$. The study details several properties of $Q_I$, demonstrates its density, and provides a linear-time algorithm for recognizing these words. Furthermore, a lower bound on the number of $n$-length ins-robust primitive words is established, and it is proven that the language of primitive words that are not ins-robust is not context-free.

Here's a breakdown:

Basic Definitions and Properties:

• A word is defined as a sequence of symbols from a finite alphabet $V$.
• A word $w$ is primitive if it is not a power of any other word (i.e., $w = v^n$ implies $n = 1$ and $w = v$).
• An ins-robust primitive word $w$ of length $n$ remains primitive after inserting any letter $a \in V$ at any position $i \in \{0, 1, \ldots, n\}$.
• The language of ins-robust primitive words ($Q_I$) is a subset of the language of primitive words ($Q$).
• The paper references prior work showing that if $u_1 u_2 \neq a^n$, then at least one of the words among $u_1 u_2$ and $u_1 a u_2$ is primitive.

Characterization of Ins-Robust Primitive Words:

• A central theorem states that a primitive word $w$ is not ins-robust if and only if it can be expressed as $w = u^{r} u_1 u_2 u^{s}$, where $u = u_1 c u_2 \in Q$, $u_1, u_2 \in V^*$, $c \in V$, and $r, s \geq 0$ with $r + s \geq 1$.
• The set of non-ins-robust primitive words is denoted as $Q_{\overline{I}} = Q \setminus Q_I$.
• If $u, v \in Q$ and $u^m = u_1 u_2$ and $v = u_1 c u_2$ for some $c \in V$, then $u^m v^n \in Q_{\overline{I}}$ for $m, n \geq 2$.
• If $w \in Q_I$, then $rev(w) \in Q_I$, where $rev(w)$ is the reverse of $w$.
• The language $Q_I$ is reflective; that is, if $uv \in Q_I$, then $vu \in Q_I$.

Density of Ins-Robust Primitive Words:

• A language $L$ is dense if for every $w \in V^*$, there exist $x, y \in V^*$ such that $xwy \in L$.
• If $|w| = n$ and $wa^n \in Q_{\overline{I}}$ where $w \notin a^*$, then $wa^n = u^2 u_1 u_2$, where $u = u_1 b u_2$ for $b \neq a$.
• If $wa^n \in Q_{\overline{I}}$, then for $b \neq a$, $wb^n \in Q_I$.
• The language $Q_I$ is dense over the alphabet $V$.

Relation to Formal Languages:

• The paper investigates the relationship between $Q_I$ and traditional languages in the Chomsky hierarchy.
• The language $Q_I$ is shown to be non-regular. This is proven using a pumping lemma argument and leveraging the result that for any fixed integer $k$, there exists a positive integer $m$ such that the equation system $(k-j)x_j + j = m$, $j = 0, 1, 2, \ldots, k-1$ has a nontrivial solution with positive integers $x_1, x_2, \ldots, x_j > 1$.
• The language of non-ins-robust primitive words, $Q_{\overline{I}}$, is proven to be non-context-free for a binary alphabet. This is shown via contradiction using the pumping lemma for context-free languages.

Counting Ins-Robust Primitive Words:

• Let $Z(k) = V^k \setminus Q$ be the set of $n$-length non-primitive words.
• Let $m \in \mathbb{N}$ and $m = {m_1}^{r_1} {m_2}^{r_2} \ldots {m_t}^{r_t}$ be the factorization of $m$, where all $m_i, 1 \leq i \leq t$, are prime and $m_i \neq m_j$ for $i \neq j$, then the number of primitive words of length $m$ is equal to

  $$\begin{array}{rl} |V|^m - & \sum\limits_{1\le i\le t}|V|^{\frac{m}{m_i}} + \sum\limits_{1\le i\le j \le t}|V|^{\frac{m}{m_i m_j}} \ - & \sum\limits_{1\le i\le j \le k \le t}|V|^{\frac{m}{m_i m_j m_k}} + \cdots + (-1)^{t-1} |V|^{\frac{m}{m_1 m_2 \cdots m_t}} \end{array}$$

• An upper bound on the number of non-ins-robust primitive words of length $n \geq 2$ is found as:

  $$|Q_{\overline{I}(n)}| \leq (n+1) \cdot (|Z(n+1)| - |V|).$$

• The number of ins-robust-primitive words of length $n$, $Q_I(n)$, over an alphabet $V$ is equal to $|Q_n| - |Q_{\overline{I}(n)}|$.

Algorithm for Recognizing Ins-Robust Primitive Words:

• A linear-time algorithm is presented to determine if a given primitive word $w$ is ins-robust.
• The algorithm leverages the property that a word $u$ is non-ins-robust if and only if $uu$ contains at least one periodic word of length $|u|$ with period $p$ such that $p$ divides of length $|u| + 1$ and $p \leq |u|$.
• The algorithm utilizes an existing algorithm for finding maximal repetitions in linear time.

Open Problems:

The paper concludes by posing questions for future research:

1. Is $Q_I^i$ for $i \geq 2$ regular?
2. Is the language of ins-robust primitive words $Q_I$ a deterministic context-free language?