Related papers: Growing words in the free group on two generators
For finitely generated subgroups $H$ of a free group $F_m$ of finite rank $m$, we study the language $L_H$ of reduced words that represent $H$ which is a regular language. Using the (extended) core of Schreier graph of $H$, we construct the…
We study the equality problem for infinite words obtained by iterating morphisms. In particular, we give a practical algorithm to decide whether or not two words generated by primitive morphisms are equal.
A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $\textit{zigzag word}$. A…
In this work, we explore the following question: If two words in a finitely generated free group have identical images as word maps on every finite group, must they be endomorphic to each other? In this regard, we introduce weak profinite…
We investigate properties of evolving linguistic networks defined by the word-adjacency relation. Such networks belong to the category of networks with accelerated growth but their shortest path length appears to reveal the network size…
This work is a contribution to the study of set of the representations of integers in a rational base number system. This prefix-closed subset of the free monoid is naturally represented as a highly non regular tree whose nodes are the…
Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices and let $\mathcal S$ be a generating set of $M_n(\mathbb{F})$ as an $\mathbb{F}$-algebra. The length of a finite generating set $\mathcal S$ of $M_n(\mathbb{F})$ is the smallest…
We consider a new family of factorial languages whose subword complexity grows as $\Theta(n^{\alpha})$, where $\alpha$ is the root of some transcendent equation. Analytical methods and in particular, a corollary of the Wiener-Pitt theorem,…
We investigate the number of sets of words that can be formed from a finite alphabet, counted by the total length of the words in the set. An explicit expression for the counting sequence is derived from the generating function, and…
We prove that the width of any word in a simply connected Chevalley group of rank at least 2 over the ring that is a localisation of the ring of integers in a number field is bounded by a constant that depends only on the root system and on…
Given a finite word $w$ over a finite alphabet $V$, consider the graph with vertex set $V$ and with an edge between two elements of $V$ if and only if the two elements alternate in the word $w$. Such a graph is said to be word-representable…
I describe a class of groups acting on rooted trees. The original claim was that all have intermediate word growth between polynomial and exponential. The argument constructs a functional equation on the growth formal power series, and…
We study numerically and analytically the average length of reduced (primitive) words in so-called locally free and braid groups. We consider the situations when the letters in the initial words are drawn either without or with…
We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional…
We study language generation in the limit under bounded memory. In this task, a learner observes examples from an unknown target language one at a time and must eventually output only new valid examples. Prior work assumes access to the…
In this paper we investigate the word problem of the free Burnside semigroup satisfying x^2=x^3 and having two generators. Elements of this semigroup are classes of equivalent words. A natural way to solve the word problem is to select a…
Let $A \cong k\langle X \rangle / I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a…
Given a (finite or infinite) subset $X$ of the free monoid $A^*$ over a finite alphabet $A$, the rank of $X$ is the minimal cardinality of a set $F$ such that $X \subseteq F^*$. We say that a submonoid $M$ generated by $k$ elements of $A^*$…
Zipf's law of abbreviation, namely the tendency of more frequent words to be shorter, has been viewed as a manifestation of compression, i.e. the minimization of the length of forms -- a universal principle of natural communication.…
The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects non-trivially every non-trivial subgroup. The main result of this paper is that…