Related papers: The non-abelian squares are not context-free

Context-free languages are widely used to describe the syntax of programming languages and natural languages. Usually, we describe a context-free language mathematically with the help of context-free grammar (for generation) or pushdown…

We examine words w satisfying the following property: if x is a subword of w and |x| is at least k for some fixed k, then the reversal of x is not a subword of w.

For any context-free grammar, we build a transition diagram, that is, a finite directed graph with labeled arcs, which describes the work of the grammar. This approach is new, and it is different from previously known graph models. We…

We study the avoidability of long $k$-abelian-squares and $k$-abelian-cubes on binary and ternary alphabets. For $k=1$, these are M\"akel\"a's questions. We show that one cannot avoid abelian-cubes of abelian period at least $2$ in infinite…

We survey recent results on the topological complexity of context-free omega-languages which form the second level of the Chomsky hierarchy of languages of infinite words. In particular, we consider the Borel hierarchy and the Wadge…

We prove that the only non-Archimedean strictly convex spaces are the zero space and the one-dimensional linear space over $\, \mathbb{Z}/3\mathbb{Z}$, with any of its trivial norms.

We introduce new avoidability problems for words by considering equivalence relations, k-abelian equivalences, which lie properly in between equality and commutative equality, i.e. abelian equality. For two k-abelian equivalent words the…

Cross-bifix-free sets are sets of words such that no prefix of any word is a suffix of any other word. In this paper, we introduce a general constructive method for the sets of cross-bifix-free binary words of fixed length. It enables us to…

Context free languages allow one to express data with hierarchical structure, at the cost of losing some of the useful properties of languages recognized by finite automata on words. However, it is possible to restore some of these…

We discuss examples of non-commutative spaces over non-archimedean fields. Those include non-commutative and quantum affinoid algebras, quantized K3 surfaces and quantized locally analytic p-adic groups.

We examine deterministic and nondeterministic state complexities of regular operations on prefix-free languages. We strengthen several results by providing witness languages over smaller alphabets, usually as small as possible. We next…

A word is said to be bordered if it contains a nonempty proper prefix that is also a suffix. A pair of words $(u, v)$ is said to be mutually bordered if there exists a word that is a nonempty proper prefix of $u$ and suffix of $v$, and…

Let $D$ be a square-free integer. Under certain conditions on $D$, we characterize non-constant arithmetic progressions of squares over quadratic extensions of $\mathbb{Q}(\sqrt{D})$.

In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.

Iterated hairpin completion is an operation on formal languages that is inspired by the hairpin formation in DNA biochemistry. Iterated hairpin completion of a word (or more precisely a singleton language) is always a context-sensitive…

Let $D$ be a division ring with center $F$ and $N$ a subnormal subgroup of the multiplicative group $D^*$ of $D$. Assume that $N$ contains a non-abelian solvable subgroup. In this paper, we study the problem on the existence of non-abelian…

Motivated by a question of Gordon and Wilton, we consider the question of which collections of words are "virtually geometric". In particular, we prove that some words (e.g. bbaaccabc) are not virtually geometric.

We deal with a normal form for context-free grammars, called Dyck normal form. This normal form is a syntactical restriction of the Chomsky normal form, in which the two nonterminals occurring on the right-hand side of a rule are paired…

Parikh's Theorem says that the Parikh image of a context-free language is semilinear. We give a short proof of Parikh's Theorem using the formulation of Verma, Seidl, and Schwentick in terms of Presburger arithmetic. The proof relies on an…

A group is called square-like if it is universally equivalent to its direct square. It is known that the class of all square-like groups admits an explicit first order axiomatization but its theory is undecidable. We prove that the theory…