Robinson Thamburaj | Searcharxiv

arXiv2026

2-Head 2D Returning Finite Automata

We introduce and study a family of two-head finite automata called two head returning finite automata (2-HRFA) operating on rectangular arrays of picture languages, in which both heads move in opposite directions. We show that the class of picture languages accepted by 2-HRFA is incomparable with the class of languages generated by context-free matrix grammars (CFMG), while it forms a proper subset of the class of languages accepted by returning pushdown automata (RPDA). In addition, we define a constrained variant, both head stepping two head returning finite automata (B2-HRFA), in which both heads are required to move in a synchronized, stepwise fashion. We prove that the class of languages accepted by returning finite automata (RFA) is a proper subset of the class of languages accepted by B2-HRFA, which in turn is a proper subset of the class of languages accepted by 2-HRFA. Closure properties for both the families of languages accepted by 2-HRFA and B2-HRFA are also investigated.

cs.FL↗

arXiv2025

On a Generalization of the Christoffel Tree: Epichristoffel Trees

Sturmian words form a family of one-sided infinite words over a binary alphabet that are obtained as a discretization of a line with an irrational slope starting from the origin. A finite version of this class of words called Christoffel words has been extensively studied for their interesting properties. It is a class of words that has a geometric and an algebraic definition, making it an intriguing topic of study for many mathematicians. Recently, a generalization of Christoffel words for an alphabet with 3 letters or more, called epichristoffel words, using episturmian morphisms has been studied, and many of the properties of Christoffel words have been shown to carry over to epichristoffel words; however, many properties are not shared by them as well. In this paper, we introduce the notion of an epichristoffel tree, which proves to be a useful tool in determining a subclass of epichristoffel words that share an important property of Christoffel words, which is the ability to factorize an epichristoffel word as a product of smaller epichristoffel words. We also use the epichristoffel tree to present some interesting results that help to better understand epichristoffel words.

cs.FL↗