Multidimensional Generalized Automatic Sequences and Shape-symmetric Morphic Words

TL;DR

This paper extends the concept of automatic sequences to multidimensional infinite words, establishing a connection with shape-symmetric morphic words and characterizing S-automaticity in higher dimensions.

Contribution

It introduces a multidimensional generalization of automatic sequences and proves their equivalence to shape-symmetric morphic words under certain conditions.

Findings

Multidimensional S-automatic words are characterized by shape-symmetric morphic words.

The paper establishes a necessary and sufficient condition for multidimensional automaticity.

It links automata theory with shape-symmetric morphic structures in higher dimensions.

Abstract

An infinite word is S-automatic if, for all n>=0, its (n + 1)st letter is the output of a deterministic automaton fed with the representation of n in the considered numeration system S. In this extended abstract, we consider an analogous definition in a multidimensional setting and present the connection to the shape-symmetric infinite words introduced by Arnaud Maes. More precisely, for d>=2, we state that a multidimensional infinite word x : N^d \to \Sigma over a finite alphabet \Sigma is S-automatic for some abstract numeration system S built on a regular language containing the empty word if and only if x is the image by a coding of a shape-symmetric infinite word.