Automatic Abelian Complexities of Parikh-Collinear Fixed Points

TL;DR

This paper proves that the abelian complexity function of fixed points of Parikh-collinear morphisms, including erasing ones, is automatic and provides a method to construct the automaton generating this function.

Contribution

It generalizes previous results by showing automaticity of abelian complexity for all Parikh-collinear morphisms, including erasing cases, with an effective automaton construction.

Findings

Abelian complexity of fixed points is automatic for all Parikh-collinear morphisms.

An effective procedure to generate the automaton for abelian complexity functions.

Extension of previous results to include erasing morphisms.

Abstract

Parikh-collinear morphisms have the property that all the Parikh vectors of the images of letters are collinear, i.e., the associated adjacency matrix has rank 1. In the conference DLT-WORDS 2023 we showed that fixed points of Parikh-collinear morphisms are automatic. We also showed that the abelian complexity function of a binary fixed point of such a morphism is automatic under some assumptions. In this note, we fully generalize the latter result. Namely, we show that the abelian complexity function of a fixed point of an arbitrary, possibly erasing, Parikh-collinear morphism is automatic. Furthermore, a deterministic finite automaton with output generating this abelian complexity function is provided by an effective procedure. To that end, we discuss the constant of recognizability of a morphism and the related cutting set.