Additive word complexity and Walnut

TL;DR

This paper explores additive complexity in infinite sequences, comparing it with abelian complexity, and uses the Walnut software to analyze properties of automatic sequences, expanding theoretical understanding and posing open questions.

Contribution

It introduces additive complexity to the study of infinite sequences, analyzes its properties using logic and Walnut, and compares it with abelian complexity, including conjectures on regularity.

Findings

Additive complexity differs from abelian complexity in infinite sequences.

Walnut software can decide properties related to additive complexity.

The paper proposes open questions and conjectures for future research.

Abstract

In combinatorics on words, a classical topic of study is the number of specific patterns appearing in infinite sequences. For instance, many works have been dedicated to studying the so-called factor complexity of infinite sequences, which gives the number of different factors (contiguous subblocks of their symbols), as well as abelian complexity, which counts factors up to a permutation of letters. In this paper, we consider the relatively unexplored concept of additive complexity, which counts the number of factors up to additive equivalence. We say that two words are additively equivalent if they have the same length and the total weight of their letters is equal. Our contribution is to expand the general knowledge of additive complexity from a theoretical point of view and consider various famous examples. We show a particular case of an analog of the long-standing conjecture on the regularity of the abelian complexity of an automatic sequence. In particular, we use the formalism of logic, and the software Walnut, to decide related properties of automatic sequences. We compare the behaviors of additive and abelian complexities, and we also consider the notion of abelian and additive powers. Along the way, we present some open questions and conjectures for future work.