Additive word complexity and Walnut

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.