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Additive Word Complexity and Walnut

Authors Pierre Popoli , Jeffrey Shallit , Manon Stipulanti

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Author Details

Pierre Popoli
  • Department of Mathematics, University of Liège, Belgium
Jeffrey Shallit
  • School of Computer Science, University of Waterloo, Canada
Manon Stipulanti
  • Department of Mathematics, University of Liège, Belgium

Funding

  • Popoli, Pierre: supported by ULiège’s Special Funds for Research, IPD-STEMA Program.
  • Shallit, Jeffrey: supported by NSERC grants 2018-04118 and 2024-03725.
  • Stipulanti, Manon: FNRS Research Associate supported by the Research grant 1.C.104.24F.

Acknowledgements

We thank the reviewers for providing valuable remarks leading to a better version of the paper. We also thank Matthieu Rosenfeld and Markus Whiteland for helpful discussions, and Eric Rowland for implementing useful Mathematica code.

Cite As Get BibTex

Pierre Popoli, Jeffrey Shallit, and Manon Stipulanti. Additive Word Complexity and Walnut. In 44th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 323, pp. 32:1-32:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.FSTTCS.2024.32

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics on words
Keywords
  • Combinatorics on words
  • Abelian complexity
  • Additive complexity
  • Automatic sequences
  • Walnut software

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