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Complexity of Infinite Words

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Machines, Computations, and Universality (MCU 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15270))

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Abstract

Complexity of infinite words is a widely studied field in combinatorics on words. A classical notion of a complexity of an infinite word is defined as a function counting, for each n, the number of its distinct factors (or blocks of consecutive letters) of length n. In their 1938 seminal paper on symbolic dynamics, Morse and Hedlund gave a relation between factor complexity and periodicity in infinite words; namely, they proved that each aperiodic infinite word w has factor complexity at least \(n+1\) for each length n. They further showed that an infinite word w has factor complexity \(n+1\) for each length n if and only if w is binary, aperiodic and balanced, i.e., w is a Sturmian word. In this paper, we consider various modifications of the notion of a complexity of infinite words and generalizations of Morse and Hedlund theorem.

The study was supported by Russian Science Foundation, project 23-11-00133. Conference travel expenses have been covered by Projet ’NOW’ of RISE Academy, Université Côte d’Azur.

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Authors and Affiliations

  1. Saint Petersburg State University, 7–9 Universitetskaya emb., 199034, Saint Petersburg, Russia

    Svetlana Puzynina

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  1. Svetlana Puzynina
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Correspondence to Svetlana Puzynina .

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  1. Université Côte d'Azur, Sophia Antipolis, France

    Enrico Formenti

  2. University of Orléans, Orléans, France

    Jérôme Durand-Lose

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Puzynina, S. (2025). Complexity of Infinite Words. In: Formenti, E., Durand-Lose, J. (eds) Machines, Computations, and Universality. MCU 2024. Lecture Notes in Computer Science, vol 15270. Springer, Cham. https://doi.org/10.1007/978-3-031-81202-6_1

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  • DOI: https://doi.org/10.1007/978-3-031-81202-6_1

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