Pseudoperiodic Words and a Question of Shevelev

Meleshko, Joseph; Ochem, Pascal; Shallit, Jeffrey; Shan, Sonja Linghui

doi:10.48550/arxiv.2207.10171

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2023

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“…However, these additional problems do not deal with negative bases, so they are not the focus of this paper. We simply mention briefly that his Open Problem A, also mentioned in [22], has recently been completely solved in [15].…”

Section: Final Remarksmentioning

confidence: 99%

Automatic sequences in negative bases and proofs of some conjectures of shevelev

Shallit

Shan

Yang

2023

RAIRO-Theor. Inf. Appl.

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We discuss the use of negative bases in automatic sequences. Recently the theorem-prover Walnut has been extended to allow the use of base (—k) to express variables, thus permitting quantification over ℤ instead of ℕ. This enables us to prove results about two-sided (bi-infinite) automatic sequences. We first explain the theory behind negative bases in Walnut. Next, we use this new version of Walnut to give a very simple proof of a strengthened version of a theorem of Shevelev. We use our ideas to resolve two open problems of Shevelev from 2017. We also reprove a 2000 result of Shut involving bi-infinite binary words.

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Section: Final Remarksmentioning

confidence: 99%