Automatic Sequences in Negative Bases and Proofs of Some Conjectures of Shevelev

J. Shallit, S. Shan, Kai Hsiang Yang
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引用次数: 3

Abstract

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.
负基中的自动序列及舍夫列夫若干猜想的证明
我们讨论了负碱基在自动序列中的使用。最近,定理证明器Walnut被扩展到允许使用底数(-k)来表示变量,从而允许在n而不是n上进行量化。这使我们能够证明关于双无穷自动序列的结果。我们首先解释核桃负碱基背后的理论。接下来,我们用这个新版本的Walnut给出一个非常简单的证明,证明了舍夫列夫定理的一个强化版本。我们用我们的想法来解决2017年舍夫列夫的两个开放性问题。我们还对2000年一个涉及双无限二进制词的闭的结果进行了修正。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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