A New Approach to the Paperfolding Sequences

Goc, Daniel; Mousavi, Hamoon; Schaeffer, Luke; Shallit, Jeffrey

doi:10.1007/978-3-319-20028-6_4

Cited by 5 publications

(5 citation statements)

References 10 publications

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“…The paperfolding sequences are an uncountable family of sequences originally introduced by Davis and Knuth (1970) and later studied by Dekking et al (1982). The first-order theory of the paperfolding sequences was proved decidable in Goč et al (2015). Every infinite paperfolding sequence is specified by an infinite sequence f of unfolding instructions.…”

Section: The Paperfolding Sequencesmentioning

confidence: 99%

Pseudoperiodic Words and a Question of Shevelev

Meleshko,

Ochem,

Shallit

et al. 2023

Discrete Mathematics &Amp; Theoretical Computer Science

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We generalize the familiar notion of periodicity in sequences to a new kind of pseudoperiodicity, and we prove some basic results about it. We revisit the results of a 2012 paper of Shevelev and reprove his results in a simpler and more unified manner, and provide a complete answer to one of his previously unresolved questions. We consider finding words with specific pseudoperiod and having the smallest possible critical exponent. Finally, we consider the problem of determining whether a finite word is pseudoperiodic of a given size, and show that it is NP-complete.

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Section: The Paperfolding Sequencesmentioning

confidence: 99%

Pseudoperiodic Words and a Question of Shevelev

Meleshko,

Ochem,

Shallit

et al. 2023

Discrete Mathematics &Amp; Theoretical Computer Science

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“…The first-order theory of the paperfolding sequences was proved decidable in [7]. We can use Walnut to determine the pseudoperiods of any specific paperfolding sequence, or the pseudoperiods common to all paperfolding sequences.…”

Section: The Paperfolding Sequencesmentioning

confidence: 99%

“…As an example, here is the Walnut code verifying the results for (i, n) = (4, 5): and both commands return TRUE. Now lcm (5,7,9,11,13,14,15,16) = 720720, and the residue classes above cover 614614 of the possible residues (mod 720720). So we have covered 614614/720720 .…”

Section: Binary Words With Pseudoperiods Of the Form (1 A)mentioning

confidence: 99%

Pseudoperiodic Words and a Question of Shevelev

Meleshko¹,

Ochem²,

Shallit³

et al. 2022

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“…Recently decidability results have been proved for a number of interesting infinite classes of infinite sequences. For the paperfolding sequences, see [15]. For a class of Toeplitz words, see [13].…”

Section: Introductionmentioning

confidence: 99%

The first-order theory of binary overlap-free words is decidable

Schaeffer¹,

Shallit

2023

Can. J. Math.-J. Can. Math.

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We show that the first-order logical theory of the binary overlap-free words (and, more generally, the $\alpha $ -free words for rational $\alpha $ , $2 < \alpha \leq 7/3$ ), is decidable. As a consequence, many results previously obtained about this class through tedious case-based proofs can now be proved “automatically,” using a decision procedure, and new claims can be proved or disproved simply by restating them as logical formulas.

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A New Approach to the Paperfolding Sequences

Cited by 5 publications

References 10 publications

Pseudoperiodic Words and a Question of Shevelev

Pseudoperiodic Words and a Question of Shevelev

Pseudoperiodic Words and a Question of Shevelev

The first-order theory of binary overlap-free words is decidable

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