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Pseudoperiodic Words and a Question of Shevelev

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

TL;DR

This work extends periodicity to $k$-pseudoperiodicity and develops a Walnut-based, automatic-sequence framework to study pseudoperiodicity across multiple famous sequences, including the Thue-Morse word. It provides complete solutions to several of Shevelev's problems, characterizes pseudoperiod triples via large automata, and derives both binary and larger-alphabet results on critical exponents and morphic constructions. The paper also establishes the NP-completeness of deciding the existence of a bounded-size pseudoperiod, highlighting computational hardness in the problem space. Through detailed case studies of Mephisto Waltz, (ternary) Thue-Morse, period-doubling, Rudin–Shapiro, Tribonacci, and paperfolding sequences, the work connects combinatorics on words with automated reasoning and offers new conjectures and directions for future research.

Abstract

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.

Pseudoperiodic Words and a Question of Shevelev

TL;DR

This work extends periodicity to -pseudoperiodicity and develops a Walnut-based, automatic-sequence framework to study pseudoperiodicity across multiple famous sequences, including the Thue-Morse word. It provides complete solutions to several of Shevelev's problems, characterizes pseudoperiod triples via large automata, and derives both binary and larger-alphabet results on critical exponents and morphic constructions. The paper also establishes the NP-completeness of deciding the existence of a bounded-size pseudoperiod, highlighting computational hardness in the problem space. Through detailed case studies of Mephisto Waltz, (ternary) Thue-Morse, period-doubling, Rudin–Shapiro, Tribonacci, and paperfolding sequences, the work connects combinatorics on words with automated reasoning and offers new conjectures and directions for future research.

Abstract

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.
Paper Structure (18 sections, 29 theorems, 24 equations, 3 figures, 2 tables)

This paper contains 18 sections, 29 theorems, 24 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notation
  3. Goals of this paper
  4. Basic results
  5. Shevelev's problems
  6. Other sequences
  7. The Mephisto Waltz sequence
  8. The ternary Thue-Morse sequence
  9. The period-doubling sequence
  10. The Rudin-Shapiro sequence
  11. The Tribonacci sequence
  12. The paperfolding sequences
  13. Critical exponents
  14. Binary words with pseudoperiods of the form $(1,a)$
  15. Larger alphabets
  16. ...and 3 more sections

Key Result

Proposition 2

An infinite word $\bf s$ is pseudoperiodic if and only if there exists a bound $B< \infty$ such that two consecutive occurrences of the same letter in $\bf s$ are always separated by distance at most $B$.

Figures (3)

  • Figure 1: Pseudoperiods of size $2$ for the Fibonacci word.
  • Figure 2: Automaton recognizing all pseudoperiods of size $3$ for the vtm sequence.
  • Figure 3: Automaton accepting base-$2$ representations of pseudoperiod triples common to all paperfolding sequences.

Theorems & Definitions (61)

  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 51 more