Pseudoperiodic Words and a Question of Shevelev

TL;DR

This paper introduces a new concept of pseudoperiodicity in sequences, revisits and simplifies Shevelev's earlier results, and explores computational complexity issues related to pseudoperiodic words.

Contribution

It generalizes periodicity to pseudoperiodicity, provides simplified proofs of Shevelev's results, and resolves an open question while analyzing the complexity of recognizing pseudoperiodic words.

Findings

Reproved Shevelev's results in a simpler manner

Solved an open problem regarding pseudoperiodicity

Proved recognizing pseudoperiodic words is NP-complete

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.