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.

翻译：我们推广了序列中熟悉的周期性概念，提出了一种新的伪周期性，并证明了其基本性质。我们重新审视了Shevelev在2012年论文中的结果，以更简单统一的方式重新证明了他的结论，并完整回答了他先前未解决的一个问题。我们考虑了寻找具有特定伪周期且临界指数尽可能小的单词。最后，我们考虑了判断有限单词是否为给定大小伪周期的问题，并证明该问题是NP完全的。