---
title: "Cyclic language"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Cyclic_language"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:32:26.009151+00:00"
instance: "kb-cron"
---

In computer science, more particularly in formal language theory, a cyclic language is a set of strings that is closed with respect to repetition, root, and cyclic shift.


== Definition ==
If A is a set of symbols, and A* is the set of all strings built from symbols in A, then a string set L ⊆ A* is called a formal language over the alphabet A.
The language L is called cyclic if

∀w∈A*. ∀n>0. w ∈ L ⇔ wn ∈ L, and
∀v,w∈A*. vw ∈ L ⇔ wv ∈ L,
where wn denotes the n-fold repetition of the string w, and vw denotes the concatenation of the strings v and w.


== Examples ==
For example, using the alphabet A = {a, b }, the language

is cyclic, but not regular.
However, L is context-free, since M = { an1bn1 an2bn2 ... ank bnk : ni ≥ 0 } is, and context-free languages are closed under circular shift; L is obtained as circular shift of M.


== References ==