Cyclic language
# Cyclic language

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 ⇔ $w^n$ ∈ L, and
* $∀v,w∈A^*.$ vw ∈ L ⇔ wv ∈ L,

where $w^n$ denotes the n-fold repetition of the string w, and vw denotes the concatenation of the strings v and $w.^{:Def.1}$

## Examples

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

| L = | { | | $a^pb^{n1}$ | $a^{n2}b^{n2}$ | ... | $a^{nk}b^{nk}$ | $a^q$ | : | $n_i$ ≥ 0 and p+q = $n_1$ } |
| - | - | - | - | - | - | - | - | - | - |
| ∪ | { | $b^p$ | $a^{n1}b^{n1}$ | $a^{n2}b^{n2}$ | ... | $a^{nk}$ $b^q$ | | : | $n_i$ ≥ 0 and p+q = $n_k$ } |

is cyclic, but not $regular.^{:Exm.2}$ However, L is context-free, since M = { $a^{n1}b^{n1}$ $a^{n2}b^{n2}$ ... $a^{nk}$ $b^{nk}$ : $n_i$ ≥ 0 } is, and context-free languages are closed under circular shift; L is obtained as circular shift of M.

