Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

If language L* (Kleene Star) is regular, does it imply that L is also regular?

share|cite|improve this question
up vote 7 down vote accepted

No. Pick a nonregular language L over an alphabet A that contains every letter of A as one-letter words. (for example, A = {0,1}, L = {w such that $|w|_0 - |w|_1 = \pm 1$}).

Then L* = A* so L* is regular, but L is not.

share|cite|improve this answer
Generalizing, you can even pick $L$ non-recursive. – Yuval Filmus Feb 27 '11 at 20:49

A nice exercise is to show that, for any $L \subseteq \{0\}^*$, $L^{*}$ is regular.

(Picking $SQR = \{0^{n^2}\ |\ n \in \mathbb{N}\}$ provides a counterexample to your statement. That $SQR$ is not regular, can be proved using the Pumping Lemma.)


This follows from the Frobenius problem for two coins (which was discoverd by J.J Sylvester).

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.