Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular

Question

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular.

These are the things that I would use to start.

As $L_1L_2$ is regular, then the homomorphism $h(L_1L_2)$ is regular.
Let $h(L_1) = L_2$ and $h(L_2) = L_1$, then $h(L_1L_2) = L_2L_1$ is regular (we already know that) or $h(L_2) = \epsilon$ and we get $L_1$
By reflexing, $L_1L_2 = (L_2L_1)^{R}$, same result.

But i don't know how to, for example, intersect something that gives me $L_1$ in order to preserve closure and finally $L_1$ be regular.

Any help?

score 7 · Accepted Answer · answered Apr 27 '13 at 04:52

7

Here is a counterexample. Let $L_1$ be any language over some alphabet $\Sigma$ containing the empty string, and let $L_2 = \Sigma^*$. Then $L_2 = L_1L_2 = L_2L_1 = \Sigma^*$ are all regular, but $L_1$ need not be, in fact it could even be uncomputable!

answered Apr 27 '13 at 04:52

Yuval Filmus

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I agree. I never looked for counterexamples because the question was "positive" (the author never says prove or refute) Again, nice and simple counterexample. Thanks! – Alejandro Sazo Apr 27 '13 at 10:56

Alejandro Sazo · Answer 2 · 2013-04-29T18:36:57.910

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Another counterexample (special case of the first answer) is find just one non regular language in specific and another regular language where $L_1 \subseteq L_2$.

Let $L_1 = \{a^{n^2}, n \geq 0\}$ (it includes $\epsilon$) and $L_2 = \{a^n, n \geq 0\}$. It's concatenation $L_1L_2 = L_2L_1 = L_2$ since the subset property established at the beggining. Then $L_1$ is not necessarily regular.

edited Apr 29 '13 at 18:36

answered Apr 29 '13 at 17:27

Alejandro Sazo

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This is a special case of my example with $\Sigma = {a}$. – Yuval Filmus Apr 29 '13 at 18:33
Yes, I will edit my answer to clarify that! – Alejandro Sazo Apr 29 '13 at 18:35

Prove that $L_1$ is regular if $L_2$, $L_1L_2$, $L_2L_1$ are regular

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