Proving a specific language is regular

Question

In my computability class we were given a practice final to go over and I'm really struggling with one of the questions on it.

Prove the following statement:

If $L_1$ is a regular language, then so is

$L_2 = \{ uv |$ $u$ is in $L_1$ or $v$ is in $L_1 \}$.

You can't use the pumping lemma for regular languages (I think), so how would you go about this? I'm inclined to believe that it's false because if $u$ is in $L_1$, what if $v$ is non-regular? Then it would be impossible to write a regular expression for it. The question is out of 5 marks though and that doesn't seem like enough of an answer for it.

Answer 1

You can use the pumping lemma to show that a language is not regular. Here, you're trying to prove that if $L_1$ is regular then $L_2$ is regular. The pumping lemma can't be used to prove this implication. It could be used to prove the contraposite (i.e. you might be able to prove that if $L_2$ is not regular, then $L_1$ is not regular by the pumping lemma). However, even assuming this can work (I haven't checked), it would be a lot more complicated than necessary for this result.

You write “what if $v$ is non-regular?” But this doesn't make sense: the concept of regularity applies to a language, not a word. For a given $u$, what language does $v$ have to belong to, in order for $uv$ to be in $L_2$?

If that's not enough of a hint, try reasoning with regular expressions. If $L_1$ is regular, then there is a regular expression that characterizes it. How can you use this regular expression to characterize $L_2$?

Answer 2

Write $L_2$ as $$\{uv \mid u \in L_1, v\in \Sigma^*\} \cup \{uv \mid u\in \Sigma^*, v\in L_1\}$$ (convince yourself it is OK to do so)

Then check about right/left quotient. With that, the proof is trivial.

EDIT: actually, quotient is not needed. Only concatenation.

Answer 3

There is another question about proving that languages are not regular. My personal favorite way to do it is, essentially, equivalent to the Myhill-Nerode Theorem.

For proving that a language is regular, you can either show it is expressible by a DFA, NFA, Regular Expression, etc. or you can use the fact that regular languages are closed under:

• Intersection (easiest to show with DFAs)
• Union (easy with regular expressions)
• Reversal (easy with NFAs)
• Taking prefixes, as in $L_3$ from @sdvvc's comment (easy with DFAs)
• Suffixes (can be obtained from reversal and prefixes)

Some combinations of these should deal with the language in the question.