Is The Following Language Regular?

Question

Let $L_{1}$ and $L_{2}$ be 2 languages over the same alphabet $\Sigma$.

$$A(L_1,L_2)=\{x\in \Sigma^*|\exists y,z\in L_2\text{ such that } yxz\in L_1\}$$

Assume that $L_{1}$ is regular and $L_{2}$ is context-free. The language $A(L_{1},L_{2})$:

is always a regular language
is always not a regular language
can sometimes be a regular language
cannot be context free

They say that the correct answer is 1.

check http://cs.stackexchange.com/questions/1547/closure-against-right-quotient-with-a-fixed-language — Ran G., Feb 28 '13 at 02:50
Wow, thanks ! I'm shocked to see the answer to that question. — Robert777, Feb 28 '13 at 06:45

score 2 · Accepted Answer · answered Mar 01 '13 at 03:38

2

Hint: Take a DFA for $L_1$. Check which states are reachable from the initial state via a word in $L_2$. Check from which states a final state can be reached via a word in $L_2$.

answered Mar 01 '13 at 03:38

Yuval Filmus

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Is The Following Language Regular?

1 Answers1