[Solved]: Is The Following Language Regular?

Problem Detail: 

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 $A(L_{1},L_{2})$:

1. is always a regular language
2. is always not a regular language
3. can sometimes be a regular language
4. cannot be context free

They say that the correct answer is 1.

Asked By : Robert777

Answered By : Yuval Filmus

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 $L_2$.

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Question Source : http://cs.stackexchange.com/questions/10147

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