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Is $L=\{ xyx \mid x,y \in \{a,b\}^* \text {and } |x| \ge 1 \}$ context-free?

 

If yes, please explain how we can write grammar or create a PDA for it. If not a CFL, then prove it through pumping lemma.

I have tried to apply the pumping lemma with $w = a^nb^naba^nb^n$ as the word in $L$, but without success.

Is $L=\{ xyx \mid x,y \in \{a,b\}^* \text {and } |x| \ge 1 \}$ context-free?

If yes, please explain how we can write grammar or create a PDA for it. If not a CFL, then prove it through pumping lemma.

I have tried to apply the pumping lemma with $w = a^nb^naba^nb^n$ as the word in $L$, but without success.

Is $L=\{ aba \mid a,b \in \{0,1\}^* \text {and } |x| \ge 1 \}$ context-free?

If yes, please explain how we can write grammar or create a PDA for it. If not a CFL, then prove it through pumping lemma.

I have tried to apply the pumping lemma with $w = a^nb^naba^nb^n$ as the word in $L$, but without success.

Is $L=\{ xyx \mid x,y \in \{a,b\}^* \text {and } |x| \ge 1 \}$ context-free?

If yes, please explain how we can write grammar or create a PDA for it. If not a CFL, then prove it through pumping lemma.

I have tried to apply the pumping lemma with $w = a^nb^naba^nb^n$ as the word in $L$, but without success.