Determine if this language is regular

Problem Detail: 

Let $L = \{xyx \mid \text{ for some }x,y \in \{0,1\}^+\}$. Is this language regular?

So I was trying to construct a DFA, but I don't how to do this with this language.

Asked By : user678392

Answered By : Timot

You can indeed use the pumping lemma to answer this. Assume $L$ is regular, and let $p$ be its pumping length. Let $w = 0^p1 \cdot 1 \cdot 0^p1 \in L$. By the pumping lemma (well, a strong form of it), you can pump from the first $p$ letters, and obtain $w' = 0^{q}1\cdot 1 \cdot 0^p1$ with $q\neq p$, hence $w'\not\in L$.

@Gilles: This came after an even number of cups of coffee :-).

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Question Source : http://cs.stackexchange.com/questions/14159