# Determining if language is context free

Is {xayb : x,y in {a,b}* and |x|=|y|} a context free language? My natural instinct would be to say that the answer is no, but can someone show me how to prove this?

-

It is context free; $|x| = |y|$ is not a very stringent requirement.
too leniently indeed! Two things to keep in mind: 1. Note that you are assuming that $|su|$ is just one character in the first portion of the expression. It may be multiple characters, and it may cross regions. 2. Although you correctly state that $\{a,b\}^{m+1} a \{a,b\}^m b$ is not in $L$, I'm not convinced you thought that through. I say it's not in $L$ because every word in $L$ is of even length. You're probably looking at the form of the word, and saying it doesn't match that given. The same word can be broken down in many ways. – Dustan Levenstein May 30 '12 at 12:14
And to show that it pumps: let $s$ be one character in the first region $\{a,b,\}^m$, and $u$ be a character in the third region. Then pumping in or out preserves $|x| = |y|$, leaving the required $a$ and $b$ interspersed untouched. – Dustan Levenstein May 30 '12 at 12:16