# How to Find the minimum permanent which is pumping lemma for language context-free?

Suppose we have context-free language and we know the Context-Free-Grammar, how do i found the minimum permanent which is pumping lemma for language ?

for example let sat i have this Grammar: \begin{align} S&\rightarrow AA\mid B\\ A&\rightarrow 0A\mid A0∣1\\ B&\rightarrow 00B0\mid 1\\ \end{align} whats is the best way to find this Minimum permanent?

• what is a "minimum permanent"? – Hendrik Jan Jan 6 '17 at 16:59
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I presume by "permanent" you mean the integer of the Pumping Lemma, that is the integer $p>0$ such that for any string $s$ in the language with length $p$ or greater we can express $s=uvxyz$ with . . . [blah blah blah, I'm not going to write the statement here]. There are two other of the right-hand form, but because of the way your grammar is formed, any string of length greater than or equal to 3 generated by the grammar must be formed by extending one of these two. In other words, the integer of the pumping lemma can be 3 or greater. It can't be 1 or 2, since while the strings $1$ and $11$ are in the language, there are no repeated variables in their parse trees (meaning they can't be pumped). Finally, then, the integer you want is 3.
• @Joe. If we decompose 0010 with $u=\epsilon, v=00, x=1, y=0, z=\epsilon$ then pumping down yields the string 1, which is in the language. I don't see the problem here. – Rick Decker Jan 11 '17 at 17:37