What is the minimum pumping length of $(01)^*$
The solutions says 1, but can someone explain why that is? I understand this language accepts the empty string, but the minimum pumping length cannot be $0$. But, why is it 1? A general explanation of how to calculate the minimum pumping length would be much appreciated as well.
Thanks.
Look at the minimal DFA for $(01)^*$: it has three states(one garbage or dead state), so any non-empty word of the language must cause it to repeat a state and therefore can be pumped. The minimum length of a non-empty word of the language is $2$, so the minimum pumping length is at most $2$. Note, though, that the language contains no word of length $1$, so if $w$ is in the language, and $|w|\ge 1$, then automatically $|w|\ge 2$, and $w$ can be pumped. Thus, $1$ is also a pumping length for the language.