What is the the pumping length for the regular expression (0+0001)((1111)*+(00)*)

Question

I have this assignment question to find the pumping length of a regular language (L). The regular expression for the L is given as

$(0+0001)((1111)^*+(00)^*)$

1. What is the length of the longest string that cannot be pumped?
2. What is the length of the shortest string that can be pumped, I think this will come naturally when we find the $p$ (the pumping length).

The pumping length of a regular language $L$ is the minimal $p$ such that every word $w \in L$ of length at least $p$ can be split as $w = xyz$ such that (i) $|xy| \leq p$, (ii) $y \neq \epsilon$, (iii) $xy^iz \in L$ for every $i \geq 0$.

As per the answer https://cs.stackexchange.com/a/83727/33673, can we chose $y$ (the middle term in $xy^iz$ to be $1111$ in this case?

Answer 1

1. It will be convenient to distribute the concatenation to get the equivalent RE $$0(1111)^\ast + 0(00)^\ast + 0001(1111)^\ast + 0001(00)^\ast.$$ Now note that any such string can be writen as $s = s_i(s_r)^\ast$ where $s_i$ denotes the initial part (i.e. $0$ or $0001$) and $s_r$ denotes the repeated part (i.e. $1111$ or $00$). We see that $s$ can be pumped if and only if $s_r$ is non-empty and hence, the longest non-pumpable string is $0001$ and the answer is $4$. Indeed any string of length $5$ or more must have a repeatable part as defined above.
2. The two shortest strings in the language are $0$ and $000$. We see that $0$ clearly cannot be pumped (as $\varepsilon$ is not in the language) but $000$ can be pumped by repeating the last two zeroes arbitrarily often (and importantly we can also delete them to get $0$). Hence the answer is $3$.