Minimum pumping length of a regular language

Question

Consider the language

L = { a3n + 5 | n ≥ 0 }

What is the minimum pumping length of L ?

Answer 1

By the minimum pumping length for a regular language L , I understand the smallest p such that every string u ∈ L of length at least p can be written u = xyz where |xy| ≤ p |xy| ≤ p , y ≠ λ and xy i z ∈ L for all i ≥ 0 . In your case, every string a 3n + 5 ∈ L with n > 0 can be written:

a3n + 5 = a5(a3)ia3

where i ≥ 0 . This decomposition satisfies the above conditions, so the minimum pumping length of L is 8 . Note that n ≥ 0 does not work because the string a 5 cannot be pumped. Note also that the minimal DFA for L has 8 states, although in general the minimum pumping length for a regular language can be less than the number of states in its minimal DFA.