Below is the Pumping lemma as stated in Automata and Computability by (Dexter C. Kozen)
Let A be a regular set. Then the following property holds of A:
There exist k≥0 such that for any string x,y,z with xyz \in A and |y|≥k, there exist strings u,v,w such that y=uvw,v\not=ε , and for all i≥0, the string xuv^iwx\in A
Here are my questions:
Let ∑=\{1\} A=\{1\}. Clearly A, is regular. Now, applying the pumping lemma, let x=1,y=ε,z=ε and so 1εε=1 \in A, and |y|=0 so k=0 but there is no way to split y such that v \not = ε. So, how is the pumping lemma valid for this regular set?
Also, we can think of k as being the number of states, but a DFA cannot have 0 states. So why is k not required to be strictly greater than 0.
Perhaps you can help my understanding by applying the pumping lemma above to the language \{1\} as well the language of the empty string. What do value does k have in these situations? What about x,y,z and u,v,w?