I am learning about finite automata for the first time. I am having trouble understanding the purpose of ε-transitions in an NFA, which seem to be crucial to counting the number of states in an NFA and therefore an equivalent DFA.
Here is an example question that confuses me: What is the minimum number of states a DFA recognizing the language of a(bc)*d can have?
To answer this question, I first drew this NFA (dashed line indicates acceptance):
Because I think the above NFA is already a DFA, I thought the answer was "5".
However, the correct NFA and equivalent DFA look like this:
Which means the answer is "4". I understand why these are correct. But, I have some questions:
1) Is my original drawing actually an NFA? If not, why?
2) If the original drawing is an NFA, does it describe the language a(bc)*d? If not, why?
3) If the original drawing is an NFA that describes the language a(bc)*d, is it also a DFA? If not, why?
4) If the original drawing is an NFA and DFA that describes the language a(bc)*d, why should I have known to draw the NFA with ε-transitions instead?