How does the epsilon conversion work for NFA to DFA conversion? - dfa

Question

This is the NFA:
Here are two tables I made for DFA and then my attempt for the DFA equivalent:
(source: yimg.com)
The problem is that it doesn't account for the epsilon, because I don't know how to convert when there are epsilon arrows.

Answer 1

This is an epsilon-NFA, just transform the epsilon-NFA into an equivalent NFA without the epsilon transitions. Make a table as you would do while transforming an NFA to a DFA, and instead of just checking where that state goes with an input, first, check the where it can go with epsilon transitions then with the input then again with the epsilon transition (this is called the epsilon-closure ). This way you will have sets of states you will reach with your inputs. The only thing you need to do is to mark any state that can reach a final state using only epsilon transition as final states as well. Then you can build an NFA without epsilon transitions and then you can use your knowledge to transform it into a DFA. Just an example; In your table, you show B has an empty set for the input 0 but it can actually take an epsilon transition to C and take a 0 there so it is actually not an empty set.

Answer 2

To account for the epsilon transitions you can do any number of epsilon transitions before and after your read the next symbol. So you don't only consider where you can go when you read the symbol 0 (as an example) but also consider where you can go when you do epsilon transitions before and after, like ε* 0 ε*.
This means that when you start at the state {A} and read the symbol 0 you can go to the following states:
A --0--> B
A --0--> B --ε--> C
A --0--> C
A --ε--> B --ε--> C --0--> C
And for reading the symbol 1 you can go to the following states:
A --1--> A
A --1--> A --ε--> B
A --1--> A --ε--> B --ε--> C
A --ε--> B --1--> B
A --ε--> B --1--> B --ε--> C
A --ε--> B --ε--> C --1--> C
So in your resulting DFA, the transition will look as follow:
+-------+---------+---------+
| state | 0 | 1 |
+-------+---------+---------+
| {A} | {B,C} | {A,B,C} |
+-------+---------+---------+