A DFA recognizing my name How can I know if my DFA is implemented correctly?
For example, I need to build a DFA, and then minimize it which will recognize my name.
Language which describe my name is:
L = {pustai, marius}

I came with this solution:

&gt;(0)--p--&gt;(1)--u--&gt;(2)--s--&gt;(3)--t--&gt;(4)--a--&gt;(5)--i--&gt;(6)(goto state 0)
  |
 (7)--m--&gt;(8)--a--&gt;(9)--r--&gt;(10)--i--&gt;(11)--u--&gt;(12)--s--&gt;((13)) (final state)


How can I know if this is implemented correctly?
And about minimization of this automate I think is impossible.
Write down all accepting paths, that is paths from the starting state to final states. These are all the words the given automaton can accept; if the set is the desired one, q.e.d.

This is only possible and a rigorous proof for (co-)finite languages. Once you have cycles you have to apply induction. See here for an in-depth explanation on how to do this for grammars; it's similar for automata.