Problem
Design a deterministic finite automata (dfa) that satisfies the following:
{ w | w has 'abab' as a substring}
Hence, w can be ε, abab, abababab, etc.
Attempt
This was my first trial. It creates strings with ab as a substring, but not necessarily abab.
This is my second trial. I, in short, don't know if I'm right.
I apologize for the major space in the drawings, hope they are legible.
Notes
I'd also greatly appreciate it if you could provide the quintuple, M = (Q, Σ, s, F, δ) for this dfa.
Thank you in advance!
I don't see why, $\epsilon$ is in $L$, where $L = \{ w \in \sum^* \mid \exists (u_0, u_1, u_2, u_3, u_4) \in \sum^{5*}, w = u_0au_1bu_2au_3bu_4\}$.
Hence for me, there is a problem with your $DFA$, because $q_0$ is a final state (so it accepts $\epsilon$).
A solution could be the following $NFA = (\{q_0, q_1, q_2, q_3, q_4\}, \{a, b\}, q_0, \{q_4\}, \delta)$ :
If your language is not $\{a, b\}$, but $L = \sum \cup \{a, b\}$ then the $DFA$ is the same, just replace the $(a,b)$ by $ \sum^*$.
If you want the $DFA$, just apply Glushkov algorithm on the $NFA$.