Consider the following deterministic finite automaton (DFA) defined over the alphabet, $\Sigma = \{a, b\}$. Identify which of the following language(s) is/are accepted by the given DFA.

Let the four states from left to right be denoted as $q_0$ (start state), $q_1$, $q_2$, and $q_3$ (accepting state).

Let us trace the meaning of each state based on the longest suffix of the input processed so far:

• $q_0$: Represents strings ending with an $a$ (not part of the suffix $ba$), or the empty string $\epsilon$.

  • On input $a$, stays at $q_0$.

  • On input $b$, transitions to $q_1$.

• $q_1$: Represents strings whose longest relevant suffix is $b$.

  • On input $b$, stays at $q_1$.

  • On input $a$, transitions to $q_2$ (suffix is now $ba$).

• $q_2$: Represents strings whose longest relevant suffix is $ba$.

  • On input $b$, transitions to $q_3$ (suffix is now $bab$).

  • On input $a$, transitions back to $q_0$ (suffix becomes $baa$, resetting pattern tracking to $a$).

• $q_3$ (Accepting state): Represents strings whose longest relevant suffix is $bab$.

  • On input $a$, suffix becomes $baba$, whose longest suffix matching a prefix of $bab$ is $ba$, so it transitions back to $q_2$.

  • On input $b$, suffix becomes $babb$, whose longest suffix matching a prefix of $bab$ is $b$, so it transitions back to $q_1$.

Since $q_3$ is the only accepting state, a string is accepted if and only if the automaton ends in state $q_3$ after processing the entire string. This occurs precisely when the string ends with the substring $bab$.

Therefore, the DFA accepts the set of all strings ending with the pattern $bab$ (Option C).