Question 25

The language is $L = \{w_1 a w_2 \mid w_1, w_2 \in \{a, b\}^*, |w_1| = 2, |w_2| \ge 3\}$.
This requires:

1. Exactly 2 arbitrary characters of $w_1$ (positions 1 and 2).
2. The 3rd character must be '$a$'.
3. Followed by at least 3 arbitrary characters of $w_2$ (positions 4, 5, 6 and beyond).

Construct the DFA states based on progress through the required string structure:

• State $q_0$: Start state (0 symbols read).
• State $q_1$: 1st symbol read from $\{a, b\}$.
• State $q_2$: 2nd symbol read from $\{a, b\}$.
• From $q_2$:
  • On '$a$': Successfully reads the mandatory '$a$' at position 3; transitions to $q_3$.
  • On '$b$': The 3rd symbol is '$b$', which violates the definition since $|w_1|$ must be exactly 2; transitions to dead/trap state $q_{\text{trap}}$.
• State $q_3$: 3 symbols read (including '$a$' at position 3). Needs at least 3 symbols to accept.
• State $q_4$: 4 symbols read (1 symbol of $w_2$).
• State $q_5$: 5 symbols read (2 symbols of $w_2$).
• State $q_6$: 6 or more symbols read (at least 3 symbols of $w_2$). This is the accepting state; self-loops on $\{a, b\}$.
• State $q_{\text{trap}}$: Dead state; self-loops on $\{a, b\}$.

Total number of states in this minimal DFA is: