The minimum possible number of states of a deterministic finite automaton that accepts the regular language $L = \{w_1 a w_2 \mid w_1, w_2 \in \{a, b\}^*, |w_1| = 2, |w_2| \ge 3\}$ is ________.

The given language requires strings that:

1. Have $|w_1| = 2$, so the $3^{\text{rd}}$ character must be specifically 'a'.

2. Have $|w_2| \ge 3$, so the total length of the string must be at least $2 + 1 + 3 = 6$.

Let us construct the minimal DFA states tracking the relevant prefix length and matching condition:

• $q_0$: Start state (length $0$)

• $q_1$: Length $1$

• $q_2$: Length $2$

• From $q_2$:

  • On input 'a', transition to $q_3$ (length $3$, $3^{\text{rd}}$ symbol is 'a').

  • On input 'b', transition to a dead/trap state $q_{\text{trap}}$ (length $3$, $3^{\text{rd}}$ symbol is not 'a', string can never be accepted).

• $q_3$: Length $3$, valid $3^{\text{rd}}$ symbol

• $q_4$: Length $4$

• $q_5$: Length $5$

• $q_6$: Length $\ge 6$ (Final/accepting state, with self-loops on $a, b$)

• $q_{\text{trap}}$: Dead state (with self-loops on $a, b$)

Counting all states:

Total number of states = 8.