1. **Problem Statement:** We need to design a Deterministic Finite Automaton (DFA) for an automatic door lock system with states Locked (L), Unlocked (U), Alarm (A), and later Temporary Lock (T). 2. **Initial DFA Design (i):** - States: $\{L, U, A\}$ - Alphabet: $\{P, I, R\}$ where $P$ = correct password, $I$ = incorrect password, $R$ = manual reset - Transitions: - From $L$, input $P$ transitions to $U$ - From $U$, inactivity (modeled as a special input or timeout) transitions to $L$ - From $U$, three consecutive $I$s transition to $A$ - From $A$, input $R$ transitions to $L$ 3. **Checking string acceptance (ii):** String: 'PW-1 PW-1 PW-1 R' interpreted as $I I I R$ (three incorrect passwords then reset) - Start at $L$ - Input $I$: no transition defined from $L$ on $I$, so remain in $L$ (assuming no change) - Input $I$: same as above - Input $I$: same as above - Input $R$: no transition from $L$ on $R$ Since no transitions from $L$ on $I$ or $R$, the string is **not accepted** by the initial DFA. 4. **Extended DFA with Temporary Lock (iii):** - Add state $T$ (Temporary Lock) - New transitions: - From $L$, two consecutive $I$s lead to $T$ - From $T$, input $P$ leads to $U$ 5. **Explanation of changes:** - The system now tracks consecutive incorrect attempts from $L$. - After two $I$s, it moves to $T$ instead of staying in $L$. - From $T$, a correct password $P$ unlocks the door. 6. **Summary:** - Initial DFA states: $L, U, A$ - Extended DFA states: $L, U, A, T$ - The string $I I I R$ is not accepted in the initial DFA.