Theory of computation : Create a finite automaton $($DFA or NFA$)$ with four states $($Idle$,$ Lid Up$,$ Lid Down, Flush$)$ to model the behaviour of an automated toilet. These states are influenced by the following events:

In the Idle state, the presence of a detected person triggers the lifting of the toilet lid, which will remain open as long as a person is present. The lid will close upon the person's departure.

Once the lid is closed, if no waste is detected, the toilet will return to the Idle state. If waste is detected, the toilet will initiate a flush and transition back to the Idle state once the waste is gone.

The accepting state is Idle, indicating that the toilet is clean and ready for subsequent use. The events should be represented using the following symbols:

n: No Presence Detected

p: Presence Detected

w: Waste

c: Clean