Question $4$:

A discrete logic controller for an oven operates as follows: If the oven on switch is activated, the oven door is closed, and the temperature is below a desired value, the controller turns the heater on$.$ If the heater is on$,$ or when the temperature is above the desired value and the door is closed, the controller turns the fan on$.$ If the oven light switch is on or the door is open, the controller turns the oven light on$.$ The oven has an oven door limit switch whose status is denoted by DLS$,$ where $\backslash (\backslash$mathrm$\{$DLS$\}=0\backslash )$ means the door is open and $\backslash (\backslash$mathrm$\{$DLS$\}=1\backslash )$ means the door is closed. The oven also has a temperature limit switch whose status is denoted by TLS$,$ where TLS$=0$ when the temperature is below the desired value and TLS$=1$ when the temperature is at or above the desired value. The status of the heater, fan, light, oven on switch, and light switch is denoted by H$,$ F$,$ L$,$ S$,$ and LS$,$ respectively.

$(1)$ Define the inputs, outputs, and states of the controller.

$(2)$ Generate a state diagram for each state variable of the controller.

$(3)$ Using your result in part $(2),$ generate Boolean logic equations for the control action. Express the equations in the simplest possible form.