: In this problem, we will design a synchronous reactive component to set

the cruising speed of a vehicle. This component corresponds to the SetSpeed component in the

cruise$-$control system CruiseController. The SetSpeed component takes the following inputs:

$$ event cruise: This event models the user turning the cruise control on or off.

$$ nat speed: This input models the speed input from the vehicle $($corresponds to the current

speed of the vehicle$).$

$$ event inc: This input models the user requesting an increase in the cruising speed.

$$ event dec: This input models the user requesting a decrease in the cruising speed.

It has one output:

$$ event$($nat$)$ cruiseSpeed: this output, if presents, contains the current cruising speed.

The component maintains two state variables:

$$ an enumerated type variable called mode $=\{$on$,$ off$\}$ that keeps track of whether the

controller is turned on or off.

$$ a current desired $($nat$)$ speed s$.$

There are two constants called minSpeed and maxSpeed that we will use within the component.

The operation of the component is as follows. The component updates the state variable on

according to the following rule: every time the event cruise occurs, the variable mode is set to on$.$

Then the component outputs the cruising speed using the cruiseSpeed output variable. If the current

speed s is within the legal range minSpeed and maxSpeed, then the cruiseSpeed variable is set to

s; if not, it is set to the closest legal value $($i$.$e$.$ minSpeed or maxSpeed$).$ Also, when mode is on$,$ if

a dec event is received, then the cruising speed is decremented $($restricting the lowest speed to

minSpeed$),$ and when the inc event is received, the cruising speed is incremented $($restricting the

highest speed to maxSpeed$).$ If the mode variable is set to off, all input events except cruise are

ignored, and there is no output.

a$.$ Show the synchronous reactive component that implements this functionality.

b$.$ Suppose we want to add another input control for the driver, pause, with the following desired

behavior. When the cruise controller is on$,$ if the driver presses pause, then the controller is

temporarily turned off. In the resulting paused state, the output cruiseSpeed should be absent$,$

and the events inc and dec should be ignored. Pressing pause again in this paused state should

resume the operation of the cruise controller, restoring the desired speed on pausing. Pressing

cruise in the paused state should switch the system off, and when the controller is off, pressing

pause should have no effect. Redesign the component SetSpeed with this additional input event

pause to capture the above specification. Hint: the component SetSpeed now maintains a state

variable mode that can be either on$,$ paused, or off, i$.$e$.,$ mode $=\{$on$,$ paused, off$\}.$