In this question, you are going to model a system with state machines. When you draw a state

machine, please include a legend of inputs, outputs, and any additional variables you use in the

top left corner, following the labeling conventions from the textbook.

a$)$ Draw a deterministic FSM $($call it PWM$)$ that has one pure input $($tick$)$ and two pure

outputs $($up and down$).$ PWM outputs up for $5$ ticks and then outputs down for $3$ ticks, and

repeats. This FSM should NOT be an extended state machine.

$($The output of such a FSM could be used to control pulse width modulation, where an up

signal corresponds to a high level and a down signal corresponds to a low level$)$

b$)$ For long sequences of $\text{'}$up$\text{'}$ and 'down' outputs, the FSM in part $($a$)$ quickly becomes

unwieldy. Redesign the FSM from part $($a$)$ as an EXTENDED state machine.

c$)$ The PWM state machine in part $($b$)$ expects a clock input 'tick'. Suppose we want to

model a crystal oscillator as a timed automaton.

a$.$ Draw a hybrid state machine CLOCK that generates a pure output signal 'tick' every $1$

millisecond.

b$.$ Suppose you manufacture your real$-$time clock device, but the crystal resonator is

faulty and resonates at half of its expected frequency. How would you update your

hybrid state machine CLOCK to reflect this change? Draw the updated state machine.

What is the new period of the clock?

d$)$ Suppose you wanted to compose state machines PWM and CLOCK such that FSM

CLOCK generates the 'tick' input for FSM PWM$.$ What kind of composition would you use,

and why?

e$)$ Draw a nondeterministic FSM that can be used to model a button that is pressed at a

random time and then released at a random time. This FSM has no inputs but has two pure

outputs $($pressed and released$).$