Consider the two transition systems below that both model a beverage

vending machine.

The left transition system models a beverage machine that after insertion of a coin

nondeterministically chooses to either provide soda or beer. The right one, however, has two

selection buttons $($one for each beverage$),$ and after insertion of a coin, nondeterministically blocks

one of the buttons. In either case, the user has no control over the beverage obtained$$the choice

of beverage is under full control of the vending machine. Let AP $=\{$pay$,$ soda, beer$\},$

a$.$ Prove that the two systems are trace equivalence.

b$.$ Give an example of a finite execution fragment of these transition systems where soda

appears two times and beer appear one time.

c$.$ Give an example of a finite execution fragment of these transition systems where each

proposition appears at least three times.

d$.$ Give an example of a trace of these transition systems where each proposition appears at least two times Problem $1(30$ points$)$: Consider the two transition systems below that both model a beverage

vending machine.

The left transition system models a beverage machine that after insertion of a coin

nondeterministically chooses to either provide soda or beer. The right one, however, has two

selection buttons $($one for each beverage$),$ and after insertion of a coin, nondeterministically blocks

one of the buttons. In either case, the user has no control over the beverage obtained$-$the choice

of beverage is under full control of the vending machine. Let pay, soda, beer