Problem $2.($Asynchronous Processes$)[25$ points$]$ For this problem, we ex$-$

pect you to explain your answers $($for example by displaying execution traces of

the process$).$ Consider the following asynchronous process that has the following

internal tasks:

nat x $=0$; y $=0$; z $=0$;T$1$ : $($y $=2)-->$ x :$=($x$+1)$ mod $3$; y:$=0$; z:$=0$

T$2$ : true $-->$ y :$=($y$+1)$ mod $3$

T$4$ : $($z $=1)-->$ x :$=2$;

$($a$)[5$ points$]$ Can there be an execution where $x+y=4?$ If yes, what is it$?$ If

no$,$ why not?

$($b$)[5$ points$]$ Can there be an execution where $x+y+z=6?$ If yes, what is

it$?$ If no$,$ why not?

$($c$)[5$ points$]$ How many distinct states are reachable from initial state

$($d$)[10$ points$]$ We will now redesign the asynchronous process such that it is

impossible to reach a state where x $+$ y $+$ z $=5,$ subject to:

$1.$ You can change the guard of just one task.

$2.$ You cannot change anything else $($updates$,$ other guards, initialization, etc.$)$

$3.$ Your new process must have executions reaching a state where x$+$y$+$z $=3.$