Calculate denotational semantics for the following nondeterministic programs.

a$.$ Let $S-=$ if $x>yx$:$=x-1x>yy$:$=y+1x+y=4x$:$=\frac{y}{x}x+y=4x$:$=$

$\frac{x}{y}$ fi$,$ and let $=\{x=3,y=1\}.$ Calculate $M(S,)$ and show your work.

b$.$ Let $W-=$ do $x>yx$:$=x-1x>yy$:$=y+1x+y=4x$:$=\frac{y}{x}x+y=4x$:$=$

$\frac{x}{y}$ od$,$ and let $=\{x=3,y=1\}.$ Calculate $M(W,)$ and show your work.

Let $b$ be an array of size $n1,$ and $bbb[n]=100nb{0}^{\text{'}}sJ-=dob[k_0]=1k_0$:$=k_0+1b[k_1]=0k_1$:$=k_1+1od{k}_0{k}_{1}J{k}_0{k}_{1}b[k_0]=0b[k_1]=1JM(S,)S|==\{p\}S\{q\}|==p|==\{p\}S\{q\}M(S,)|==q\{p\}S\{q\}{}_{\text{t}\text{o}\text{t}}\{p\}S\{q\}AA0i.$ Decide which number $(0or1)is$ the

majority $inb$ without counting their quantities.

Write a program named MAJORITY $in$ our language that can solve the above problem and bind the majority

among $0$ and $1to$ variable major. You can assume that $bis$ written $in$ the memory state; and $to$ simplify the

question, $we$ artificially define $b[n]=100,so$ you don't need $to$ worry about a possible runtime error when the

array index reaches $n.$ Your program doesn't have $tobe$ deterministic. $Be$ careful $of$ the grammar $in$ our

programming language.

Here are some hints:

$We$ can use the following linear$-$search$-$like algorithm: scan the array $bto$ pair $up$ each $0$ with $a1.$ Once $we$

have some $1\text{'}s$ left over, then $1is$ the majority; once $we$ have some $0^{\text{'}}s$ left over, then $0is$ the majority; $if$

$we$ can pair $up$ all numbers, then either $of$ them can $be$ the majority.

A student Jason wrote a partial solution that can $be$ useful $to$ solve the above problem:

$J-=dob[k_0]=1k_0$:$=k_0+1b[k_1]=0k_1$:$=k_1+1od$

Consider $k_0$ and $k_{1}as$ pointers for number $0$ and $1$ respectively. What program $J$ does $isto$ find the next

$k_0$ and $k_1$ such that $b[k_0]=0$ and $b[k_1]=1.$ You can use $J$ inside $of$ your program.

True $or$ false. Justify your answer briefly.

$a.IfM(S,)$ contains exactly one state, then $S$ must $be$ a deterministic statement.

$b.If|==\{p\}S\{q\},$ then $|==p.$

$d.If|==\{p\}S\{q\},$ then $M(S,)|==q.$

$e.If\{p\}S\{q\},$ then ${}_{\text{t}\text{o}\text{t}}\{p\}S\{q\}.$