$(30$pt$)$ Consider a language where assignments can appear in the same context as expressions; the value

of $a=b=c$ equals the value of $c.$ The following grammar, $G,$ generates such expressions that includes

assignments in addition to additions and multiplications:

$($a$)(3$pt$)$ Show a parse tree for the string: id $=id**($id $=id+id**id.$

$($b$)(10$pt$)$ For each production Alongrightarrow$,$ compute FIRST$()$ and FOLLOW$(A)$ using the algorithm below;

FIRST $()$ is computed by string$\_$FIRST $().$ For each token added, indicate the pair $($step$,$ prod$)$ used

to add it$,$ where $0$ step $3$ is the step in the algorithm $($marked as $0,1,2,3$ below$)$ and

$0$prod$9$ is the production involved; indicate $(0,-)$ when step $0$ is used for terminals.

$($c$)(5$pt$)$ For each production $i,1i9,$ compute PREDICT$(i).$

$($d$)(2$pt$)$ Using the information computed above, show that this grammar is not LL$(1).($See definition

on the slide $19$ of the LR$-$parsing chapter.$)$

$($e$)(10$pt$)$ Modify this grammar to make it LL$(1).$ Explain clearly your changes and prove it is LL$(1).$