$1.$ Briefly explain why the following expressions induce a "type error"? $($a$)(1/$ point$)[("$I$",4\mathrm{.}0)$ ; $("$R$",0\mathrm{.}0)$ ; $("$S$",1)]$

$2$

$($b$)(1/$ point$)2$

type $$a option $=$ Some of $$a $|$ None letfa$=$

if a $<0\mathrm{.}0$ then None else a

$2.$ Provide expressions $($without type annotations$)$ that have the following types: $($a$)(1/$ point$)$ int $>$ int list $>$ bool list

$2$

$($b$)(1/$ point$)$a $>($ a$>$b$)>$b $2$

$($c$)(1/$ point$)($a$>$b$)>$a $>$b $>$ bool $2$

$($d$)(1/$ point$)$a $>($a $>$a$)>$a $>$a $2$

CS$314$

Final, Page $3$ of $1216$

th

Dec, $2023$

$($e$)(1/$ point$)($a $>$ int$)>$a $>$ int $2$

$($f$)(1/$ point$)$a $>($a $>$b$)$ list $>$b list $.2$

$($g$)(1$point$)$ Defineafunctionfthatwhenusedinthefollowingexpressionwillnotproduce any type errors:

fold$\_$left f $([],0)[5$;$4$;$3$;$2$;$1]$

The implementation and type of fold$\_$left are given for reference, below.

let rec fold$\_$left f a l $=$ match l with $|[]>$ a

$|$ h::t $>$ fold$\_$left f $($f a h$)$ t

$($h$)(1$point$)$ Whatwouldyouputinplacefortheblanksuchthatthefollowingcodereturns $44?$

let f $=$ fun x y z $>$

List.fold$\_$left $($fun acc e $>$ acc $+$ e $+($y z$))0$ x in

f $[1$;$1$;$1$;$1]\_\_\_\_\_\_\_\_\_4$

CS$314$ Final, Page $4$ of $1216$

th

Dec, $2023$

$($i$)(1$ point$)$ What would you put in place for the blank such that sublist $[1]0=[]$ and sublist $[1$; $0$; $2$; $3]1=[1$; $0]?$

let sublist l t $=$

fold$\_$right $($fun x a $>\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_)$ l $[]$

The implementation of fold$\_$right is given for reference, below.

let rec fold$\_$right f l acc $=$ match l with $|[]>$ acc

$|$ x::xs $>$ f x $($fold$\_$right f xs acc$)$

$($j$)(1$ point$)$ What would you put in place for the blank such that the following program returns $(10\mathrm{.}0,40\mathrm{.}0)?$

type taxpayer $=$

$|$ Person of float

$|$ Company of float

let tax y $=$

let income$\_$tax a x $=\_\_\_\_\_\_\_\_\_\_$ in

match y with

$|$ Person i $>$ income$\_$tax i $0\mathrm{.}1$

$|$ Company j $>$ income$\_$tax j $0\mathrm{.}2$ in

$($ tax $($ Person $100\mathrm{.}0),$ tax $($Company $200\mathrm{.}0))$