Computer Science ConnectionsCurryingFor a binary predicate P$($x$,$ y$)$ in an expression like $8$y : $8$x : P$($x$,$ y$),$ we can think of this expression as frst pluggingin a value for y$,$ which then yields a unary predicate $8$x : P$($x$,$ y$)($which takes an argument x$).$ Theres an interestingparallel between this view of nested quantifers and a way of writing functions in some programming languages$\mathrm{.}1(*$ M L $*)2$ fun sum a b $=$ a $+$ b;A quick note on ML syntax:fun is a keyword that sayswere defning a function; sumis the name of it; a b is thelist of arguments; and thatfunction is defned to returnthe value of a$+$b$\mathrm{.}3$ sum $32$; $(*$ returns $5*)4$ sum $9912$; $(*$ returns $111*)1$ # Python$2$ def sum$($a$,$b$)$:$3$ return a$+$b$4$ sum$(3,2)$ # returns $55$ sum$(99,12)$ # returns $1111$ ; Scheme$2($define sum$3($lambda $($a b$)(+$ a b$)))$For Scheme: $($lambda argsbody$)$ denotes the functionthat takes arguments args andreturns the value of thefunction body body. Also,Scheme is a prefx language,so applying the function f toarguments arg$1,$ arg$2,,$argN is written $($f arg$1$ arg$2...$ argN$)$; for example, $(+12)$ has the value $3\mathrm{.}4($sum $32)$ ;; returns $55($sum $9912)$ ;; returns $111$Figure $3\mathrm{.}30$ A sum function, implemented in three languages.For concreteness, lets think about a smallfunction that takes two arguments and justreturns their sum. Figure $3\mathrm{.}30$ shows implementations of this function in three differentprogramming languages, and then uses sumto actually compute $3+2$ and $99+12.$But now suppose that we want to defne afunction that takes one argument and adds $3$to it$.$ Can we make use of the sum functionto do so$?($The analogy to predicates is thattaking a two$-$argument predicate and applying it to one argument gives one$-$argumentpredicate; here were trying to take a twoargument function in a programming language and apply it to one argument to yield aone$-$argument function.$)$ The answer is yesand it turns out that creating the add $3$function using sum is very easy in ML: we simply apply sum to one argument, and the result is a function thatstill wants one more argument. To do this, we execute val add$3=$ sum $3$; which defnes a value $($thats the valpart of the syntax$)$ whose name is add$3$ and whose value is sum applied to $3.$ That makes add$3$ a one$-$argumentfunction, and now add$30$ has the value $3$; add$3108$ has the value $111$; and add$3199$ has the value $202\mathrm{.}1$ fun sum a b $=$ a $+$ b;$2$ val add$3=$ sum $3$;$34$ sum $32$; $(*$ returns $5*)5$ add$32$; $(*$ returns $5*)1$ def sum$($a$)$:$2$ def sumA$($b$)$:$3$ return a$+$b$4$ return sumA$56$ add$3=$ sum$(3)78$ sum$(3)(2)$ # returns $59$ add$3(2)$ # returns $51($define sum $($lambda $($a$)2($lambda $($b$)(+$ a b$)))$Defne sum to be a functionthat takes an argument a andreturns $......$ the function that takes anargument b and returns a$+$b$.($This function $[$taking b$,$returning a$+$b$]$ is called sumAin the Python code, and isunnamed in the Schemecode.$)34($define add$3($sum $3))56(($sum $3)2)$ ;; returns $57($add$32)$ ;; returns $5$Figure $3\mathrm{.}31$ Curried implementations of sum.A function like sum in ML$,$ which takes itsmultiple arguments one at a time, is said tobe Curriedafter the logician Haskell Curry$(19001982).($As usual, theres a richer history than a concept named after one personwould suggest: the idea dates back at least toMoses Sch$$nfnkel $(18881942)$ and GottlobFrege $(18481925),$ two earlier logicians$.)$The programming language Haskell is alsonamed in Currys honor. Thinking about Curried functions is a classical topic in the studyof programming languages. $($Currying canbe very handy in programmingparticularlywhen you have one parameter to a functionthat stays the same for many different calls.......This is a discrete math class, we have to show computer science is related to this