Understanding Currying in Scala - scala

Question

I'm getting problems to understand the currying concept, or at least the SCALA currying notation.
wikipedia says that currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument.
Following this explanation, are the two next lines the same for scala?
def addCurr(a: String)(b: String): String = {a + " " + b}
def add(a:String): String => String = {b => a + " " + b}
I've run both lines with the same strings a and b getting the same result, but I don't know if they are different under the hood
My way of thinking about addCurr (and currying itself) is that it is a function that receives a string parameter a, and returns another function that also receives a string parameter b and returns the string a + " " + b?
So if I'm getting right, addCurr is only syntactic sugar of the function add and both are curryed functions?
According to the previous example, the next functions are also equivalent for scala?
def add(a: String)(b: String)(c: String):String = { a + " " + b + " " + c}
def add1(a: String)(b: String): String => String = {c => a + " " + b + " " + c}
def add2(a:String): (String => (String => String)) = {b => (c => a + " " + b + " " + c)}

Answer 1

They have a bit different semantics, but their use-cases are mostly the same, both practically and how it looks in the code.
Currying
Currying a function in Scala in that mathematical sense is a very straightforward:
val function = (x: Int, y: Int, z: Int) => 0
// function: (Int, Int, Int) => Int = <function3>
function.curried
// res0: Int => (Int => (Int => Int)) = <function1>
Functions & methods
You seem to be confused by the fact that in Scala, (=>) functions are not the same as (def) methods. Method isn't a first-class object, while function is (i.e. it has curried and tupled methods, and Function1 has even more goodness).
Methods, however, can be lifted to functions by an operation known as eta expansion. See this SO answer for some details. You can trigger it manually by writing methodName _, or it will be done implicitly if you give a method to where a function type is expected.
def sumAndAdd4(i: Int, j: Int) = i + j + 4
// sumAndAdd4.curried // <- won't compile
val asFunction = sumAndAdd4 _ // trigger eta expansion
// asFunction: (Int, Int) => Int = <function2>
val asFunction2: (Int, Int) => Int = sumAndAdd4
// asFunction2: (Int, Int) => Int = <function2>
val asFunction3 = sumAndAdd4: (Int, Int) => Int
// asFunction3: (Int, Int) => Int = <function2>
asFunction.curried
// res0: Int => (Int => Int) = <function1>
asFunction2.curried
// res1: Int => (Int => Int) = <function1>
asFunction3.curried
// res2: Int => (Int => Int) = <function1>
{sumAndAdd4 _}.tupled // you can do it inline too
// res3: Int => (Int => Int) = <function1>
Eta expansion of multiple parameter list
Like you might expect, eta expansion lifts every parameter list to its own function
def singleArgumentList(x: Int, y: Int) = x + y
def twoArgumentLists(x: Int)(y: Int) = x + y
singleArgumentList _ // (Int, Int) => Int
twoArgumentLists _ // Int => (Int => Int) - curried!
val testSubject = List(1, 2, 3)
testSubject.reduce(singleArgumentList) // Int (6)
testSubject.map(twoArgumentLists) // List[Int => Int]
// testSubject.map(singleArgumentList) // does not compile, map needs Int => A
// testSubject.reduce(twoArgumentLists) // does not compile, reduce needs (Int, Int) => Int
But it's not that currying in mathematical sense:
def hmm(i: Int, j: Int)(s: String, t: String) = s"$i, $j; $s - $t"
{hmm _} // (Int, Int) => (String, String) => String
Here, we get a function of two arguments, returning another function of two arguments.
And it's not that straightforward to specify only some of its argume
val function = hmm(5, 6) _ // <- still need that underscore!
Where as with functions, you get back a function without any fuss:
val alreadyFunction = (i: Int, j: Int) => (k: Int) => i + j + k
val f = alreadyFunction(4, 5) // Int => Int
Do which way you like it - Scala is fairly un-opinionated about many things. I prefer multiple parameter lists, personally, because more often than not I'll need to partially apply a function and then pass it somewhere, where the remaining parameters will be given, so I don't need to explicitly do eta-expansion, and I get to enjoy a terser syntax at method definition site.

Answer 2

Curried methods are syntactic sugar, you were right about this part. But this syntactic sugar is a bit different. Consider following example:
def addCur(a: String)(b: String): String = { a + b }
def add(a: String): String => String = { b => a + b }
val functionFirst: String => String = add("34")
val functionFirst2 = add("34")_
val functionSecond: String => String = add("34")
Generaly speaking curried methods allows for partial application and are necessary for the scala implicits mechanism to work. In the example above i provided examples of usage, as you can see in the second one we have to use underscore sign to allow compiler to do the "trick". If it was not present you would receive error similar to the following one:
Error:(75, 19) missing argument list for method curried in object XXX
Unapplied methods are only converted to functions when a function type
is expected. You can make this conversion explicit by writing curried_ or curried(_)(_) instead of curried.

Answer 3

Your question interested me so I tried this out my self. They actually desugar down to some very different constructs. Using
def addCurr(a: String)(b: String): String = {a + " " + b}
This actually compiles to
def addCurr(a: String, b: String): String = {a + " " + b}
So it completely removes any currying effect, making it a regular arity-2 method. Eta expansion is used to allow you to curry it.
def add(a:String): String => String = {b => a + " " + b}
This one works as you would expect, compiling to a method that returns a Function1[String,String]