✎ pp:2023:scala:l03 [books]

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====== Lab 2. High order functions ======

Objectives:
  * implement and use **higher-order** functions. A **higher-order** function takes other functions as parameter or returns them
  * implement **curry** and **uncurry** functions, and how they should be properly used (review lecture).

** Create a new Scala worksheet to write your solutions **

===== 2.1 Intro. Functions as parameters =====
**2.1.1** Write a function ''apply'' that takes an integer and return the result of the applied function on the given integer. Start from the code stub below:
<code scala>
def apply(n: Int, f: Int => Int): Int = {
   ???
}
</code>

**2.1.2** Write a function ''doubler'' that returns a function that doubles the input it receives (an integer). Start from the code stub below:
<code scala>
def doubler(): Int => Int = {
   ???
}
</code>

===== 2.2 Custom high order functions =====
**2.2.1** Define the function ''foldWith'' which uses an operation ''op'' to reduce a range of integers to a value. For instance, given that ''op'' is addition (+), the result of folding the range 1 to 3 will be 1+2+3=6. ''foldWith'' should be curried (it will take the operation and return another function which expects the bounds).
<code scala>
def foldWith (op: (Int,Int) => Int)(start: Int, stop: Int): Int = {
  def tail_fold(crt: Int, acc: Int): Int  = ???
  ??
}
</code>

**2.2.2** Define the function ''foldConditional'' which extends ''foldWith'' by also adding a predicate ''p: Int => Int''. ''foldConditional'' will reduce only those elements of a range which satisfy the predicate.

<code scala>
def foldConditional(op: (Int,Int) => Int, p: Int => Boolean)(start: Int, stop: Int): Int = ???
</code>

**2.2.3** Write a function ''foldMap'' which takes values $math[a_1, a_2, \ldots, a_k] from a range and computes $math[f(a_1)\;op\;f(a_2)\;op\;\ldots f(a_k)].
Use the ''apply'' and ''foldWith'' methods
<code scala>
def foldMap(op: (Int,Int) => Int, f: Int => Int)(start: Int, stop: Int): Int = ???
</code>

**2.2.4** Write a function which computes $math[1 + 2^2 + 3^2 + \ldots + (n-1)^2 + n^2] using ''foldMap''.
<code scala>
def sumSquares(n: Int): Int = ???
</code>

**2.2.5** Write a function ''hasDivisor'' which checks if a range contains a multiple of k. Use ''foldMap'' and choose ''f'' carefully.
<code scala>
def hasDivisor(k: Int, start: Int, stop: Int): Boolean = ???
</code>

**2.2.6** We can compute the sum of an area defined by a function within a range a,b (the integral of that function given the range), using the following recursive scheme:
  * if the range is small enough, we treat f as a line (and the area as a trapeze). It's area is $math[(f(a) + f(b))(b-a)/2].
  * otherwise, we compute the mid of the range, we recursively compute the integral from a to mid and from mid to b, and add-up the result.

Implement the function ''integrate'' which computes the integral of a function f given a range:
<code scala>
def integrate(f: Double => Double)(start: Double, stop: Double): Double = ?
??
</code>

===== 2.3 Curry vs Uncurry =====
**2.3.1** Modify the function below so that it's curry and use it to calculate ''5*3''
<code scala>
def multiply(x:Int, y:Int): Int => x * y
</code>

**2.3.2** Modify the function below so that it's curry and use it to compare 3 numbers and return the maximum
<code scala>
def compare(x: Int, y: Int, z: Int): Int =
{
  if x > y && x > z then 
    x
  else if y > x && y > z then
    y
  else
    z
}
</code>

**2.3.3** Modify the function from **2.2.1** so that i
t's uncurry

===== 2.4 Characteristics set function =====
A characteristic set function is a function that describes a set of integers. Mathematically, it is defined as followed: f(a) is true if and only if a is in the set, false otherwise
For example, for the set {1,5,6} can have the following function associated: <code scala> def f(x: Int): Boolean = x == 1 || x == 5 || x == 6 </code>

**2.4.1** Write a characteristic functions for the numbers below 20

**2.4.2** Write a characteristic functions for the numbers between 2 parameters

**2.4.3** Write a function that return the characteristic function of the reunion of two sets

**2.4.4** Write a function that return the characteristic function of the intersection of two sets