Lab 3. Higher-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).

3.1. Define the function foldWith which uses an operation op, and an initial value b 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 with b = 0 will be ( ( 0 + 1 ) + 2 + 3 = 6. foldWith should be curried (it will take the operation and return another function which expects the bounds). Note that foldWith is curry. 

def foldWith (b: Int)(op: (Int,Int) => Int)(start: Int, stop: Int): Int = {
  def tail_fold(crt: Int, acc: Int): Int  = ???
  ??
}

3.2. 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. 

def foldConditional(b: Int)(op: (Int,Int) => Int, p: Int => Boolean)(start: Int, stop: Int): Int = ???

3.3. Implement the function foldRight which has the same behaviour as foldWith, but the order in which the operation is performed is now: 1+(2+(3+0))=6. What is the simplest way to implement it?

3.4. Write a function foldMap which takes values $ a_1, a_2, \ldots, a_k$ from a range and computes $ f(a_1)\;op\;f(a_2)\;op\;\ldots f(a_k)$ . 

def foldMap(op: (Int,Int) => Int, f: Int => Int)(start: Int, stop: Int): Int = ???

3.5. Write a function which computes $ 1 + 2^2 + 3^2 + \ldots + (n-1)^2 + n^2$ using foldMap. 

def sumSquares(n: Int): Int = ???

3.6. Write a function hasDivisor which checks if a range contains a multiple of k. Use foldMap and choose f carefully. 

def hasDivisor(k: Int, start: Int, stop: Int): Boolean = ???

3.7. 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 $ (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: 

def integrate(f: Double => Double)(start: Double, stop: Double): Double = ???

3.8. We define Line2D to be lines in a 2-dimensional space, and we represent them as functions. Write a function which takes a Line2D and translates it up on the Ox axis by a given offset. For instance, translateOx of 2 on y=x+1 will return y=x+3. 

type Line2D = Int => Int
def translateOx(offset: Int)(l: Line2D): Line2D = ???

3.9. (!) Write a function which takes a Line2D and translates it up on the Oy axis by a given offset. 

def translateOy(offset: Int)(l: Line2D): Line2D = ???

3.10. Write a function which takes two lines, and checks if there exist integer coordinates within a given range for x, where the lines intersect. 

def intersect(l1: Line2D, l2: Line2D)(start: Int, stop: Int): Boolean = ???

3.11. Write a function which takes two lines and a range of integers, and checks if l1 has larger y values than l2 over the entire range. 

def larger(l1: Line2D, l2: Line2D)(start: Int, stop: Int): Boolean = ???