Homework 4

Due on Monday, 19 November at 10:15am sharp (right before the class).

NOTE: It is your responsibility to ask us if any part of the problem formulation is unclear.

Problem 1

Determine for the following program its output with static and dynamic scoping. Explain the difference, if there is any.

object MyClass {
 val x = 5
 def foo(z: Int): Int = {
   x + z   
 def bar(y: Int): Int = {
   val x = 1
   val z = 2
 def main() {
   val x = 7

Problem 2

Consider the following grammar:

E ::= E+E
E ::= E*E
E ::= E=E
E ::= Num | true | false

where Num represents integers.

a) Consider the generalized CYK algorithm from

Run the algorithm on input:


checking whether it can be parsed. Show the content of the 'chart', that is, the set of triples $(A,i,j)$, after the algorithm finishes.

b) How many parse trees can you extract from this chart?

c) Write down those parse trees that are correct according to the following types:

+ : Int x Int→Int

* : Int x Int→Int

= : Bool x Bool → Bool

= : Int x Int → Bool

d) Can you modify the CKY algorithm following the ideas from:

to introduce a semantic function $\mathbf{f}$ that computes only those trees that are correct according to the above types of operators?

e) What does the new algorithm return on the above input?

Problem 3

Consider the following class:

class Rectangle {
  var width: Int
  var height: Int
  var xPos: Int
  var yPos: Int
  def area: Int = {
    if(width > 0 && height > 0) 
      width * height
  def resize(maxSize: Int) {
    while(area > maxSize) {
      width = width / 2
      height = height / 2

a) Determine the environment of this class.

b) By giving the type derivation trees for each method show that this class type checks.

Problem 4

Determine if the following piece of codes type check according to the type rules.

Show all steps and give type derivation trees where applicable.

a) The class Array has a field length in which the length of the array is stored.

def swap(lst: Array[Int], a: Int, b: Int): Array[Int] = {
  if (a >= lst.length || b >= lst.length) lst else {
    val swap = lst(a)
    lst(a) = lst(b)
    lst(b) = swap


Recall that we have the two type rules:

  • If $C$ is a declared subclass of $D$, then $C <: D$
  • \begin{equation*}
      \frac{\Gamma \vdash t: C \quad\quad C<:D}{\Gamma \vdash t: D}

i) Give the type rule corresponding to new.
ii) Show this code typechecks.

class Shape
class Rectangle(width: Int, length: Int) extends Shape {
  def area : Int = width * length
class Square(length: Int) extends Rectangle(length,length)
new Square(5).area