Functions and Data

Let’s see how functions create and encapsulate data structures.

We want to design a package for doing rational arithmetic.

A rational number x / y is represented by two integers:

• its numerator x, and
• its denominator y.

Rational Addition

Suppose we want to implement the addition of two rational numbers.

def addRationalNumerator(n1: Int, d1: Int, n2: Int, d2: Int): Int
def addRationalDenominator(n1: Int, d1: Int, n2: Int, d2: Int): Int

It would be difficult to manage all these numerators and denominators!

A better choice is to combine the numerator and denominator of a rational number in a data structure.

Classes

In Scala, we do this by defining a class:

class Rational(x: Int, y: Int) {
  def numer = x
  def denom = y
}

This definition introduces two entities:

• A new type, named Rational.
• A constructor Rational to create elements of this type.

Scala keeps the names of types and values in different namespaces. So there's no conflict between the two definitions of Rational.

Objects

We call the elements of a class type objects.

We create an object by prefixing an application of the constructor of the class with the operator new.

new Rational(1, 2)

Members of an Object

Objects of the class Rational have two members, numer and denom.

We select the members of an object with the infix operator . (like in Java).

val x = new Rational(1, 2) // x: Rational = Rational@2abe0e27
x.numer // 1
x.denom // 2

Rational Arithmetic

We can now define the arithmetic functions that implement the standard rules.

n1 / d1 + n2 / d2 = (n1 * d2 + n2 * d1) / (d1 * d2)
n1 / d1 - n2 / d2 = (n1 * d2 - n2 * d1) / (d1 * d2)
n1 / d1 * n2 / d2 = (n1 * n2) / (d1 * d2)
n1 / d1 / n2 / d2 = (n1 * d2) / (d1 * n2)
n1 / d1 = n2 / d2 iff n1 * d2 = d1 * n2

Implementing Rational Arithmetic

def addRational(r: Rational, s: Rational): Rational =
  new Rational(
    r.numer * s.denom + s.numer * r.denom,
    r.denom * s.denom
  )

def makeString(r: Rational) =
  s"${r.numer}/${r.denom}"

And then:

makeString(addRational(new Rational(1, 2), new Rational(2, 3)))

Methods

One can go further and also package functions operating on a data abstraction in the data abstraction itself.

Such functions are called methods.

Rational numbers now would have, in addition to the functions numer and denom, the functions add, sub, mul, div, equal, toString.

Here's a possible implementation:

class Rational(x: Int, y: Int) {
  def numer = x
  def denom = y
  def add(r: Rational) =
    new Rational(numer * r.denom + r.numer * denom, denom * r.denom)
  def mul(r: Rational) = ...
  ...
  override def toString = s"$numer/$denom"
}

Note that the modifier override declares that toString redefines a method that already exists (in the class java.lang.Object).

Here is how one might use the new Rational abstraction:

val x = new Rational(1, 3)
val y = new Rational(5, 7)
val z = new Rational(3, 2)
x.add(y).mul(z)

Data Abstraction

In the above example rational numbers weren't always represented in their simplest form.

One would expect the rational numbers to be simplified:

• reduce them to their smallest numerator and denominator by dividing both with a divisor.

We could implement this in each rational operation, but it would be easy to forget this division in an operation.

A better alternative consists of simplifying the representation in the class when the objects are constructed:

class Rational(x: Int, y: Int) {
  private def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
  private val g = gcd(x, y)
  def numer = x / g
  def denom = y / g
  ...
}

gcd and g are private members; we can only access them from inside the Rational class.

In this example, we calculate gcd immediately, so that its value can be re-used in the calculations of numer and denom.

It is also possible to call gcd in the code of numer and denom:

class Rational(x: Int, y: Int) {
  private def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
  def numer = x / gcd(x, y)
  def denom = y / gcd(x, y)
}

This can be advantageous if it is expected that the functions numer and denom are called infrequently.

It is equally possible to turn numer and denom into vals, so that they are computed only once:

class Rational(x: Int, y: Int) {
  private def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
  val numer = x / gcd(x, y)
  val denom = y / gcd(x, y)
}

This can be advantageous if the functions numer and denom are called often.

The Client's View

Clients observe exactly the same behavior in each case.

This ability to choose different implementations of the data without affecting clients is called data abstraction.

It is a cornerstone of software engineering.

Self Reference

On the inside of a class, the name this represents the object on which the current method is executed.

Add the functions less and max to the class Rational.

class Rational(x: Int, y: Int) {
  ...
  def less(that: Rational) =
  numer * that.denom < that.numer * denom

  def max(that: Rational) =
    if (this.less(that)) that else this
}

Note that a simple name x, which refers to another member of the class, is an abbreviation of this.x. Thus, an equivalent way to formulate less is as follows.

def less(that: Rational) =
  this.numer * that.denom < that.numer * this.denom

Preconditions

Let's say our Rational class requires that the denominator is positive.

We can enforce this by calling the require function.

class Rational(x: Int, y: Int) {
  require(y > 0, "denominator must be positive")
  ...
}

require is a predefined function. It takes a condition and an optional message string. If the condition passed to require is false, an IllegalArgumentException is thrown with the given message string.

Assertions

Besides require, there is also assert.

Assert also takes a condition and an optional message string as parameters. E.g.

val x = sqrt(y)
assert(x >= 0)

Like require, a failing assert will also throw an exception, but it's a different one: AssertionError for assert, IllegalArgumentException for require.

This reflects a difference in intent

• require is used to enforce a precondition on the caller of a function.
• assert is used as to check the code of the function itself.

Constructors

In Scala, a class implicitly introduces a constructor. This one is called the primary constructor of the class.

The primary constructor:

• takes the parameters of the class
• and executes all statements in the class body (such as the require a couple of slides back).

Auxiliary Constructors

Scala also allows the declaration of auxiliary constructors.

These are methods named this.

Adding an auxiliary constructor to the class Rational:

class Rational(x: Int, y: Int) {
  def this(x: Int) = this(x, 1)
  ...
}

Classes and Substitutions

We previously defined the meaning of a function application using a computation model based on substitution. Now we extend this model to classes and objects.

How is an instantiation of the class new C(e1, …, en) evaluated?

The expression arguments e1, …, en are evaluated like the arguments of a normal function. That's it.

The resulting expression, say, new C(v1, …, vn), is already a value.

Now suppose that we have a class definition,

class C(x1, …, xn) {
  …
  def f(y1, …, ym) = b
  …
}

where:

• The formal parameters of the class are x1, …, xn.
• The class defines a method f with formal parameters y1, …, ym.

(The list of function parameters can be absent. For simplicity, we have omitted the parameter types.)

How is the following expression evaluated?

new C(v1, …, vn).f(w1, …, wm)

The following three substitutions happen:

• the substitution of the formal parameters y1, …, ym of the function f by the arguments w1, …, wm,
• the substitution of the formal parameters x1, …, xn of the class C by the class arguments v1, …, vn,
• the substitution of the self reference this by the value of the object new C(v1, …, vn).

Operators

In principle, the rational numbers defined by Rational are as natural as integers.

But for the user of these abstractions, there is a noticeable difference:

• We write x + y, if x and y are integers, but
• We write r.add(s) if r and s are rational numbers.

In Scala, we can eliminate this difference because operators can be used as identifiers.

Thus, an identifier can be:

• Alphanumeric: starting with a letter, followed by a sequence of letters or numbers
• Symbolic: starting with an operator symbol, followed by other operator symbols.
• The underscore character '_' counts as a letter.
• Alphanumeric identifiers can also end in an underscore, followed by some operator symbols.

Examples of identifiers:

x1     *     +?%&     vector_++     counter_=

Operators for Rationals

So, here is a more natural definition of class Rational:

class Rational(x: Int, y: Int) {
  private def gcd(a: Int, b: Int): Int = if (b == 0) a else gcd(b, a % b)
  private val g = gcd(x, y)
  def numer = x / g
  def denom = y / g
  def + (r: Rational) =
    new Rational(
      numer * r.denom + r.numer * denom,
      denom * r.denom
    )
  def - (r: Rational) = ...
  def * (r: Rational) = ...
  ...
}

and then rational numbers can be used like Int or Double:

val x = new Rational(1, 2)
val y = new Rational(1, 3)
x * x + y * y

Precedence Rules

The precedence of an operator is determined by its first character.

The following table lists the characters in increasing order of priority precedence:

(all letters)
|
^
&
< >
= !
:
+ -
* / %
(all other special characters)

Abstract Classes

Consider the task of writing a class for sets of integers with the following operations.

abstract class IntSet {
  def incl(x: Int): IntSet
  def contains(x: Int): Boolean
}

IntSet is an abstract class.

Abstract classes can contain members which are missing an implementation (in our case, incl and contains).

Consequently, no instances of an abstract class can be created with the operator new.

Class Extensions

Let's consider implementing sets as binary trees.

There are two types of possible trees: a tree for the empty set, and a tree consisting of an integer and two sub-trees.

Here are their implementations:

class Empty extends IntSet {
  def contains(x: Int): Boolean = false
  def incl(x: Int): IntSet = new NonEmpty(x, new Empty, new Empty)
}

class NonEmpty(elem: Int, left: IntSet, right: IntSet) extends IntSet {

  def contains(x: Int): Boolean =
    if (x < elem) left contains x
    else if (x > elem) right contains x
    else true

  def incl(x: Int): IntSet =
    if (x < elem) new NonEmpty(elem, left incl x, right)
    else if (x > elem) new NonEmpty(elem, left, right incl x)
    else this
}

Empty and NonEmpty both extend the class IntSet.

This implies that the types Empty and NonEmpty conform to the type IntSet

• an object of type Empty or NonEmpty can be used wherever an object of type IntSet is required.

IntSet is called the superclass of Empty and NonEmpty.

Empty and NonEmpty are subclasses of IntSet.

In Scala, any user-defined class extends another class.

If no superclass is given, the standard class Object in the Java package java.lang is assumed.

The direct or indirect superclasses of a class C are called base classes of C.

So, the base classes of NonEmpty are IntSet and Object.

Implementation and Overriding

The definitions of contains and incl in the classes Empty and NonEmpty implement the abstract functions in the base trait IntSet.

It is also possible to redefine an existing, non-abstract definition in a subclass by using override.

abstract class Base {
  def foo = 1
  def bar: Int
}

class Sub extends Base {
  override def foo = 2
  def bar = 3
}

Object Definitions

In the IntSet example, one could argue that there is really only a single empty IntSet.

So it seems overkill to have the user create many instances of it.

We can express this case better with an object definition:

object Empty extends IntSet {
  def contains(x: Int): Boolean = false
  def incl(x: Int): IntSet = new NonEmpty(x, Empty, Empty)
}

This defines a singleton object named Empty.

No other Empty instances can be (or need to be) created.

Singleton objects are values, so Empty evaluates to itself.

Dynamic Binding

Object-oriented languages (including Scala) implement dynamic method dispatch.

This means that the code invoked by a method call depends on the runtime type of the object that contains the method.

Dynamic dispatch of methods is analogous to calls to higher-order functions.

Can we implement one concept in terms of the other?

• Objects in terms of higher-order functions?
• Higher-order functions in terms of objects?

Traits

In Scala, a class can only have one superclass.

But what if a class has several natural supertypes to which it conforms or from which it wants to inherit code?

Here, you could use traits.

A trait is declared like an abstract class, just with trait instead of abstract class.

trait Planar {
  def height: Int
  def width: Int
  def surface = height * width
}

Classes, objects and traits can inherit from at most one class but arbitrarily many traits:

class Square extends Shape with Planar with Movable …

On the other hand, traits cannot have (value) parameters, only classes can.

Scala's Class Hierarchy

Top Types

At the top of the type hierarchy we find:

• Any
  • The base type of all types
  • Methods: ==, !=, equals, hashCode, toString
• AnyRef
  • The base type of all reference types
  • Alias of java.lang.Object
• AnyVal
  • The base type of all primitive types

Bottom Type

Nothing is at the bottom of Scala's type hierarchy. It is a subtype of every other type.

There is no value of type Nothing.

Why is that useful?

• To signal abnormal termination
• As an element type of empty collections

The Null Type

Every reference class type also has null as a value.

The type of null is Null.

Null is a subtype of every class that inherits from Object; it is incompatible with subtypes of AnyVal.

val x = null         // x: Null
val y: String = null // y: String
val z: Int = null    // error: type mismatch

Exercise

The following Reducer abstract class defines how to reduce a list of values into a single value by starting with an initial value and combining it with each element of the list: