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:
x
, andy
.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.
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:
Rational
.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
.
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)
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
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
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)))
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)
In the above example rational numbers weren't always represented in their simplest form.
One would expect the rational numbers to be simplified:
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 reused
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 val
s, 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.
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.
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
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.
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.In Scala, a class implicitly introduces a constructor. This one is called the primary constructor of the class.
The primary constructor:
require
a couple of slides back).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)
...
}
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:
x1, …, xn
.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:
y1, …, ym
of the function f
by the
arguments w1, …, wm
,x1, …, xn
of the class C
by the class
arguments v1, …, vn
,this
by the value of the
object new C(v1, …, vn)
.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:
x + y
, if x
and y
are integers, butr.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:
'_'
counts as a letter.Examples of identifiers:
x1 * +?%& vector_++ counter_=
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
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)
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
.
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 subtrees.
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
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 userdefined 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
.
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, nonabstract
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
}
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.
Objectoriented 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.
Empty contains 1 shouldBe res0
new NonEmpty(7, Empty, Empty) contains 7 shouldBe res1
Dynamic dispatch of methods is analogous to calls to higherorder functions.
Can we implement one concept in terms of the other?
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 trait
s.
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.
At the top of the type hierarchy we find:
Any
==
, !=
, equals
, hashCode
, toString
AnyRef
java.lang.Object
AnyVal
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?
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
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:
abstract class Reducer(init: Int) {
def combine(x: Int, y: Int): Int
def reduce(xs: List[Int]): Int =
xs match {
case Nil => init
case y :: ys => combine(y, reduce(ys))
}
}
object Product extends Reducer(1) {
def combine(x: Int, y: Int): Int = x * y
}
object Sum extends Reducer(0) {
def combine(x: Int, y: Int): Int = x + y
}
val nums = List(1, 2, 3, 4)
Product.reduce(nums) shouldBe res0
Sum.reduce(nums) shouldBe res1