Definitions And Evaluation

Naming Things

Consider the following program that computes the area of a disc whose radius is 10:

3.14159 * 10 * 10

To make complex expressions more readable we can give meaningful names to intermediate expressions:

val radius = 10
val pi = 3.14159

pi * radius * radius

Besides making the last expression more readable it also allows us to not repeat the actual value of the radius.

Evaluation

A name is evaluated by replacing it with the right hand side of its definition

Example

Here are the evaluation steps of the above expression:

pi * radius * radius
3.14159 * radius * radius
3.14159 * 10 * radius
31.4159 * radius
31.4159 * 10
314.159

Methods

Definitions can have parameters. For instance:

def square(x: Double) = x * x

square(3.0) shouldBe res0

Let’s define a method that computes the area of a disc, given its radius:

def square(x: Double) = x * x

def area(radius: Double): Double = 3.14159 * square(radius)

area(10) shouldBe res0

Multiple Parameters

Separate several parameters with commas:

def sumOfSquares(x: Double, y: Double) = square(x) + square(y)

Parameters and Return Types

Function parameters come with their type, which is given after a colon

def power(x: Double, y: Int): Double = ...

If a return type is given, it follows the parameter list.

Val vs Def

The right hand side of a def definition is evaluated on each use.

The right hand side of a val definition is evaluated at the point of the definition itself. Afterwards, the name refers to the value.

val x = 2
val y = square(x)

For instance, y above refers to 4, not square(2).

Evaluation of Function Applications

Applications of parametrized functions are evaluated in a similar way as operators:

  1. Evaluate all function arguments, from left to right
  2. Replace the function application by the function's right-hand side, and, at the same time
  3. Replace the formal parameters of the function by the actual arguments.

Example

sumOfSquares(3, 2 + 2)
sumOfSquares(3, 4)
square(3) + square(4)
3 * 3 + square(4)
9 + square(4)
9 + 4 * 4
9 + 16
25

The substitution model

This scheme of expression evaluation is called the substitution model.

The idea underlying this model is that all evaluation does is reduce an expression to a value.

It can be applied to all expressions, as long as they have no side effects.

The substitution model is formalized in the λ-calculus, which gives a foundation for functional programming.

Termination

Does every expression reduce to a value (in a finite number of steps)?

No. Here is a counter-example:

def loop: Int = loop

loop

Value Definitions and Termination

The difference between val and def becomes apparent when the right hand side does not terminate. Given

def loop: Int = loop

A definition

def x = loop

is OK, but a definition

val x = loop

will lead to an infinite loop.

Changing the evaluation strategy

The interpreter reduces function arguments to values before rewriting the function application.

One could alternatively apply the function to unreduced arguments.

For instance:

sumOfSquares(3, 2 + 2)
square(3) + square(2 + 2)
3 * 3 + square(2 + 2)
9 + square(2 + 2)
9 + (2 + 2) * (2 + 2)
9 + 4 * (2 + 2)
9 + 4 * 4
25

Call-by-name and call-by-value

The first evaluation strategy is known as call-by-value, the second is is known as call-by-name.

Both strategies reduce to the same final values as long as

  • the reduced expression consists of pure functions, and
  • both evaluations terminate.

Call-by-value has the advantage that it evaluates every function argument only once.

Call-by-name has the advantage that a function argument is not evaluated if the corresponding parameter is unused in the evaluation of the function body.

Scala normally uses call-by-value.

Exercise

Complete the following definition of the triangleArea function, which takes a triangle base and height as parameters and returns its area:

def triangleArea(base: Double, height: Double): Double =
  base * height / res0

triangleArea(3, 4) shouldBe 6
triangleArea(5, 6) shouldBe res1