The final results are different. We conclude that x and y are not the same.

Establishing Operational Equivalence

On the other hand, if we define

val x = new BankAccount
val y = x

then no sequence of operations can distinguish between x and y, so x and y are the same in this case.

Assignment and Substitution Model

The preceding examples show that our model of computation by substitution cannot be used.

Indeed, according to this model, one can always replace the name of a value by the expression that defines it. For example, in

val x = new BankAccount
val y = x

the x in the definition of y could be replaced by new BankAccount.

But we have seen that this change leads to a different program!

The substitution model ceases to be valid when we add the assignment.

It is possible to adapt the substitution model by introducing a store, but this becomes considerably more complicated.

Imperative Loops

In the first sections, we saw how to write loops using recursion.

While-Loops

We can also write loops with the while keyword:

def power (x: Double, exp: Int): Double = {
  var r = 1.0
  var i = exp
  while (i > 0) { r = r * x; i = i - 1 }
  r
}

As long as the condition of a while statement is true, its body is evaluated.

For-Loops

In Scala there is a kind of for loop:

for (i <- 1 until 3) { System.out.print(i.toString + " ") }

This displays 1 2.

For-loops translate similarly to for-expressions, but using the foreach combinator instead of map and flatMap.

foreach is defined on collections with elements of type A as follows:

def foreach(f: A => Unit): Unit =
  // apply `f` to each element of the collection

Example:

for (i <- 1 until 3; j <- "abc") println(s"$i $j")

translates to:

(1 until 3) foreach (i => "abc" foreach (j => println(s"$i $j")))

Exercise

Complete the following imperative implementation of factorial:

Until now, our programs have been side-effect free.

Therefore, the concept of time wasn't important.

For all programs that terminate, any sequence of actions would have given the same results.

This was also reflected in the substitution model of computation.

Reminder: Substitution Model

Programs can be evaluated by rewriting:

• a name is evaluated by replacing it with the right-hand side of its definition,
• function application is evaluated by replacing it with the function’s right-hand side, and, at the same time, by replacing the formal parameters by the actual arguments.

Say you have the following two functions iterate and square:

def iterate(n: Int, f: Int => Int, x: Int): Int =
  if (n == 0) x else iterate(n-1, f, f(x))
def square(x: Int) = x * x

Then the call iterate(1, square, 3) gets rewritten as follows:

iterate(1, square, 3)
if (1 == 0) 3 else iterate(1-1, square, square(3))
iterate(0, square, square(3))
iterate(0, square, 3 * 3)
iterate(0, square, 9)
if (0 == 0) 9 else iterate(0-1, square, square(9))
9

Rewriting can be done anywhere in a term, and all rewritings which terminate lead to the same solution.

This is an important result of the λ-calculus, the theory behind functional programming.

For instance, these two rewriting will eventually lead to the same result:

if (1 == 0) 3 else iterate(1 - 1, square, square(3))
iterate(0, square, square(3))

// OR
if (1 == 0) 3 else iterate(1 - 1, square, square(3))
if (1 == 0) 3 else iterate(1 - 1, square, 3 * 3)

Stateful Objects

One normally describes the world as a set of objects, some of which have state that changes over the course of time.

An object has a state if its behavior is influenced by its history.

Example: a bank account has a state, because the answer to the question “can I withdraw 100 CHF ?” may vary over the course of the lifetime of the account.

Implementation of State

Every form of mutable state is constructed from variables.

A variable definition is written like a value definition, but with the keyword var in place of val:

var x: String = "abc"
var count = 111

Just like a value definition, a variable definition associates a value with a name.

However, in the case of variable definitions, this association can be changed later through an assignment:

x = "hi"
count = count + 1

State in Objects

In practice, objects with state are usually represented by objects that have some variable members.

Here is a class modeling a bank account:

class BankAccount {
  private var balance = 0
  def deposit(amount: Int): Int = {
    if (amount > 0) balance = balance + amount
    balance
  }
  def withdraw(amount: Int): Int =
    if (0 < amount && amount <= balance) {
      balance = balance - amount
      balance
    } else throw new Error("insufficient funds")
}

The class BankAccount defines a variable balance that contains the current balance of the account.

The methods deposit and withdraw change the value of the balance through assignments.

Note that balance is private in the BankAccount class, it therefore cannot be accessed from outside the class.

To create bank accounts, we use the usual notation for object creation:

val account = new BankAccount

Working with Mutable Objects

Here is a program that manipulates bank accounts.

val account = new BankAccount       // account: BankAccount = BankAccount
account deposit 50                  //
account withdraw 20                 // res1: Int = 30
account withdraw 20                 // res2: Int = 10
account withdraw 15                 // java.lang.Error: insufficient funds

Applying the same operation to an account twice in a row produces different results. Clearly, accounts are stateful objects.

Identity and Change

Assignment poses the new problem of deciding whether two expressions are "the same"

When one excludes assignments and one writes:

val x = E; val y = E

where E is an arbitrary expression, then it is reasonable to assume that x and y are the same. That is to say that we could have also written:

val x = E; val y = x

(This property is usually called referential transparency)

But once we allow the assignment, the two formulations are different. For example:

val x = new BankAccount
val y = new BankAccount

Are x and y the same?

Operational Equivalence

To respond to the last question, we must specify what is meant by “the same”.

The precise meaning of “being the same” is defined by the property of operational equivalence.

In a somewhat informal way, this property is stated as follows:

• Suppose we have two definitions x and y.
• x and y are operationally equivalent if no possible test can distinguish between them.

Testing for Operational Equivalence

To test if x and y are the same, we must

• Execute the definitions followed by an arbitrary sequence S of operations that involves x and y, observing the possible outcomes.

val x = new BankAccount
val y = new BankAccount
f(x, y)

• Then, execute the definitions with another sequence S' obtained by renaming all occurrences of y by x in S:

val x = new BankAccount
val y = new BankAccount
f(x, x)

• If the results are different, then the expressions x and y are certainly different.
• On the other hand, if all possible pairs of sequences (S, S') produce the same result, then x and y are the same.

Based on this definition, let's see if the expressions

val x = new BankAccount
val y = new BankAccount

Let's follow the definitions by a test sequence:

val x = new BankAccount
val y = new BankAccount
x deposit 30
y withdraw 20                // java.lang.Error: insufficient funds

Now rename all occurrences of y with x in this sequence. We obtain: