Unit 33: Monad

Learning Objectives

After this lecture, students should:

Understand what are functors and monads
Understand the laws that a functor and monad must obey and be able to verify them

Generalizing `Loggable<T>`

We have just created a class Loggable<T> with a flatMap method that allows us to operate on the value encapsulated inside, along with some "side information". Loggable<T> follows a pattern that we have seen many times before. We have seen this in Maybe<T> and Lazy<T>, and InfiniteList<T>. Each of these classes has:

an of method to initialize the value and side information.
have a flatMap method to update the value and side information.

Different classes above have different side information that is initialized, stored, and updated when we use the of and flatMap operations.

Maybe<T> stores the side information of whether the value is there or not there.
Lazy<T> stores the side information of whether the value has been evaluated or not.
InfiniteList<T> stores the side information that the values in the list may or may not be evaluated.
Loggable<T> stores the side information of a log describing the operations done on the value.

These classes that we wrote follow certain patterns that make them well behaved when we create them with of and chain them with flatMap. Such classes that are "well behaved" are examples of a programming construct called monads. A monad must follow three laws, to behave well. Let's examine the laws below.

Identity Laws

Before we list down the first and second laws formally, let's try to get some intuition over the desired behavior first.

The of method in a monad should behave like an identity. It creates a new monad by initializing our monad with a value and its side information. For instance, in our Loggable<T>,

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public static <T> Loggable<T> of(T value) {
  return new Loggable<>(value, "");
}

The logger is initialized with empty side information (e.g., empty string as a log message).

Now, let's consider the lambda that we wish to pass into flatMap -- such a lambda takes in a value, compute it, and wrap it in a "new" monad, together with the correponding side information. For instance,

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Loggable<Integer> incrWithLog(int x) {
  return new Loggable<>(incr(x), "incr " + x + "; ");
}

What should we expect when we take a fresh new monad Loggable.of(4) and call flatMap with a function incrWithLog? Since Loggable.of(4) is new with no operation performed on it yet, calling

1	`Loggable.of(4).flatMap(x -> incrWithLog(x))`

should just result in the same value exactly as calling incrWithLog(4). So, we expect that, after calling the above, we have a Loggable with a value 5 and a log message of "incr 4".

Our of method should not do anything extra to the value and side information -- it should simply wrap the value 4 into the Loggable. Our flatMap method should not do anything extra to the value and the side information, it should simply apply the given lambda expression to the value.

Now, suppose we take an instance of Loggable, called logger, that has already been operated on one or more times with flatMap, and contain some side information. What should we expect when we call:

1	`logger.flatMap(x -> Loggable.of(x))`

Since of should behave like an identity, it should not change the value or add extra side information. The flatMap above should do nothing and the expression above should be the same as logger.

What we have just described above is called the left identity law and the right identity law of monads. To be more general, let Monad be a type that is a monad and monad be an instance of it.

The left identity law says:

Monad.of(x).flatMap(x -> f(x)) must be the same as f(x)

The right identity law says:

monad.flatMap(x -> Monad.of(x)) must be the same as monad

Associative Law

Let's now go back to the original incr and abs functions for a moment. To compose the functions, we can write abs(incr(x)), explicitly one function after another. Or we can compose them as another function:

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int absIncr(int x) {
  return abs(incr(x));
}

and call it absIncr(x). The effects should be exactly the same. It does not matter if we group the functions together into another function before applying it to a value x.

Recall that after we build our Loggable class, we were able to compose the functions incr and abs by chaining the flatMap:

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Loggable.of(4)
      .flatMap(x -> incrWithLog(x))
      .flatMap(x -> absWithLog(x))

We should get the resulting value as abs(incr(4)), along with the appropriate log messages.

Another way to call incr and then abs is to write something like this:

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Loggable<Integer> absIncrWithLog(int x) {
  return incrWithLog(x).flatMap(y -> absWithLog(y));
}

We have composed the methods incrWithLog and absWithLog and grouped them under another method. Now, if we call:

1 2	`Loggable.of(4) .flatMap(x -> absIncrWithLog(x))`

The two expressions must have exactly the same effect on the value and its log message.

This example leads us to the third law of monads: regardless of how we group that calls to flatMap, their behaviour must be the same. This law is called the associative law. More formally, it says:

monad.flatMap(x -> f(x)).flatMap(x -> g(x)) must be the same as monad.flatMap(x -> f(x).flatMap(y -> g(y)))

A Counter Example

If our monads follow the laws above, we can safely write methods that receive a monad from others, operate on it, and return it to others. We can also safely create a monad and pass it to the clients to operate on. Our clients can then call our methods in any order and operate on the monads that we create, and the effect on its value and side information is as expected.

Let's try to make our Loggable misbehave a little. Suppose we change our Loggable<T> to be as follows:

// version 0.3 (NOT a monad)
class Loggable<T> {
  private final T value;
  private final String log;

  private Loggable(T value, String log) {
    this.value = value;
    this.log = log;
  }

  public static <T> Loggable<T> of(T value) {
    return new Loggable<>(value, "Logging starts: ");
  }

  public <R> Loggable<R> flatMap(Transformer<? super T, ? extends Loggable<? extends R>> transformer) {
    Loggable<? extends R> logger = transformer.transform(this.value);
     return new Loggable(logger.value, logger.log + this.log + "\n");
  }

  public String toString() {
    return "value: " + this.value + ", log: " + this.log;
  }
}

Our of adds a little initialization message. Our flatMap adds a little new line before appending with the given log message. Now, our Loggable<T> is not that well behaved anymore.

Suppose we have two methods foo and bar, both take in an x and perform a series of operations on x. Both returns us a Loggable instance on the final value and its log.

Loggable<Integer> foo(int x) {
  return Loggable.of(x)
      .flatMap(...)
      .flatMap(...)
       :
  ;
}
Loggable<Integer> bar(int x) {
  return Loggable.of(x)
      .flatMap(...)
      .flatMap(...)
       :
  ;
}

Now, we want to perform the sequence of operations done in foo, followed by the sequence of operations done in bar. So we called:

1	`foo(4).flatMap(x -> bar(x))`

We will find that the string "Logging starts" appears twice in our logs and there is now an extra blank line in the log file!

Functors

We will end this unit with a brief discussion on functors, another common abstraction in functional-style programming. A functor is a simpler construction than a monad in that it only ensures lambdas can be applied sequentially to the value, without worrying about side information.

Recall that when we build our Loggable<T> abstraction, we add a map that only updates the value but changes nothing to the side information. One can think of a functor as an abstraction that supports map.

A functor needs to adhere to two laws:

preserving identity: functor.map(x -> x) is the same as functor
preserving composition: functor.map(x -> f(x)).map(x -> g(x)) is the same as functor.map(x -> g(f(x)).

Our classes from cs2030s.fp, Lazy<T>, Maybe<T>, and InfiniteList<T> are functors as well.

Monads and Functors in Other Languages

Such abstractions are common in other languages. In Scala, for instance, the collections (list, set, map, etc.) are monads. In pure functional languages like Haskell, monads are one of the fundamental building blocks.