Monad

So far we’ve looked at monoids and functors. The next algebraic data structure we’ll cover is a monad. If you’ve wondered what a monad is but never really understood it, this is the post for you. I am sure that you’ve used it without realizing it. So let’s get to it.

Definition

A monad has more structure than a functor. This means that you can call map on it and that it obeys all the functor laws. In addition, a monad has a flatMap function which you can use to chain monads together. In essence, monads represent units of computation that you can chain together and the result of this chaining is also a monad.

Let’s look at a few examples.

Example

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val first = List(1, 2)
val next = List(8, 9)

for {
  i <- first
  j <- next
}
yield(i * j)

The above code[1] uses a for comprehension to muliply elements of the list together. Under the hood, this gets translated to:

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first flatMap {
  f => next map {
    n => f * n
  }
}

The compiler is making use of the List monad to chain operations together. Let’s break this down. 

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next map {
  n => f * n
}

This part of the code will return a List since that is what calling map on a List does. Since we have two elements in first list, the result of mapping will generate two lists of two elements each. This isn’t what we want. We want a single list that combines the results together. 

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first flatMap {
  ...
}

The flattening of results is what flatMap does - it takes the two lists and squishes them into one.

Monad Laws

For something to be a monad, it has to obey the monadic laws. There’s three monad laws:

  1. Left identity
  2. Right identity
  3. Associativity

Left Identity

This law means that if we take a value, put it into a monad, and then flatMap it with a function f, that’s the same as simply applying the function f to the original value. Let’s see this in code:

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scala> def f(x: Int): List[Int] = { List(x * 2) }
f: (x: Int)List[Int]

// left identity
List(2).flatMap(f) == f(2)
res5: Boolean = true

Right Identity

This law means that if we take a monad, flatMap it, and within that flatMap we try to create a monad out of it, then that’s the same as original monad. Let’s see this in code:

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// right identity
scala> List(1, 2, 3).flatMap({ x => List(x) }) == List(1, 2, 3)
res6: Boolean = true

Let’s walkthrough this. The function to flatMap gets the elements of the original list, List(1, 2, 3), one-by-one. The result is List(List(1), List(2), List(3)). This is then flattened to create List(1, 2, 3), which is the original list.

Associativity

This law states that if we apply a chain of functions to our monad, that’s the same as the composition of all the functions. Let’s see this in code:

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scala> def f(x: Int): List[Int] = { List(x + 1) }
f: (x: Int)List[Int]

scala> def g(x: Int): List[Int] = { List(x + 1) }
g: (x: Int)List[Int]

scala> List(1, 2, 3).flatMap(f).flatMap(g) == List(1, 2, 3).flatMap(x => f(x).flatMap(g))
res8: Boolean = true

Conclusion

This brings us to the end of the post on monads and their laws. List isn’t the only monad in your arsenal. Options and Futures are monads, too. I suggest going ahead and constructing examples for monadic laws for them.