 Mount St. Helens from Mount Adams, WA. July 2011

The three Monad laws may seem pretty abstract at first, but they’re quite practical. Let’s try to internalize the laws by running through two of Scala’s most popular monads and making sure they adhere. We could use something like ScalaCheck to more rigorously check these laws, but the purpose of this post is to help internalize them, so we’ll only be manually verifying them using our intuition.

1. Left identity: `return a >>= f ≡ f a`
2. Right identity: `m >>= return ≡ m`
3. Associativity: `(m >>= f) >>= g ≡ m >>= (\x -> f x >>= g)`

In Haskell, `return` is used to “inject a value into the monadic type”. In Scala, we do this via constructors (unless you’re using Scalaz, in which case you probably already know everything this post has to offer).

The `>>=` operator in Haskell corresponds to Scala’s `flatMap` method.

## List

The List Monad deals with the context of non-determinism—that is, it represents multiple values. When we run multiple lists through a sequence comprehension we end up with the all combinations of values from each list.

``````for (x <- List(1, 2, 3); y <- List('a', 'b', 'c')) yield (x, y)
=> List((1,a), (1,b), (1,c), (2,a), (2,b), (2,c), (3,a), (3,b), (3,c))
``````

Now the laws. Let’s setup two simple functions `f` and `g`, both of type ```Int => List[Int]```.

``````// Let f be a function that takes an Int and produces a List of its
// neighboring Ints along with itself:
val f: (Int => List[Int]) = x => List(x - 1, x, x + 1)

// Let g be a function that takes an Int x
// and produces a List containing +x and -x
val g: (Int => List[Int]) = x => List(x, -x)
``````

### Left identity

``````val a = 2
val lhs = List(a).flatMap(f)
=> List(1, 2, 3)

val rhs = f(a)
=> List(1, 2, 3)

lhs == rhs
=> true
``````

### Right identity

``````val m = List(2)

val lhs = m.flatMap(List(_))
=> List(2)

val rhs = m
=> List(2)

lhs == rhs
=> true
``````

### Associativity

``````val m = List(1, 2)

val lhs = m.flatMap(f).flatMap(g)
=> List(0, 0, 1, -1, 2, -2, 1, -1, 2, -2, 3, -3)
// Sidenote: now do you see what is meant by non-determinism?

val rhs = m.flatMap(x => f(x).flatMap(g))
=> List(0, 0, 1, -1, 2, -2, 1, -1, 2, -2, 3, -3)

lhs == rhs
=> true
``````

Looks good to me.

## Option

Let’s create new test functions `f` and `g` of type ```Int => Option[Int]```. Given the type signature, it’s natural to think of `f` and `g` as partial functions that are only defined on certain inputs.

``````// If x is not less than 10, return 2x
val f: (Int => Option[Int]) = x => if (x < 10) None else Some(x * 2)

// If x is reater than 50, return x + 1
val g: (Int => Option[Int]) = x => if (x > 50) Some(x + 1) else None
``````

For the sake of testing our laws, the implementations of these functions really don’t matter as long as the types line up.

### Left identity

``````val a = 30
val lhs = Option(a).flatMap(f)
=> Some(60)

val rhs = f(a)
=> Some(60)

lhs == rhs
=> true
``````

### Right identity

``````val m = Option(30)

val lhs = m.flatMap(Option(_))
=> Some(30)

val rhs = m
=> Some(30)

lhs == rhs
=> true
``````

### Associativity

``````val m = Option(30)

val lhs = m.flatMap(f).flatMap(g)
=> Some(61)

val rhs = m.flatMap(x => f(x).flatMap(g))
=> Some(61)

lhs == rhs
=> true
``````

## The end

I hope this post helped you internalize the Monad laws. If you need more practice, continue this exercise in your REPL for the `Try` and `Either` monads, or better yet: create your own Monad and verify that it obeys the laws! 