Tuesday, September 18, 2007

Monads are Elephants Part 1

Introductions to monads are bit of cottage industry on the Internet. So I figured, "why buck tradition?" But this article will present Scala's way of dealing with monads.

An ancient parable goes that several blind men were experiencing their first elephant. "It's a tree," one said while wrapping his arms around its legs. "A large snake," another said while holding its trunk. A third said...um, something about a broom or a fan or whatever.

From this parable we can conclude this: the ancients believed that the visually impaired like to fondle large mammals. Fortunately we live in a more enlightened age.

We're also supposed to learn something about how our limitations can prevent us from grasping the whole picture and that we're all blind in some way. It's so Zen.

I think there's a third lesson to be learned - the opposite of the main intent: that it's possible to learn much about the big picture by getting a series of limited explanations. If you had never seen an elephant and people told you things like "it has legs as thick as tree trunks," "a nose like a snake," "a tail like a broom," "ears like fans," etc. then you'd soon have a pretty good understanding. Your conception wouldn't be perfect, but when you finally saw an elephant it would fit neatly into the mental picture you had slowly built up. Just as the elephant was about to step on you, you'd think "wow, the legs really are like trees."

Monads are Container Types

One of the most commonly used container types is List and we'll spend some time with it. I also mentioned Option in a previous article. As a reminder, an Option is always either Some(value) or None. It might not be clear how List and Option are related, but if you consider an Option as a stunted List that can only have 0 or 1 elements, it helps. Trees and Sets can also be monads. But remember that monads are elephants, so with some monads you may have to squint a bit to see them as containers.

Monads are parameterized. List is a useful concept, but you need to know what's in the List. A List of Strings (List[String]) is pretty different from a List of Ints (List[Int]). Obviously it can be useful to convert from one to the other. Which leads us to the next point.

Monads Support Higher Order Functions

A higher order function is a function that takes a function as a parameter or returns a function as a result. Monads are containers which have several higher order functions defined. Or, since we're talking about Scala, monads have several higher order methods.

One such method is map. If you know any functional languages then you're probably familiar with map in one form or another. The map method takes a function and applies it to each element in the container to return a new container. For instance

def double(x: Int) = 2 * x
val xs = List(1, 2, 3)
val doubles = xs map double
// or val doubles = xs map {2 * _}
assert(doubles == List(2, 4, 6))

Map does not change the kind of monad, but may change its parameterized type...

val one = Some(1)
val oneString = one map {_.toString}
assert(oneString == Some("1"))

Here the {_.toString} notation means that the toString method should be called on the element.

Monads are Combinable

Now let's say we have a configuration library that let's us fetch parameters. For any parameter we'll get back an Option[String] - in other words we may or may not get a string depending on whether the parameter is defined. Let's say we also have a function, stringToInt, which takes a String and returns Some[Int] if the string is parseable as an integer or None if it's not. If we try to combine them using map we run into trouble.

val opString : Option[String] = config fetchParam "MaxThreads"
def stringToInt(string:String) : Option[Int] = ...
val result = opString map stringToInt

Unfortunately, since we started with an Option and mapped its contained element with a function that results in another Option, the variable "result" is now an Option that contains an Option, ie result is an Option[Option[Int]]. That's probably not terribly useful in most cases.

To motivate a solution, imagine if instead of Option we'd used List and ended up with List[List[Int]]] - in other words a list containing some number of lists. Given that, we just need "flatten" - a function which takes a list of lists (List[List[A]]) and returns a single list (List[A]) by concatenating everything together.1

A flatten function for Option[Option[A]] works a bit differently.

def flatten[A](outer:Option[Option[A]]) : Option[A] = 
   outer match {
     case None => None 
     case Some(inner) => inner 
   }

If the outer option is None, then result is None. Otherwise the result is the inner Option.

These two flatten functions have similar signatures: they take an M[M[A]] and turn it into an M[A]. But the way they do it is quite different. Other monads would have their own ways of doing flatten - possibly quite sophisticated ways. This possible sophistication is why explanations of monads will often use "join" instead of "flatten." "Join" neatly indicates that some aspect of the outer monad may be combined (joined) with some aspect of the inner monad. I'll stick with "flatten," though, because it fits with our container analogy.

Now, Scala does not require you to write flatten explicitly. But it does require that each monad have a method called flatMap.2. What's flatMap? It's exactly what it sounds like: doing a map and then flattening the result.

class M[A] {
  private def flatten[B](x:M[M[B]]) : M[B] = ...
  def map[B](f: A => B) : M[B] = ...
  def flatMap[B](f: A => M[B]) : M[B] = flatten(map(f))
}

With that, we can revisit our problematic code...

val opString : Option[String] = config fetchParam "MaxThreads"
def stringToInt(string:String) : Option[Int] = ...
val result = opString flatMap stringToInt

Because of flatMap we end up with "result" being an Option[Int]. If we wanted, we could take result and flatMap it with a function from Int to Option[Foo]. And then we could faltMap that with a function from Foo to Option[Bar], etc.

If you're keeping score, many papers on monads use the word "bind" instead of "flatMap" and Haskell uses the ">>=" operator. It's all the same concept.

Monads Can Be Built In Different Ways

So we've seen how the flatMap method can be built using map. It's possible to go the other way: start with flatMap and create map based on it. In order to do so we need one more concept. In most papers on monads the concept is called "unit," in Haskell it's called "return." Scala is an object oriented language so the same concept might be called a single argument "constructor" or "factory." Basically, unit takes one value of type A and turns it into a monad of type M[A]. For List, unit(x) == List(x) and for Option, unit(x) == Some(x).

Scala does not require a separate "unit" function or method, and whether you write it or not is a matter of taste. In writing this version of map I'll explicitly write "unit" just to show how it fits into things.

class M[A](value: A) {
  private def unit[B] (value : B) = new M(value)
  def map[B](f: A => B) : M[B] = flatMap {x => unit(f(x))}
  def flatMap[B](f: A => M[B]) : M[B] = ...
}

In this version flatMap has to be built without reference to map or flatten - it will have to do both in one go. The interesting bit is map. It takes the function passed in (f) and turns it into a new function that is appropriate for flatMap. The new function looks like {x => unit(f(x))} meaning that first f is applied to x, then unit is applied to the result.

Conclusion for Part I

Scala monads must have map and flatMap methods. Map can be implemented via flatMap and a constructor or flatMap can be implemented via map and flatten.

flatMap is the heart of our elephantine beast. When you're new to monads, it may help to build at least the first version of a flatMap in terms of map and flatten. Map is usually pretty straight forward. Figuring out what makes sense for flatten is the hard part.

As you move into monads that aren't collections you may find that flatMap should be implemented first and map should be implemented based on it and unit.

In part 2 I'll cover Scala's syntactic sugar for monads. In part 3 I'll present the elephant's DNA: the monad laws. Finally, in part 4 I'll show a monad that's only barely a container. In the meantime, here's a cheat sheet for translating between computer science papers on monads, Haskell, and Scala.

GenericHaskellScala
Mdata M a
or
newtype M a
or
instance Monad (M a)
class M[A]
or
case class M[A]
or
trait M[A]
M aM aM[A]
unit vreturn vnew M(v)
or
M(v)
map f mfmap f mm map f
bind f mm >>= f
or
f =<< m
m flatMap f
joinjoinflatten
 dofor

Footnotes

1. The Scala standard library includes a flatten method on List. It's pretty slick, but to explain it I would have to go into implicit conversions which would be a significant distraction. The slick part is that flatten makes sense on List[List[A]] but not on List[A], yet Scala's flatten method is defined on all Lists while still being statically type checked.

2. I'm using a bit of shorthand here. Scala doesn't "require" any particular method names to make a monad. You can call your methods "germufaBitz" or "frizzleMuck". However, if you stick with map and flatMap then you'll be able to use Scala's "for comprehensions"

17 comments:

adriaan said...

Type constructor polymorphism, a fairly new addition to Scala (in 2.5) now supports defining monads more conveniently. For a real life example, see my first take on a library for variable binding that uses a state monad to generate fresh variable names. I intend to eventually integrate this with Scala's library of parser combinators, which are also monads.

Lionel said...

Excellent explanation that doesn't need you to already know the answer to make sense of the text.
I wait eagerly for the next part.
Beside a good proof of scale power and expressiveness.

Anonymous said...

I understand Monads as Computations, but understanding Monads as Containers is more difficult. Most articles on the topic discuss List as a Monad which is easy to understand but how does this result extend to other "collections" (for example, Tree etc)?

Chuck said...

If the blind men had said that an elephant had legs like tree trunks or a nose like a snake, that would be fine. The problem is that they said it the whole elephant was like a tree trunk or a snake or a fan. So when you're told to look out for a
"trunk/snake/fan/broom feeling thing", you wouldn't know where to begin. And that's where the analogies of Monads as spacesuits, assembly lines, containers, and so on really fall down. Call it an imperfect analogy or a leaky abstraction, but it's just a dissonant clamor of conflicting mental models that confuse understanding of the whole.

I finally "get" monads now that I see them as a collection of behaviors (i.e. their laws and operators), but I still find myself fighting the poor mental models I was saddled with from the start. Of course it also doesn't help that I have to fight the type system to do anything complex with Monads, but that's more Haskell's "fault" (or at least my beef with Haskell) than it is with monads qua monads.

Xanana Gusmao said...

In the table comparing theory, haskell & scala;

map f m

So m stands for "monad" ?

James Iry said...

Gang, sorry I've been away from the blog for awhile and so I'm a bit behind on my comments.

Adriaan, thanks for the links. I don't want to get too far down the path of Scala's type system. Trying to explain higher kinded types while explaining monads might cause sensory overload.

Lionel, thanks for the compliment. I do agree that Scala is powerful, but it's possible to implement monads in many languages. It's just that Scala has built in support for them.

I'll take the other two comments individually.

James Iry said...

Anonymous,

"Monads as Computations" is the more powerful understanding and I plan to work my articles in that direction. Unfortunately, it's not a very natural fit for collection classes in a (mostly) strict language like Scala. For them the container approach I outline works better.

Still, I'll give it a shot. If you can see List as a computation, then you might see generalized Tree as a computation that when given a traversal order returns a List (or other sequence type). The tree monad then is a computation that can be bound to tree producing functions in order to produce a new computation.

Hmmm, seems easier to think of a Tree monad based on map and flatten.

James Iry said...

chuck,

You're mostly right that once you "get" monads then most of the other analogies fall away. So my answer is another analogy ;-) Monad analogies are the scaffolding you use to create a tall building. Once the building can stand on it's own you don't need the scaffolding any more.

I think the exception is container. I think it still has some applicability even when you work with more computational monads like continuations, parser combinators and IO. It's just that, as I hint in the article, you have to squint more to see those as containers.

James Iry said...

Xanana,

Haskellers and mathemeticians both like to use short variable names. In this case the names are meant to be a reminder of the types that map expects.

Specifically, for "map f m", f is a function of the appropriate type and m is an instance of a monad. Technically, m could be any functor, but "map f f" wouldn't be very helpful in most cases. :-)

James Iry said...

I've made some updates to this article. First, I had planned on doing both "for" and the laws in one article but they each deserve their own treatment plus I've added a plan for a 4th installment. Second, I fixed the embarrassing and repeated use of "outter" instead of "outer." :) Third, I've posted a forward link to part 2.

Ricky Clarkson said...

The code in this suggests that 1.toString gives "one", when it gives "1".j

James Iry said...

LOL. Fixed.

Raoul Duke said...

many thanks for your article. i do believe it is helping.

Felix Palludan said...

I like the artivle, monads seem strange to me. I have no idea what it is (like you say) but I want to learn more for some reason. Love the way you motivate the subject, computer science doesnt have to be dry and dusty, i didn't realize when you got into the technical stuff :)

Dustin Ted Whitney said...

That was so much easier to understand than the description of monads on the scala website. Thank you!

Erik Westra said...

After 5 years this is still an awesome article. Thank you

Nishan said...

what exactly does "monad" term mean ?? is it another kind of an interface that must implement abstract methods?? In short , if some one asked what is "monad" , what would be the shortest answer?