Functors, Applicatives, and Monads in Pictures (2013)

The reason that the Monad structure is interesting, in my view, is that it's simple and neatly captures the notion of a "computation" that we can compose in different ways.

The core useful operator for monads in Haskell is >>=. It has the following type:

    m a -> (a -> m b) -> m b

If you squint, this is like function application with a few extra m's thrown in. Here's normal function application for comparison:

    a -> (a -> b) -> b

So what is this extra m useful for? It's like a hole where we get to plug in some custom logic. In a sense, it lets us change what it means to "apply" a "function". This turns out to be useful for a whole bunch of things, not just IO. (Honestly, from a pedagogical standpoint, I think IO is a bit of a distraction!)

For example, take the Maybe type. It's Haskell's nullable: a Maybe a means you either have an a or Nothing.

    data Maybe a = Just a | Nothing

Remembering the signature of >>= above, what's the most natural way to implement "application"? If we break the function into cases, it becomes pretty straightforward. Here's the specialized function signature we want to implement:

    (>>=) :: Maybe a -> (a -> Maybe b) -> Maybe b
    x >>= f = ...

If x is Nothing then we don't have anything to pass into f so the whole result has to be Nothing. If x has a value, we can just get that value out and pass it into f normally, returning the final result.

    Nothing >>= f = Nothing
    Just x  >>= f = f x

(In case you're not familiar with Haskell syntax, the above is actually a valid definition of (>>=) for Maybe!)

For Maybe, being a monad gives us a standard way of working with values while automatically dealing with Nothing. It abstracts over repetitive null checking and lets us easily build up Maybe values based on other Maybe values.

Other examples of monads are the same in spirit. The list monad, for example, lets us handle any number of inputs in a way that's similar to Maybe. The State monad similarly lets us combine values while carrying along an implicit state internally.

The rest of the Monad structure (namely the return function and the laws) are just a formal way of codifying behavior behavior that's already intuitive.

So how does this all apply to doing input and output?

Well, the problem in Haskell is that it's a language of evaluating expressions at heart: executing effects makes no sense any more than it would in arithmetic. To work with effects we instead have a special, opaque type IO; normal expressions get evaluated to IO actions that can be run to produce the desired effect—namely the IO type.

Critically, the IO type does not have to be a monad. It could be completely self-contained and have custom functions for doing one action after the other. We could imagine something like:

    after :: IO a -> IO b -> IO b

which would let you run an IO statement then run a second one and only return the value of the last one. It's like an imperative block of code!

However, we would also like some way of using the results of an IO statement, perhaps assigning them to a name. We can't do this normally because the IO statements are run separately from expression evaluation. We'd have to have some sort of function that could take an IO value, unwrap it and do something with it. And how would we express an interface like this? With a normal function!

    doSomething :: IO a -> (a -> IO b) -> IO b

Hey, doesn't that look familiar? It's exactly (>>=)!

I'm hand-waving a bit again, but the rest of the monad structure comes up when you try to make sure after behaves consistently and intuitively.

So IO being a monad emerges naturally from the desire to be able to compose actions and depend on their results in a way that's separate from normal variable bindings and expression evaluation.

The causation here is important: it's not that IO is a monad, but rather the IO type (which could exist on its own) happens to naturally and usefully form a monad. But it does a lot of other things too, including some specific capabilities (like spawning threads) that are hard to generalize.

So my point, I suppose, is twofold: monads are useful for combining some notion of computation and IO happens to be an interesting example, but the fact that we wrap statements with external effects in a custom type called IO does not inextricably depend on the idea of a monad.

Did that explanation help? I wrote a blog post on a similar topic that might be interesting too: http://jelv.is/blog/Haskell-Monads-and-Purity

> wrap I/O while preserving functional purity

So I think that expanding your idea of what monads do might help. Monads enforce a sequencing, and let later computations in the sequence depend on the result of an earlier one. Have a look at the definition of the Monad instance for Maybe:

    instance  Monad Maybe  where
        return x            = Just x
        (Just x) >>= k      = k x
        Nothing  >>= _      = Nothing

Now consider (because it's a contrived but simple example) that you have some `Map` type (from Strings to Strings, just for convenience), a value of that type `myMap :: Map` and a function `lookup :: Map -> String -> Maybe String`. Let's do the equivalent of `myMap[myMap[myMap["a"]]]`:

    case lookup myMap "a" of
      Nothing -> Nothing
      Just v -> case lookup myMap v of
                  Nothing -> Nothing
                  Just v' -> lookup myMap v'

This pattern of 1. do a thing, 2. check its result, 3. feed the result into the next step of the computation is what's abstracted over by the monad. We can write the same lookup using `do`-notation:

    do
      v <- lookup myMap "a"
      v' <- lookup myMap v
      lookup myMap v'

If this is making sense, I'd suggest repeating the exercise with `Either`, which is often used to pass an error message on its `Left` constructor (bypassing the rest of the computation). Actual results are stored on the `Right` constructor. If that makes sense, then I would then look at `Reader` (which lets you do computations with some value (like an environmental context) at-hand, and then maybe `State`.

If all the functional stuff is clear, then I'd look at the `STM` monad, which implements Software Transactional Memory. STM is IO-like in the sense that you are manipulating shared state, but you only have a restricted set of tools to do it with - the type `STM a` means "a transaction that fiddles with some shared memory, then returns a value of type `a`". To actually execute the transaction, you have to turn it into an `IO` action using the function `atomically :: STM a -> IO a` and put it in a side-effecting computation somewhere.

Hopefully that clears things up a bit: `IO` is just a special case of this sequencing strategy, but the really cool things happen because we can define what sequencing computations means for different data types. I think this is what some haskellers mean when they say "monads let you overload semicolons".

> But that doesn't tell me what are they good for. We invent things for a reason, and I don't see the reason. Now I know the reason, it's been told to me many times (some way to wrap I/O while preserving functional purity),

That's an excessively narrow reason. Its certainly a factor of why monads are front-and-center in Haskell, given the goals of the language, but its not really the reason monads are interesting or useful. Monads (and Functors and Applicatives, as well, as more general constructs) are interesting constructs in programming because they are powerful abstractions that unite disparate, useful data types and which, therefore, allow code that works across those data types. They therefore enable library code in circumstances where, in languages without such abstractions, fill-in-the-blanks template code patterns would be required, so they promote code reuse over copy-and-paste coding.

That IO operations are among the things that can be represented by monads is certainly part of their usefulness, but if IO was all they were good for, they wouldn't be all that interesting.

You kind of just have to use them to “get it”, but here’s an analogy: if you write a data structure, you naturally want to make it “iterable” in your language’s usual way, so you can take advantage of a library of generic functions for working with iterable things.

Same goes for monads. If we have N data types and M functions, instead of writing N×M implementations, we can write just N monad instances + M generic implementations.

So it’s kind of tautological, but monads are basically useful because lots of useful things happen to form monads—exceptions, loggers, parsers, dependency injection, persistent state operations, continuations, futures, STM transactions, I/O actions, and so on.

If you can write a data type that represents an API, and implement a couple of interfaces, you get a complete, expressive EDSL for free.

With Facebook’s Haxl, for example, you can write ordinary serial-looking I/O code, and with just a few typeclass instances, instantly get concurrent/async data fetching without changing a single line of business logic. You can’t readily do that without the kind of first-class effects that monads provide.

Have you used jQuery? The way that selectors and method chaining works has a monadic flavor which is very convenient. In the alternate reality where haskell is running in your browser, if you were creating jQuery from scratch you would recognize that the thing you were making was monadic, you would make use of the large number of useful functions that fall out from Monad here: http://hackage.haskell.org/package/base-4.8.1.0/docs/Control..., and you would also recognize that much of what you needed in your jQuery API is already provided by those functions, so no need to re-implement things in an ad hoc way and force your users to learn new useless things. You might even find that what you're trying to create is a mashup, or "stack", of two or three different monadic things that have already been defined. You'd know that because your new jQuery structure obeys a couple simple rules that it is well-behaved and reasonable in the presence of those functions (and others that don't exist yet), and you may even notice some optimizations you can confidently make.

Whoever told you it has to do with IO and functional purity vastly oversimplified things. That is just one use case.

Monads take care of a very common computation pattern: wrapping and unwrapping data from a container to do stuff with it. This process is error-prone and leads to code bloat if done by hand whenever needed.

I mean, how often have you had to apply a function to every element in a vector of values? fmap takes care of that without having to go into the list, take a value one at a time, and apply the function you want to use.

Rather than have to unwrap your container to use functions, a monad offers the means to transform functions to use a monadic container as input instead. No wrapping or unwrapping by hand required; using bind, fmap, or ap(ply), you transform lots of functions to use the monadic container and build a seamless pipeline for data to traverse.

How often have you had to deal with a value that might or might not exist--e.g. searching a string for a substring's position? If you had that problem, you might say, well, I'm going to return an integer, but a negative number means no match. And then in all code thereafter, I have to check if the integer is negative and do different stuff. A simple Maybe/Optional monad would take care of all that for you.

How often have you had to write verbose output alongside a computation? You could pepper your code with print statements, but maybe you want that output controlled by some runtime parameter. Would you want to check at every print statement whether that parameter is true? Or you could use a Writer monad, pass along the verbose output through the computation, with an easy means to transform regular functions into using Writers, and then decide what to do with the output at the end.

Sure, it's not convenient to try to use monads if the language doesn't offer it, or if there's not a library to facilitate it (trust me, I know this; I implemented a slew of templates to use monads in C++).

But programmers do this stuff all the time: unwrap data, do stuff with it, pack it back up. They do this over and over, repeating themselves, and such repetitive code is a maintenance trap waiting to happen.

Monads help you avoid that trap, and they help you take your program and reduce it to "one thing in, one thing out." Now maybe those "one things" are containers, but linearizing the flow of data this way only helps make the program's concept easier to understand.

I tried covering it (and showing there's a better way) in this talk: https://www.youtube.com/watch?v=449j7oKQVkc

Railway oriented programming; Error handling in functional languages

https://www.youtube.com/watch?v=UvD1VjRvGIk

It's not about I/O. It's about being able to handle cross-cutting concerns like logging, or error handling, or dependency injection, or async, with ordinary refactor-safe functions rather than global variables. http://m50d.github.io/2013/01/16/generic-contexts.html was my effort.

The `Maybe` monad presented here lets you create a language without null references. That's good enough reason for me. Rust doesn't have HKTs like Haskell does, but it still uses the same things that are called monads in Haskell. They just don't want to call them monads.

because in haskell functions are curried,

  function_name :: param1 -> param2 -> return

is a function that if you only give it param1 it returns a function that takes an argument of param2Type and returns returnType. This way you can easily compose functions

Because you write functions like this in mathematics as well.

What we usually see is something like this:

    f(x) = y

But if you look at it carefully the function f is a mapping of the domain x to the range y (the arrow is f), written like so:

    x -> y

Of course, the domain, being a set of all permissible values, is the type. So we write the type instead:

    type1 -> type2

Everything in Haskell is a function, and pure functions only ever take ONE parameter. Therefore the commas make no sense. The name in front just aids in naming stuff.

If you have a function that takes 2 parameters, they'd have to be curried. How would you represent curried functions?

    f::type1 -> type2 -> type3 

To make it more concrete, let's look at add instead of f.

So we can define a function like this (let's use Python for its readability):

    def add(a, b): return a + b

But remember, in haskell, functions are pure. Meaning they only map one input to one output. In order to make this happen, we need to split the function up into parts that only take one parameter.

Let's start with the plus operator as a function (it is one in Haskell just made into an infix). To think about it, it'd be something like this:

    plus(a)(b)

Where plus(a) is defined as:

    def plus(a): return plusA

Hence the first part would become a curried function like so, which takes another parameter:

    plusA(b)

Where plusA() is defined as such:

    def plusA(b): return b + a # a is a constant

So if you look at it from the types it was being transformed from:

    plus() # takes a Real, returns plusA.

    plusA(___) # takes a Real, returns a Real. written as (Real -> Real)

So if you put it together, the function signature for plus() is:

plus() takes Real -

    plus :: Real->

plus() returns plusA (which is Real->Real) -

    plus :: Real-> (Real-> Real)

Or in other words we can write it as such

    add :: Real -> Real -> Real

A function has ONE parameter and ONE return value. The trick is that these parameters and returned values may be tuples or functions themselves.

So we can have a function which takes a single parameter which is a tuple :

    function_name :: (Param1Type, Param2Type) -> ReturnType

Or we can have a function which takes a single parameter and returns another function:

    function_name :: Param1Type -> (Param2Type -> ReturnType)

Using implied operator precedence, the later is written:

    function_name :: Param1Type -> Param2Type -> ReturnType

And has to be distinguished from a function which takes a function as parameter:

    function_name :: (Param1Type -> Param2Type) -> ReturnType

Two things to add to the other comments: the language is named after Haskell Curry, from whom we get the term "currying," which is what we use to describe "the process of transforming a function that takes multiple arguments into a function that takes just a single argument and returns another function if any arguments are still needed".

Secondly, note that in Haskell a space represents function application -- it's not just there to separate terms.

This stuff might seem crazy, but once you start using it for useful things you come to really miss it in other languages. :)

My take, with no pretension of being accurate:

Say we have a function with signature

    f :: Int -> Int -> Int -> Int

Let's say this just add those Ints together.

Now let's create another function by partially applying the initial function: f1 = f 42

f1 signature will be

    f1 :: Int -> Int -> Int

Kind of "consuming" the signature. Repeat this until using up all the parameters. You end up with a function with no parameters, which map to a single Int.

You could say that f3 -> Int (note, not a valid notation for the sake of example). No need for a different symbol.