Lost in Technopolis

by John Wiegley

Monads in Pictures

Posted by John Wiegley on August 20, 2012 with labels:

This is not a tutoral on monads, nor will I use any math terms here. This is for people who have learned enough about monads to use them, but want to get a better picture of what they’re doing and why they exist.

Functions

One way to develop a first intuition about monads is to follow the progression of abstractions from functions to monads. Here is a simple picture of what a function does. I’ve put Haskell syntax for calling the function on top, and a graphical representation of the operation on the bottom:

Function use

Function use

A function maps some value a to another value, here shown as b. What happens along the way between input and output is anybody’s guess, although it’s usually some form of computation. My own programming background led me to view every function as requiring some kind of processing work, but that’s only one way to implement a function. Functions, in the abstract, are simply a way to go from one value to another.

Functors

The next step up the abstraction ladder is the Functor. Why do you need functors? Because sometimes you don’t just want to go from a to b. Instead, you know that a wraps (or contains, or provides) another value, and what you really want is to apply your function to “the inner value of a”.

Lists do this. If you have a list of integers, there are times when you want to apply a function to the integers in the list rather than to the list itself. So Lists are great candidate for being Functors, which they are.

To represent the “context” around the value you really want, I’ve used brackets in the following diagrams. But these brackets do not mean lists, they just mean “context”:

Functor use

Functor use

Here we use the same function as before, only instead of mapping from a to b directly with a function call, the function fmap:

  1. “Unboxes” the value from the incoming Functor;

  2. Calls f to turn that value to a new value;

  3. “Boxes” the result in another Functor of the same shape and kind.

NOTE: Although I use physical metaphors here, like boxing, shape, etc., do not be misled into thinking of Functors as always being like physical things. It’s possible for a Functor to itself be a function, in which case the “context” models computation, rather than containership. The best way of thinking about a Functor depends entirely on how it’s implemented.

Monads

Believe it or not, Monads are just a simple tweak on Functors. Browsing the Internet might leave you thinking they are a highly specialized entity only properly understood by mathematicians. But the real truth is that if you grok Functors, you’re only one step away from comprehending Monads in all their glory.

Up above we said that fmap does three things: it unboxes a Functor, applies your function to the value that was in the box, and then boxes the result into a new Functor of the same shape and kind. This is the very soul of Functors.

Monads do almost exactly the same thing! They just make one little change: they don’t re-box the result value. Boxing the result still needs to happen, but that job gets moved from the Monad to your function.

Here’s a picture showing a Monad at work:

Monad use

Monad use

What?! This looks just like the Functor picture! Only look closely: instead of the grey application arrow going from a to b, it now goes from a to the context of b. Also, instead of calling fmap f [a], we use an infix function that swaps the arguments: [a] >>= f.

This is all that makes Monads special. But what the picture doesn’t tell you is why they’re awesome, and what the implications of such a change are.

Because our function now returns a new context, it can decide to change that context, a service Functors cannot provide. If I map a function over a Functor, the result is always a new Functor of the same shape and kind. But if I bind a function over a Monad (note the difference in terminology), the result can be a new Monad of the same kind but a different shape. It’s this potentional for difference that provides the power of Monads.

Consider this chain of Monadic binds:

Monad chain

Monad chain

At each step along this chain, context can change. It can be used to carry mutating state, a token stream, an auxiliary result value, an error code, etc. All because the functions involved now participate directly in the binding operation, by having job of boxing intermediate results in new monads.

This new responsibility can be a burden. You can’t bind a pure function over a Monad that knows nothing about the Monad. At the very least, you have to call liftM on your function, to get a new function that does know about the Monad. There are times when all this lifting, and having to be conscious of the “monadic context” can get wearisome.

But there you have it: Functions give you the ability to associate values; Functors give you the ability to associate values within contexts; and Monads let you carry that context through a sequence of binding operations.

Arrows

Sometimes, the Monad abstraction keeps its context in the wrong place: around the values passed between functions. There are times when you’d rather have context surround the operation, rather than the data. This is what Arrows provide. Put simply, they add context to the concept of value mapping (i.e., the service that functions provide, although this is not the only way to map values). In fact, any function can be turned into an arrow with the arr function.

Here is the function call from above again, this time upgraded to an Arrow operation:

Arrow use

Arrow use

Note the use of run<Arrow>. Each arrow provides its own method for executing it – or it may not expose this functionality at all. It’s quite possible for a library to provide completely opaque arrows, which only get executed under controlled conditions. Thus, the input and output types to an arrow are all the user of an arrow needs to know about. There could be all kinds of other information there, including other functions that get called when arrows of such type are composed.

So what can arrows be used for? Any time you want context passed around with your function. Take, for example, a database query function you want to pass to another function. Ordinarily (and thanks to lazy evaluation), you’d just invoke the query and pass the result, and the query would only happen if the results were actually needed. But what if the function needs to execute the query repeatedly? In that case, the callee must perform the query.

With regular functions, you’d need to pass both the query function and the database handle for it to execute the query on. Or you could use a reader Monad, infecting the code performing the query with knowledge of that Monad. What would be preferable would be to bundle that database handle with the query, creating an enriched query function that knows itself which database to talk to. Enter the arrow.

Even more interesting use cases for arrows typically involve rich compositions. You can take one arrow with context, and another arrow with context, and compose them in various ways to create a composed arrow with composed context. What that composition means depends entirely on the Arrow type involved.

Applicative Functors

As a bonus – though it probably won’t help you grok Monads any better – I want to mention Applicative Functors.

Applicatives upgrade our use of Functors in one special way: Whereas fmap only accepts functions that go from one value to another, Applicative lets you map functions that take any number of arguments over an equal number of Applicative Functors:

Applicative use

Applicative use

In this example, rather than applying an f that goes from a to b over a Functor that provides an a, we get to apply an f that takes four arguments over four separate Functors, all at once.

This isn’t the fully story of Applicatives, by any means, but it is the crux. The Control.Applicative module provides a lot of helper functions to help take advantage of currying, composing and sequencing applicative applications, similar to what you do with regular function applications. The key intuition is the ability to transform any function into a function that operates in the realm of Functors, no matter how many arguments it takes. Once you can do that, your functions can be made to operate freely within the context of Applicatives, without needing to knowing anything about that context.


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