# Synchronised Asynchronous Operation: My Journey to Category Theory

April 13, 2017

In the last few days I’ve been dabbled in the world of abstract algebra with the goal of understanding and make sense the domain that I’ve been doing at office. In the world of distributed system, parallelism and asynchrony is inevitable. In the server, we have things like NodeJS or Vert.x. All client app, like web, native mobile, are all built on top of the common theme of event loop and asynchronous event handling.

It does not take me long time to recognise that it happens everywhere. When I open the home screen of my app these things happen:

The app will build the state of the home screen by asynchronously fetching bits of data from product catalogue to get the product information. It also need something from product recommendation to decide where to put that information on screen. There are also banners from marketing services. To show that little number of item in the cart, it needs to get that data from order processing. Not to mention all the interaction and animation. Altogether build the state of the home screen.

What will happen if the marketing service is down, or timed out? What will happen if the internet is dead, or when it connected 3 seconds later. It is increasingly difficult to make sense how to build the home screen with all edge cases included. The code in MainActivity class becomes mishmash of state management and transition and hard to reason about without running it.

## Callbacks and their hell

Pattern like callbacks in Javascript starts to emerge which basically states:

For every operation in which the result can be a success or failure, provide an anonymous or named function for both success and failure.

Platforms like Web Browser, NodeJS, or even Vert.x prohibit long computation and blocking thread. I/O will take sometime to finish, so rather than blocking the whole event loop, we submit operations and give success and failure callbacks when it happens. Consider this pseudocode

function fetchFromDatabase(connection, success, failure) {
connection.doQuery("SELECT FROM Order WHERE id= ?",
113243, success, failure);
}


So far so good, but how about after getting successful result, we want to also publish to the event queue and consider successful if all of the operation succeeded.

function fetchFromDatabaseAndSendToQueue(connection, queue, success, failure) {
connection.doquery("SELECT FROM Order WHERE id =?",
113243,
function(result) {
queue.send(result, success, failure)
},
failure);
}


We see that on every synchronous operation, We need to go one level deeper. The more operations we want, the deeper the nesting and indentation is. This is known affectionally by programmers as the callback hell.

## One Step to the Promise-land

To fix this problem ECMAScript 6 introduces Promise. Promise is a structure to express an asynchronous operation. It says that it will solve the callback hell.

let timeoutPromise = new Promise((resolve, reject) => {
setTimeout(() => resolve("success"), 250);
});

timeOutPromise.then((msg) => console.log(msg));


Seems promising, the arrow function syntax on ES6 helps so much in cleaning up the syntax. However because Promise is modelled after operation, it’s still hard to compose them. Say we want some synchronous and asynchronous operation as follows:

Let get files from directory, open and process them, and then write its result.

read_dir('/home/Documets')
.then(files => Promise.all(files.map(file => {
.then(data => write_file(data.name, convert(data.data)))
})))
.then(() => console.log("success"),
e => console.log("error", e))


If we want to do writing after we succeeded listing all the files, then you’ll have nested function. The more imperative the domain is the Promise will be in deep then structure. For example: if we try to retry on error, we’d need to put our retry on the reject part of the then. The same thing if we want to transform the result of the promise and send them to different operation. There’s no escape from hell. We just move from one kind of hell to another.

## Ascension to Functional Nirvana

Let’s digress a little bit to a structure that we know and love: array. In ECMAScript 6 we can do some basic operations operation on them:

  const numbers = [1, 2, 4, 88, 97, 22, 11];

// this will create new list
// of square of all elements
const squared = numbers.map( x => x * x );

// this will create new list with odd numbers
const odd     = numbers.filter( x => x % 2 == 1);

// this will sum the whole array to one number
const sum     = numbers.reduce( (a, x) => a + x, 0);


The method of Array.prototype.map(), Array.prototype.filter(), and Array.prototype.reduce() create a new Array:

• Map creates a new array with each element the function $f(x) = x^2$ denoted by closure of x => x * x applied.
• Filter creates a new array with only elements that matches the predicate $x \mathbin{mod} 2 = 1$ denoted by closure of x => x % 2 == 1;
• Reduce apply sum function of $f(a, x) = a + x$ with $a$ is the accumulator and $x$ is the current value. This will reduce the the whole list to one, single value. This operation also known as Folding .

This way we can combine and compose their operation.

  // this will sum the squared odd numbers
const sumoddsq = numbers
.filter( x => x % 2 == 1 )
.map(x => x * x)
.reduce( (a, x) => a + x, 0);


Which combine map, filter, and reduce to one very nice fluentFluent method means that readibility of source code is close to the written prose, in this case it can be read as Given an array of numbers, let's filter the odd, square all of them, and then sum the whole thing method. Most operations to collection are mapping, filtering, and reducing of some sort. They remove the needs of having loops on your API, make the code more readable and best of all they are pure function.

If we’re going to have a free function of map, filter, and reduce instead of prototypical ones, the operation can be translated as follows.

reduce(
map(
filter(numbers, x => x % 2 == 1),
x => x * x),
(a, x) => a + x, 0)


It’s a function composition! I’m going to wear my math hat, ignoring the parameters other than numbers and if filter is $f$, map is $m$, and reduce is $r$ the composition is similar this expression.

If we draw nice diagram to build intuition for the next attempt, it will be like this:

We transform an Array of Int to Array of Int, and then Array of Int to Array of Int, and finally Array of Int to single value of Int. In other word, we transform values (Int) inside another type (Array) to new values inside another type and finally turn that value out from that type.

## Making sense of the operation attempt

A diagram showing operation that will yield a success value or a failure
Back to our original computation modeled in both callbacks and Promises. It seems to me that they have a common pattern:

A computation that takes an input and will later returns a success with some value or failure with an exception.

Let’s say we want to chain said operation by only picking up the success value. For example an operation of fetching User Id from an HTTP endpoint and then use that information to get the order of that user. We can draw the operation roughly as follows.

Thinking like this, we suddenly find that our operation is actually composable. Ignoring the errors and operations, what we did was converting one type to another. This is the basic of Map that we have learnt before.

If we imagine them to be values without operation, we basically compose two map operations, from UserName to User and from User to Order. The mapping function goes from UserName to User and then to Order.

However, because they are operations and not a simple value mapping, the $\circ$ compose operator is not enough. Lets write the math

We have problem here. How we create a composition to unwrap $User$ from $UserLogin$ so our operation compose? If don’t do that, we’re stuck with those two operations and to do the operation in serial, we need to shove operation $G$ inside the $UserLogin$ type. Let’s Shove it!

The notation is actually not correct, but for the sake of building intuition, let’s just let it be. I just want to model the callback. It’s actually shoving the next operation inside the callback of previous operation and this does not compose well. If we need new operation, we need to shove it inside $OrderRequest$ and so on. What was that? Yes, It’s the callback/promise hell. We’re going back to square one. Why can’t we compose operations as easy as transforming list?

But worry no more, let’s solve this by defining operator that will unwrap the value from previous operation and feed that the next operator and let’s name it flatMap with this symbol $\rightarrowtail$ for our made-up operation. Suddenly, our operation compose!

This composable model does not exist in Javascript but exists in Java 8 with CompletableFuture and Rx with Observable. If you ever use those library, you know how easy it is to compose operations. CompletableFuture has thenCompose method, and Rx has flatMap method.

  // this returns CompletableFuture
.thenComposeAsync(user -> orderRepo.findByUser(user))
.thenRunAsync(order -> log.debug("Order fetched: " + order.id);

// this returns Observable from RxJava
.flatMap(user -> rxOrderRepo.findByUser(user))
.subscribe(order -> log.debug("Order fetched: " + order.id);


Here, findByName and findByUser returns CompletableFuture or Observable depends on the library used. The code is nicely composable and easy to read.

## Back To Category Theory (ish)

So we’re here, discovering pattern as we went through our journey of writing more composable and readable code. So far we have recognised two patterns. First, applying function on top of value inside some context (in this case array) and second, composing operations by taking out the value inside some context and (in our case HTTP/Database call) and then use that to do another independent operation. These patterns happens a lot of time.

This is where I start to search for “higher level language” to use those patterns on my app. A language that can be used as modelling language for my problem domain. I’ve heard Category Theory and Lambda Calculus for sometime, but I avoid it. Turns out, they are what exactly I need to model my problem. I started to follow and watch Erik Meijer’s excellent talks like this one:

It turns out collection mapping and composable operation is two concepts in Category Theory called Functor and Monad . Those two are so common in programs, that I don’t even think about their name. In essence, I’m discovering the ‘Category of Programming’ bottom-up by recognizing and deducing patterns.

### Category

So let go into Category Theory. By definition a Category is a simple collections with three components: a collection of objects , a collection of morphism , a notion of compositon of said morphism, and an identity.

Morphism ties two objects together. Sometimes, it also called arrow . Given a source object $X$ and target object $Y$, a morphism $f$ defined as $f: X \rightarrow Y$.

Category Theory as collection of objects, morphism, identity, and composition of morphism with its laws
With collection of objects and morphisms, there are laws regarding category. For each Category $\mathscr C$:

A Composition between morphism of $f:X \rightarrow Y$ and $g: Y \rightarrow Z$, there must be a morphism denoted by $g \circ f: X \rightarrow Z$.

For every object there’s a morphism $id$ such that $id_X$ is the Identity of object $Y$ so that $id_Y \circ f = f$.

The composition of morphisms is Associative so that $(h \circ g) \circ f = h \circ (g \circ f)$

For programmer, Objects in Category Theory can be considered the same as Types. Morphism is the function between data types. This will help us to model our problem into categories within a problem domain.

Transforming, or morphing, between types are common when we’re creating our application. The example above we transform from a HTTP Request to User DTO to User Domain Object to Screen State. So basically we’re moving from one types to another, from one domain to another. A change of price in the catalogue can affect the order, and so on. The understanding of Category Theory may at least make us able to model and reason our problem domain.

## Summary

I’ll conclude my blog here. I hope this helps you discover the merit of having some knowledge of Category Theory to make a better software. I actually write this article just after reading and following lectures. So, if I made mistake, please tell me in the comments.