A Monad in Practicality: First-Class Failures

Published 08 December 2013

There are plenty of tutorials on what monads are out there, some times using fairly “interesting” (i.e.: weird) analogies. This is not one of them. Instead, here I’ll provide a walk through some practical use cases for specific monadic structures in the JavaScript land.

This article shows how the Maybe monad can be used for handling simple failure use cases, without the drawbacks associated with using a value like null (e.g.: NullPointerExceptions). It then extrapolates slightly from the simple case into complex failure scenarios where you want to track the reasons of the failures, and shows how these cases can be modelled in terms of the Either monad to achieve the same goals of the core exception handling functionality, but without the problems (e.g.: lack of compositionality, non-locality) associated with the try ... catch mechanism. Finally, it concludes with a variation of the Either monad called Validation (which is an applicative functor, rather than a monad), that can be used for aggregating failures in scenarios like schema validation, and how to write and compose abstract operations that can manipulate any monadic computation.

Table of Contents

1. Introduction

Failures are difficult, yet our applications tend to fail more than we would want them to. More so, failures in the presence of side-effects are specially dangerous, because we need to somehow revert the changes we’ve applied, but before we do that we need to know how much of the changes got applied, and what the correct state should be. The usual solution for impure failures is to use exceptions, JavaScript handles this through the try ... catch statement and Error objects, which is similar to what other mainstream programming languages, like Java, Python, and Ruby use.

The common usage of these mechanisms in mainstream programming, however, have a handful of problems:

Non locality

When you throw an exception, you leave the local stack and environment, and ends up God-knows-where. Maybe the recovering site will be able to handle the failure, maybe it won’t. In the latter case, your program will be running in an inconsistent state and as long as it continues to do so, bad things can happen.

Consider the case where you fail to handle a failure to connect with the database due to a temporary network unavailability, but the recover site happens to swallow the error/or is not able to react in a sensible manner. Your application goes up, and all data every user tries to save in your website is silently moved over to /dev/null.

Lack of compositionality

We want to compose computations to cut down the complexity of the application, but exceptions limit the amount of compositionality we can achieve. This happens because throwing and recovering from exceptions is a second-class — and side-effecty — construct, thus we need to write call-site specific code to compose computations that use such mechanisms.

Impaired reasoning

With side-effects, non-local exceptions and recovering, our code ends up with a confusing flow which is difficult to reason about, since now a small piece of code may affect several places, depending on how the exceptions are handled up the call stack.

Ad-hoc pattern matching galore

Due to the issue of non-locality, to properly control and recover from side-effectful exceptions, one needs to model all possible kinds of failure as specific sub-classes of the Error object, then re-throw all of the exceptions that don’t match a specific subclass in the recovering site. In JavaScript, there are two main problems with this approach: first one, most people don’t ever use specific subclasses of Error, thus you would have to PATTERN MATCH WITH REGULAR EXPRESSIONS ON NATURAL LANGUAGE, INITIALLY WRITTEN FOR THE (HUMAN) DEVELOPER in order to achieve this; and it leads to an unnecessary amount of code-bloat for deriving these classes.

What about monads, then?

Monads solve all of these problems, so they are a good fit for modelling failures that can be recovered from, in some way, programmatically. There’s no way to recover from a dead HDD, for example, thus your program should just fail as fast as possible in these cases.

This article describes how the Maybe, Either monads and the Validation applicative functor provide the necessary framework for modelling these kinds of computations and, more importantly, composing and abstracting over them, without impairing reasoning about the code. To do so, we use objects that follow the laws of algebraic structures defined in the Fantasy Land specification for JavaScript.

But what’s a monad? A monad is a wrapper for some computational context, which satisfies certain algebraic laws, and provides you a single operation to create a new monad by transforming the computational context from another monad. But don’t worry if you don’t get this now, monads are an intentionally abstract interface, so we can generalise everything! Which is why we’ll instead look at specific types of monads in this article.

2. Modelling Errors

Some computations might not be able to give you a response, but in most programming languages we still regard them as a computation where you provide some values, and get a new value back. But what happens when the computation can’t provide the value? You don’t know, unless you read the documentation or source code for that particular functionality.

Some monads allow one to make this kind of effect (potential failure) explicit, thus you’re always forced to acknowledge that things might go wrong when using the function. While this might sound like too much work at first, we should initially consider that by just making the effects of a computation explicit we gain astounding clarity about the code we’re reading — suddenly, the contracts of what a function may do are expressed in the code, rather than on the documentation. The code can’t get out of sync with itself!

None the less, since monads are a kind of structure that follows a standard representation and laws, we can easily write functions to abstract over any kind of monad — just as you could write a generic function that works in any kind of collection. This allows one to abstract over computations, avoiding the issue of repeating yourself over and over and over again.

2.1. Maybe Things Don’t Work

The simplest case of potential failure is a computation that may say: “Yes, I have a result and this is the result,” or “I am sorry, but I don’t have a result.” This effect is usually handled implicitly in major programming languages by returning something like Null. Of course this comes with its own well-known problems, but we can easily capture the effect using the Maybe monad, without the problems associated with null references!

Let’s consider a simple case: I want to extract the first item of a sequence. The problem is that sequence might not have any items, in which case it would not make sense for the operation to return any value. In JavaScript, if you use sequence[0] you’re always going to get undefined if the item doesn’t exist.

Furthermore, I want to combine the results of this applying operation to two different sequences, and some of these sequences might have no elements. A naive approach would be to just extract the first element and use the concatenation operator:

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// Array(a) -> a | undefined
function first(sequence) {
  return sequence[0]
}

var consonants = 'bcd'
var vowels     = 'aei'
var nothing    = []

var firstConsonant = first(consonants)
var firstVowel     = first(vowels)
var firstNothing   = first(nothing)

var combination1 = firstVowel + firstConsonant // 'ab', yey!
var combination2 = firstVowel + firstNothing   // 'aundefined', eugh!

Okay, so the naive approach does not work, because the sequence may have no items. One needs to check if they’ve got an answer before concatenating things (let’s disregard the fact that concatenating a string with something non-existent should have been a type error for now):

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if (firstVowel !== undefined && firstConsonant !== undefined) {
  combination1 = firstVowel + firstConsonant // 'ab', yey!
} else {
  combination1 = undefined
}

if (firstVowel !== undefined && firstNothing !== undefined) {
  combination2 = firstVowel + firstNothing // never happens
} else {
  combination2 = undefined
}

Okay, so now we have our code working greatly, but just look at how many checks we had to do just in order to combine two things! Let’s try modelling our operation in terms of the Maybe monad and see if we can get rid of all this cruft. A Maybe monad has two cases: Just(a) is a monad with the value a, and Nothing is a monad with no computational context — the null case.

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// Array(a) -> Maybe(a)
function first(sequence) {
  return sequence.length > 0?  Maybe.Just(sequence[0])
  :      /* otherwise */       Maybe.Nothing()
}

Now, for any sequence that we feed into the first function, we may either get an answer, or we may get no answer, and this is reflected on what we return from the function. We’ve made the effects (the potential failure) of this function explicit, but in doing so we’ve increased the amount of code we had to write slightly. Sadly, not only this, but our original code for combining things don’t work anymore!

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var consonants = 'bcd'
var vowels     = 'aei'
var nothing    = []

var firstConsonant = first(consonants)
var firstVowel     = first(vowels)
var firstNothing   = first(nothing)

var firstVowel + firstNothing   // doesn't make sense
var firstVowel + firstConsonant // doesn't give you 'ab'

We can’t really combine an answer with something that has no answer. It doesn’t make any sense. Likewise, we can’t straightforwardly combine firstNumber and firstLetter, because the addition operator doesn’t know how to handle a Maybe monad. Not being able to combine straight-forwardly an answer with something that doesn’t exist is a good idea, but we would of course like to have an operator that can work with the values of a maybe.

We can’t take the value out of the monad, however, so how do we combine things if we can’t extract their values? Well, every monad provides the chain operation, which allows a function to transform the value from one monad, and put the transformed value into another monad. If this sounds confusing, imagine that in this case we’ve got a cat into a box. We have no way of extracting the cat from the box, but we have a machine that will allow we to add a small top hat to the cat, and provide a new box of the same shape (lest the poor soul suffers) for it. This is basically the intuition for the following piece of code:

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// Monad(a), Monad(a) -> Monad(a)
function concatenate(monadA, monadB) {
  // We take the value of the `monadA`
  return monadA.chain(function(valueA) {
    // And the value of the `monadB`
    return monadB.chain(function(valueB) {
      // And place the concatenated value in a new monad
      // of the same type as the `monadB`
      //
      // The `of` operator allows us to put things inside
      // a monad.
      return monadB.of(valueA + valueB)
    })
  })
}

And finally, we can use the concatenate operation instead of the + operator:

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combination1 = concatenate(firstVowel, firstNothing)   // stays Nothing
combination2 = concatenate(firstVowel, firstConsonant) // Just('ab')

Great, our code is terse again, and we didn’t even have to do anything to propagate the failures when it doesn’t make sense to combine two things! Sounds like the right path to be on.

2.2. Interlude: chain-ing monads

You might have realised that I used two methods on the monad objects in the previous section, which I have hardly explained: chain and of. These are the two operations that all monads must implement to be considered a monad. More so, these operations need to follow a few algebraic laws to ensure that all monads can be composed without any edge case, or inconsistent behaviour.

Before I talk about the Either monad, it helps to keep in mind that monads are things that contain computational context (values, in most cases), and have one operation to manipulate some values of the monad (chain), and an operation to put values into a monad (of). These are the only two (low level) ways we can interact with the values, and what they do is highly dependent on the specific monad type. We can never interact with the values in a monad directly, because that would break the laws (and as you probably know if you’re of legal age in your country, breaking the laws tends to end up badly), however we can easily write any sort of high-level construct to manipulate the values just using these two functions.

Consider, for example, our concatenate operation. We’ve used chain twice, and in the second case, we used the of method to return a monad of the same type to the chain operation, since chain always expects you to return a monad. None the less, the second usage of chain is painfully similar with one operation you might be well familiar with: Array.map. Think about it, we’re just transforming the value by a function that returns a new value and placing it back in the monad! Let’s get a bit more abstract, then:

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// Monad(a), (a -> b) -> Monad(b)
function map(monad, transformation) {
  return monad.chain(function(value) {
    return monad.of(transformation(value))
  })
}

// Monad(a), Monad(a) -> Monad(a)
function concatenate(monadA, monadB) {
  return monadA.chain(function(valueA) {
    return map(monadB, function(valueB) {
      return valueA + valueB
    })
  })
}

Great, we got rid of one monadB.of call! But we could even abstract it further by realising that we just want a monad with the value computed from the value of two monads, a fairly common operation that’s called lift2M — The M stands for Monad, of course:

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// Monad(a), Monad(b), (a, b -> c) -> Monad(c)
function lift2M(monadA, monadB, transformation) {
  return monadA.chain(function(valueA) {
    return map(monadB, function(valueB) {
      return transformation(valueA, valueB)
    })
  })
}

// Monad(a), Monad(a) -> Monad(a)
function concatenate(monadA, monadB) {
  return lift2M(monadA, monadB, function(valueA, valueB) {
    return valueA + valueB
  })
}

2.3. You Either Succeed, or You Fail

While the Maybe monad is awesome for the simple cases, like “we want to find an item in a list, but it might not be there,” “we want to get the value associated with a key, but the key might not be there,” “we want to read a file, but the file might not be there.” It doesn’t really get us much farther than the “it might not be there” kind of failure.

Sometimes our failures might be a little more complex, they might require a little bit more of information to the developer, they might even encompass several different types of failures! We just can’t model these kinds of computations with the Maybe monad because the failure case doesn’t accept any additional information.

We clearly need a new monad for this: meet Either, the monad that can model a success and its associated value, or a failure and its associated value! And the best of all, since Either is a monad, we can seamlessly compose values using Either with the functions we’ve defined before for the Maybe monad.

To see how the Either monad can be useful, let’s consider the following scenario: I want to divide some integer by another integer, but one of them might be 0, and that would have been an error.

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var Fail  = Either.Left
var Right = Either.Right

// Int, Int -> Either(fError, Int)
function divide(a, b) {
  return b === 0?         Fail(new Error('Division by 0.'))
  :      /* otherwise */  Right(a / b)
}

Now we can use that function to safely divide numbers by other numbers:

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divide(4, 2)  // Right(2)
divide(5, 0)  // Left(Error('Division by 0.'))

And abusing the fact that the + operator in JavaScript can be used for either concatenating Strings or arithmetic addition, we’ve already got a function to sum 2 numbers in a monad:

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var add = concatenate // A little abuse of JavaScript's operator semantics :P

add(divide(4, 2), divide(9, 3)) // Right(5)
add(divide(3, 1), divide(4, 0)) // Left(Error('Division by 0.'))

Again, we didn’t have to do anything — no try/catch, no guards — and our failures got propagated automatically. And what’s better, because monads share a common interface, we can apply the functions we’ve defined for one monad to any type of monad whatsoever. Monads (and their friends) are the ultimate DRY tool!

Suppose now that we wanted to sum the result of dividing the elements of one list, by the elements of another list, granted both lists have the same number of elements. If we fail to achieve any of these things, we should provide a friendly error message to the user.

We can start by first defining a zip operation that takes two lists, and gives a list of pairs, where each index corresponds to the a pair of the elements of one list and the other, which is fairly straight-forward to define:

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// [a], [b] -> Either(Error, [(a, b)])
function zip(as, bs) {
  return as.length !== bs.length?
           Fail(new Error('Can\'t zip lists of different lengths.'))

  :      /* otherwise */
           Right(as.reduce(function(a, i) {
                             return [a, bs[i]]
                           }, []))
}

Now we can define an operation that takes a list of pairs, and returns a list of the result of dividing the first item in the pair by the second, which is also fairly straight-forward:

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// [(Int, Int)] -> [Either(Error, Int)]
function dividePairs(nss) {
  return nss.map(function(a, b) {
    return divide(a, b)
  })
}

And finally the sum of these numbers, which just folds over the list to perform the addition. Since all of the numbers are wrapped in an Either monad, we do need to use chain to perform operations on these numbers, and put them back in the monad, however:

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// [Either(Error, Int)] -> Either(Error, Int)
function sum(ns) {
  // We need to start from a Monad, but we can reuse our
  // previously defined `add` computation to work on
  // these new monads too!
  return ns.reduce(add, Right(0))
}

And putting it all together:

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var fives = [5, 10, 15, 20]
var odds  = [1, 3, 6, 9]
var alien = [3, 1, 0, 10, 2, 1]


map(zip(fives, odds), dividePairs).chain(sum)
// => Right(13.05...)
map(zip(fives, alien), dividePairs).chain(sum)
// => Left(Error('Can\'t zip lists of different lengths.'))
map(zip(fives, alien.slice(0, 4)), dividePairs).chain(sum)
// => Left(Error('Division by 0.')

2.4. Interlude: Recovering From Failures

The attentive reader would have noted that no errors were handled in the previous section, even though the scenario required us to display an error message to the user. There’s a reason this: as you might have noticed, the type for the Monad defines that they contain a thing of type a, and they pass this thing of type a over to the continuation fed to the chain method. The problem with the Either monad is that it has an a and a b!

We could solve this problem in two ways: both values could be projected into the chain method, wrapped in a tuple (a static list containing two elements — similar to what dividePairs works with), and the function would have to deal with both values. This would disallow us from using the concatenate function we defined for the Maybe monad, however, since that function expects to combine two things a, not a tuple of a and b.

Then there’s the approach that people usually use when implementing the Either monad: project only the successful values. This bias does pose a problem in our case because there are no rules for how to work with the value we did not project in the monad, so if we want to recover from failures we’ll need new operations that are not standardised for a monad.

My Either monad implementation, which is heavily based on Scalaz’s Disjunction provides an orElse function, which works similarly to the chain method, but projects the value of the failure, and keeps the successful values. Fantasy Land’s Either expects you to either use a catamorphism or swap the values and project with the chain method. Haskell expects you to pattern match on the algebraic data types if you want to handle the failures. And so on, and so forth. You should assume that anything that isn’t on the Fantasy Land specification refers to my monad implementations in this article.

A way to display the errors to the user would then involve using one of the library-defined methods to deal with the other failure case:

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map(zip(fives, alien), dividePairs)
  .chain(sum)
  .orElse(function(error) {
    console.log('Error when trying to sum the lists: ' + error.message)
  })

2.5. Sometimes You Fail More Than Once

One of the problems with sequencing actions with the monadic chain operation is that, as we’ve seen with the Either monad, they’re a fail-fast path, which means that the whole sequence of actions is abruptly finished with a failure in case any of the actions fail. Sometimes, however, you don’t want to sequence things in this fashion, but rather aggregate all of the failures and propagate them. A common use case for this is validating inputs, which is why our next algebraic structure is the Validation applicative functor.

A Validation applicative is almost exactly the same as the Either monad, with two differences: it has a vocabulary aimed towards error handling, with the Success and Failure constructors, rather than the generalised disjunction tags Left and Right in the Either monad; and it can propagate an aggregation of all failures through the Applicative Functor interface. Now, Applicative Functors are not something I’ll go in much detail in this article, but for the purposes of this article you can think about them as a list where every element is a function, with a map operation that, instead of mapping a function over the list, you apply (thus, applicative) the list of functions to an element or list of elements.

But let’s leave the theory aside for a second and talk about a scenario where this algebraic structure is useful: you have a sign up form where the user might provide a username and password, and this information should comply with a few rules. In this case, it would be rather annoying to have the user try signing up, then failing on the first failure, having the user correct that one and try again, and fail, correct and fail. It’s much better to report everything upfront.

For this scenario, we’ll have the following rules:

  • Usernames should contain only numbers and letters.
  • Passwords should be at least 6 characters long.
  • Passwords must contain at least one special character.

And to make things simpler, each of these rules will be encoded as a separate function, that returns a Validation monad depending on whether the input passes the rule or not:

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var Validation = require('applicatives.validation')
var Success = Validation.Success
var Failure = Validation.Failure

// String -> Validation([Error], String)
function isNameValid(name) {
  return /^[\d\w]+$/.test(name)?
           Success(name)
  :      /* otherwise */
           Failure([new Error('Username can\'t be empty.')])
}

// String -> Validation([Error], String)
function isPasswordLongEnough(password) {
  return password.length > 6?
           Success(password)
  :      /* otherwise */
           Failure([new Error('Password have to be at '
                             +'least 6 characters long.')])
}

// String -> Validation([Error], String)
function isPasswordStrongEnough(password) {
  return /[\W]/.test(password)?
           Success(password)
  :      /* otherwise */
           Failure(new Error(['Password must contain at '
                             +'least one special character.']))

And we can verify that our functions work, and provide us the correct results for whatever inputs we throw at them:

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isNameValid("")
// => Failure([Error: Username can't be empty.])
isNameValid("robotlolita")
// => Success(robotlolita)
isPasswordLongEnough("robot")
// => Failure([Error: Password have to be at least 6 characters long.])
isPasswordLongEnough("rosesarered")
// => Success(rosesarered)
isPasswordStrongEnough("rosesarered")
// => Failure([Error: Password must contain at least one special character.])
isPasswordStrongEnough("roses.are.red")
// => Success(roses.are.red)

Now, this is not much interesting. We could have done the very same with, say, the Either monad. The interesting part is when we start combining these things and aggregating the failures, rather than failing at the first thing that isn’t correct. So, for example, a password needs to attend two different conditions to be correct, and we would like the user to know all the things they need to fix up when things don’t go the way we expect.

As previously mentioned, the Validation applicative functor exposes the ap method which can be used to aggregate failures. If either of the operands for the ap method contain a validation error, then the error is propagated, just like in the monad. But if both sides contain a validation error, then both errors are combined using a semigroup — another algebraic structure which allows one to combine things. Lists are fairly straight-forward semigroups, so we’ll just use them here.

If both validations contain a success value, then the usual Applicative Functor rules apply, and the first validation is expected to have a function that will get applied to the second applicative. Our validations contain strings, however, so we can’t just apply a string to another string and expect anything sensible to happen. Instead what we can do is to start with a validation that contains a function that captures each value of the subsequent functions and returns a new function at each application: closures to the rescue!

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// String -> Validation(Array(Error), String)
function isPasswordValid(password) {
  return Success(function(a) {
           return function(b) {
             return password
           }
         }).ap(isPasswordLongEnough(password))
           .ap(isPasswordStrongEnough(password))
}

isPasswordValid("robot")
// => Failure([
//      Error: Password have to be at least 6 characters long.,
//      Error: Password must contain at least one special character.
//    ])
isPasswordValid("rosesarered")
// => Failure([Error: Password must contain at least one special character.])
isPasswordValid("roses.are.red")
// => Success(roses.are.red)

But it’s so tedious to define functions in this way, specially given the syntactical burden of function definitions in JavaScript. So, can we abstract all this cruft away? Wouldn’t it be so much better to just pass in function(a, b){ ... }? Oh, but there’s a concept in functional programming that allows us to do exactly this: currying.

So, currying is the act of transforming a function that takes multiple arguments at once, in a function that takes each argument one at a time, so if we say: curry(2, function(a, b){ return a + b }) we’re really saying: function(a){ return function(b){ return a + b }}. So, let’s define curry as one of our abstraction overlords:

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// Int, (a1, a2, ..., aN -> b) -> a1 -> a2 -> ... -> aN -> b
function curryN(n, f){
  return function _curryN(as) { return function() {
    var args = as.concat([].slice.call(arguments))
    return args.length < n?  _curryN(args)
    :      /* otherwise */   f.apply(null, args)
  }}([])
}

And finally, derive our new isPasswordValid and isAccountValid functions from this aggregate combinator:

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// String -> Validation(Array(Error), String)
function isPasswordValid(password) {
  return Success(curryN(2, function(){ return password }))
           .ap(isPasswordLongEnough(password))
           .ap(isPasswordStrongEnough(password))
}

// String, String -> Validation(Array(Error), [String, String])
function isAccountValid(name, password) {
  return Success(curryN(2, function(a, b){ return [a, b] }))
           .ap(isNameValid(name))
           .ap(isPasswordValid(password))
}

isAccountValid("", "")
// => Failure([
//      Error: Username can't be empty.,
//      Error: Password have to be at least 6 characters long.,
//      Error: Password must contain at least one special character.
//    ])
isAccountValid("someone", "password")
// => Failure(Error: Password must contain at least one special character.)
isAccountValid("robotlolita", "roses.are.red")
// => Success(robotlolita,roses.are.red)

3. Abstracting Computations

So far monads (and applicatives) have been proven to be a fairly reliable way to deal with failures, even allowing one to easily aggregate errors in computations with the Validation applicative functor. But the way they were dealt with in this article brought about quite a lot of syntactical noise, and we’ve had to define our own combinators every time we wanted to work with them in a slightly higher level.

The good thing about using monads (and other algebraic structures) is that we don’t need to care much about writing our own combinators. If anyone has written a combinator for a monad operation, you can use it with any monad whatsoever! We can even give a particular monad characteristics of another monad, using monad transformers, so that we could, for example, deal with a Monad(Array(a)) using the array methods directly on the wrapped value, regardless of the monad that has the array.

While the Fantasy Land specification is still young, and there aren’t as many combinator and abstraction libraries as in Haskell, there are a few libraries that provides the common combinators from Haskell’s Control.Applicative, and some monad transformers. Ideally, you would just require the abstractions you need and get done with your work with the least amount of code and complexity possible.

For example, if you have a list of monads and wants to sequence all of them, but you don’t want to lose yourself in a chain callback hell, just use the sequence operation on monads, which gives you a list with all the results of a sequence, regardless of which monad they come from — so you can even mix asynchronous and synchronous operations without having to do any work at all (the same can’t be said about Promises/A+ promise combinators, since they’re specific to Promises/A+):

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// This will give you a Future of an array of results
// And the actions will be performed sequentially.
sequence(Future, [doA(), doB(), doC(), doD(), doE()])
  .chain(function(results) {
    console.log(results.join('\n'))
  })

4. Conclusion

Even though the Fantasy Land specification for algebraic computations in JavaScript is still young, we can already get a lot out of it for free. More so, with monads and applicative functors we can easily model any sort of computational failure that we can recover from, without making our program flow difficult, and without losing our ability to compose those computations easily.

You can check out the example implementation of the things discussed in this article on this github repository.

5. Libraries

You can find implementations of the monads discussed in this article under the folktale organisation on Github. Bryan et al also provide their own implementations of common monads on the fantasyland organisation on Github. And you can look for several other implementations of algebraic libraries that are compatible with the Fantasy Land specification searching for “fantasy-land” on NPM.

Below I list some of my implementations for the Maybe and Either monads, and the Validation applicative functor, and a few combinator libraries:

Maybe
My implementation of the Maybe monad
Either
My implementation of the Either monad
Validation
My implementation of the Validation applicative functor
Control.Monads
My (still experimental) port of Haskell's [Control.Monad](http://hackage.haskell.org/package/base-4.6.0.1/docs/Control-Monad.html) package, which provides common combinators and operations to work with monads in a higher level.
Fantasy ArrayT
A monad transformer for monads containing JavaScript arrays.
Fantasy Combinators
A library providing common functional combinators — like compose, constant, flip, etc.

6. References and Additional Reading

Toposes, Triples and Theories, by M. Barr and C. Wells
You don't need to have a PhD in Category Theory to understand Monads, Functors and friends, but having an understanding of the basics in Category Theory really helps. M. Barr and C. Wells's book provides a fairly detailed description of the subject, and it's freely available online, although I have not finished reading the whole book yet.
Haskell Wiki on Category Theory
The Haskell Wiki page on Category Theory lists several resources that may help you to understand the concepts and foundations. Some of them are written with Haskell programmers in mind, some are purely mathematical and don't assume any previous knowledge in either Haskell or Category Theory.
Null References: The Billion Dollar Mistake, by Tony Hoare
Tony Hoare, who introduced Null references in the ALGOL programming language (and the Communicating Sequential Processes formalism, which is the basis for Go's concurrency mechanisms), talks about the issues of null references and how he consider the decision to be his "billion-dollar mistake."
Scalaz
Scalaz is a library providing purely functional data structures for functional programming. Most of my decisions for the APIs of my monadic structures are directly influenced by Scalaz's API.
Functor, Applicative Functors and Monoids chapter in LYAH
The Learn You A Haskell for Great Good! book has a fairly nice introduction to Functors, Applicative Functors, Monoids, and later on Monads and some other algebraic structures (like Zippers).
Brian Beckman: Don't Fear The Monad
Brian explains how the concept of a monad are already familiar for many programmers, by starting with functions, and generalising from monoids to monads. While this will not tell you how to use monads, and what they're good for, it'll certainly allow you to reason better about how everything fits together.
Fantasy Land Specification
The Fantasy Land specification that all monads in JavaScript should follow to allow interoperability and abstractions to be written and shared by everyone, for maximum DRY.
Applicative Programming, Disjoint Unions, Semigroups and Non-breaking Error Handling
Tony Morris provides an explanation of how Applicative Functors for a Disjoint Union (the Either monad) can be used to aggregate failures and sequencing computations without a fail-fast path, in Scala.

7. Changes and Acknowledgements

  • 16th December, 2013: Changed the article to point out that Validation is not a monad, but just an Applicative functor, since you can’t provide a true monad instance for it, as per Tony Morris, Bryan McKenna and Mauricio Scheffer clarifications. More information regarding the problem can be found in this Github issue, and in this thread on Scalaz’s discussion group.

  • 3rd October, 2014: Fixed the type of liftM2, as pointed out by Erwin in the comments.