A Layman's Guide to Applicatives in ReasonML
Other posts in this series:
Applicative functors
I'll start this post off with a tantalizing quote that I first heard from a former colleague/mentor Kris Nuttycombe:
In functional programming, applicatives are the essence of parallel processing, and monads are the essence of sequential processing.
In this post about applicative functors (aka applicatives), and my next planned post about monads, I hope to dig into this notion, and try to impart some intuition as to why this is true.
When I first learned about monads and applicatives, it took me longer to grok
applicatives, even though monads are an abstraction built on top of
applicatives. I was more used to the idea of monads from using things like
flatMap
on lists and arrays (even though flatMap
for list/array doesn't
really impart the feeling of sequential processing), and from using various
async types like Promises and Futures. Also, when writing basic application
code, you tend to do a lot of sequential processing (in both OO/imperative
and FP styles), which tend to relate more to monadic concepts  you often
have workflows where you compute a value, then use that value in the next
computation, and so on. Additionally, the apply
/ap
/<*>
function (which
we'll soon see) is not something you tend to run into as much in its base
form, so it's not as immediately recognizable. However, you do see
applicative behavior when using things like JavaScript's Promise.all
function, where you pass in an array of Promises
and get back and array of
results, assuming all the promises succeed, or a failed Promise
if any of
them fail.
The apply
/ap
/<*>
function itself is just so strangely simple yet
mysterious, and to me, it was not immediately obvious how this humble
function lends itself to parallel processing.
The plan
I'm going to approach the topic of applicatives in the way that I think would have helped me to more quickly understand and appreciate it:
 Introduce the
apply
/ap
/<*>
function and theAPPLY
typeclass to show what it is, how simple it is, and to immediately dispel any notions of magic  Introduce the
pure
function and the fullonAPPLICATIVE
typeclass  Mention why
apply
andpure
are in separate typeclasses  Show some example implementations of
APPLICATIVE
, to reiterate how simple and unmagical it is  Show how
apply
andpure
relate toFUNCTOR
'smap
 Demonstrate how
map
andapply
/ap
/<*>
functions can be reformulated into functions that makes the parallel capability more clear (tuple2
,map2
, etc.) Short but detailed walkthrough of how
apply
works
 Short but detailed walkthrough of how
 Introduce the map/ap/ap pattern, as I like to call it
 Show the variation on the map/ap/ap pattern: pure(f)/ap/ap
 Talk about applicative “effects”
 Talk about
APPLY
extensions  Talk about applicative validation
 Talk about the
APPLICATIVE
laws
Applicative Programming with Effects paper
Before we jump in, I'll just put a link to the paper where I believe the concept of applicative functors was first introduced in 2007. The paper is called Applicative Programming with Effects by Conor McBride and Ross Paterson. As far as academic papers go, it's very readable, so I'd recommend checking it out at some point. That said, if you're new to FP, it's important to remember that with academic papers you may not understand much or any of it initially. If that's the case, don't worry about it  try to plant a few seeds then come back to it in the future. If you're never are able to understand it, that's okay too! You don't need to understand the full scope of everything to make use of the parts you do understand, and you don't need to fully grok all the underlying math and academics. FP is not an allornothing paradigm.
APPLY typeclass
The APPLY
typeclass is an extension of the FUNCTOR
typeclass that I
covered in my blog post about
functors Being an
extension of FUNCTOR
simply means that APPLY
is everything that FUNCTOR
is, and must abide by the same laws as FUNCTOR
, but it also adds something
new to the mix, along with some new laws. APPLY
adds one new function which
is often called apply
, ap
, or as an operator <*>
. I'll try to use the
name apply
here, but might also refer to ap
or <*>
. I'll cover the laws
at the end of the article to avoid getting lost before we even get started.
We can define the APPLY
typeclass as a ReasonML module type like this:
module type APPLY = {
type t('a);
let map : ( 'a => 'b, t('a)) => t('b);
let apply : (t('a => 'b), t('a)) => t('b);
};
As you can see, APPLY
has the type t('a)
and the map
function we know
from FUNCTOR
, and adds the new function apply
. The apply
function is
the only difference compared to FUNCTOR
.
Using the module include
mechanism
from OCaml/ReasonML, we can define APPLY
like this too:
module type FUNCTOR = {
type t('a);
let map: ('a => 'b, t('a)) => t('b);
};
module type APPLY = {
include FUNCTOR;
let apply: (t('a => 'b), t('a)) => t('b);
};
The include
here more clearly illustrates the relationship between
FUNCTOR
and APPLY
. If you haven't seen include
, it's basically like a
modulelevel copy or spread  it takes whatever's inside the module you're
including
and spreads it into the module that has the include
. So in this
case, we're including the type t('a)
and let map = ...
from FUNCTOR
,
and then we just add our apply
after that.
If you looked closely at the first example, you
might have noticed that I aligned the 'a
and 'b
parameters to
illustrate a similarity between map
and apply
:
let map : ( 'a => 'b, t('a)) => t('b);
let apply : (t('a => 'b), t('a)) => t('b);
map
applies a pure function to a value that's inside a functor context,
while apply
can apply a function that's inside a functor context to a
value that's inside another functor context. This curiously simple function
is what unlocks the power of parallel processing, but if you don't see it
yet, that's okay, I didn't either!
APPLICATIVE typeclass
I'm going to introduce one more small concept now, because it's quite simple
and it makes sense to just explain it together with apply
. This new concept
is the pure
function. All pure
does it take a pure value of type
'a
, and stick it in a functor context t('a)
. I say “pure value” to
differentiate it from an “effectful value,” which is discussed later.
let pure: 'a => t('a);
Here, 'a
can be any type of value you want  the key point is that our
other functions just see it as an 'a
 they don't know nor care what it is.
The APPLICATIVE
typeclass is simply an extension of APPLY
that adds this pure
function, and its corresponding laws:
module type APPLICATIVE = {
// From FUNCTOR
type t('a);
let map: ('a => 'b, t('a)) => t('b);
// FROM APPLY
let apply: (t('a => 'b), t('a)) => t('b);
// For APPLICATIVE
let pure: 'a => t('a);
};
Or we can write it using include
like this:
module type FUNCTOR = {
type t('a);
let map: ('a => 'b, t('a)) => t('b);
};
module type APPLY = {
include FUNCTOR;
let apply: (t('a => 'b), t('a)) => t('b);
};
module type APPLICATIVE = {
include APPLY;
let pure: 'a => t('a);
};
pure
is a simple, but interesting function in that we now have the ability
to actually put a value into our functor context, whereas before, we could
only operate on values that were already in the context, using map
and
apply
. Also note that we do not have a way to get a value out of our
applicative functor context once it's in there, i.e. we don't have a t('a) => 'a
function here.
APPLY and APPLICATIVE historical notes
For some historical context from Haskell, I believe that when the applicative
functor was first identified as a distinct abstraction, the key parts had
already been sort of “identified” as functions or concepts in the Monad
typeclass. Some work has been done to split off an Applicative
typeclass,
which includes pure
and <*>
(the apply
operator). Languages and FP
libraries that are newer and that don't have the burden of maintaining
backwards compatibility seem to be separating the concept of APPLY
and
APPLICATIVE
up front. The reason to split these is that there are things
that can conform to APPLY
, but not APPLICATIVE
, so it makes sense to keep
those abstractions separate.
For the rest of the article, we're going to just focus on APPLICATIVE
as a
whole because it's useful to have both apply
and pure
at our disposal.
However, this is a good time to mention that when faced with a problem, you
should always try to follow the rule of least
power. Use the
abstraction that does what you need with the least amount of power  this
makes your code more abstract, which means there are fewer ways for it to do
the wrong thing or be used incorrectly, and makes it more general so that
more things can use it, or be used with it.
Let's see what APPLICATIVE
looks like in some real examples:
Option applicative
Let's implement APPLICATIVE
for option('a)
. This one is pretty easy, you
just “follow the types.” I'm going to implement all the functions at the
toplevel of Option
for convenience, then just alias the functions in the
typeclass instances. I'm also going to use include
to deal with the
hierarchy from FUNCTOR
up to APPLICATIVE
. include
works for both module
types and modules, so we can use it in both the typeclasses and the
instances.
module type FUNCTOR = {
type t('a);
let map: ('a => 'b, t('a)) => t('b);
};
module type APPLY = {
include FUNCTOR;
let apply: (t('a => 'b), t('a)) => t('b);
};
module type APPLICATIVE = {
include APPLY;
let pure: 'a => t('a);
};
module Option = {
type t(a') = option('a);
let map = (aToB: 'a => 'b, optionA: option('a)) => switch(optionA) {
 Some(a) => Some(aToB(a));
 None => None;
};
let apply = (optionAToB: option('a => 'b), optionA: option('a)) =>
switch(optionAToB, optionA) {
 (Some(aToB), Some(a)) => Some(aToB(a))
 (Some(aToB), None) => None
 (None, Some(a)) => None
 (None, None) => None
};
let pure = a => Some(a);
module Functor: FUNCTOR with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let map = map;
};
module Apply: APPLY with type t('a) = t('a) = {
include Functor;
let apply = apply;
};
module Applicative: APPLICATIVE with type t('a) = t('a) = {
include Apply;
let pure = pure;
};
};
apply
is pretty straightforward, because there aren't too many ways of
implementing it. You could of course just return None
in all cases, but
that would violate the APPLY
laws, which I'll cover at the end of the
article. You can only ever get a 'b
value if you have both the 'a => 'b
function and the 'a
value, so the cases where we only have the function or
only the value, or neither just have to return None
.
As for pure
, it just takes a value and sticks it in Some
.
Once I have map
, apply
, and pure
, I can just create my instance modules
Functor
, Apply
, and Applicative
by aliasing the functions in the right
places. As you can see, I'm defining each typeclass separately, even though
Applicative
can do everything Apply
and Functor
can do. In ReasonML,
this is not strictly necessary to do, but I like to do it anyway, because it
helps to quickly identify what a particular type can do, and it serves as a
constant reminder of what each typeclass does  kind of like incode,
typechecked documentation about a type.
In fact, the Option
module itself actually conforms to APPLICATIVE
already because it has all the necessary parts, so it's not even strictly
necessary to define Functor
, Apply
and Applicative
, but I like to do it
anyway to be explicit, and it feels more like Haskell/PureScript typeclass
instances. Also, I like to annotate the module types, just to make it clear
what my intentions are with these modules, even though the compiler can infer
the module types.
That's it for option
!
Js.Promise applicative
Now let's try implementing apply
for a more complex type: Js.Promise.t('a)
.
First think about the type signature, and substitute Js.Promise.t('a)
for t('a)
:
let apply: (t('a => 'b), t('a)) => t('b);
let apply: (Js.Promise.t('a => 'b), Js.Promise.t('a)) => Js.Promise.t('b)
If you've worked with promises, it's probably not too hard to see how this is
implemented  you just need to wait for the 'a => 'b
function and the 'a
value, and then resolve with the function applied to the value.
One quick observation here is that in theory, we should be able to fire off
both of these promises at the same time, as they are not dependent on
oneanother  the promise of 'a => 'b
doesn't care about the promise of
'a
and vice versa  neither of them need to wait for the other to do its
job. Our apply
function has to wait for both of these promises, because we
need the 'a => 'b
function and the 'a
value at the same time to use them
together to get the 'b
, but the input promises themselves are independent.
That said, these promises have already been “fired off” before we even get
our hands on them in apply
, because that's what JS promises do  they start
running as soon as you construct them! So in apply
we get two
alreadyrunning “hot” promises, and we just need to wait for both of them to
finish.
module Promise = {
let apply = (promiseAToB: Js.Promise.t('a => 'b), promiseA: Js.Promise.t('a)) => {
promiseAToB
> Js.Promise.then_(aToB =>
promiseA > Js.Promise.then_(a => Js.Promise.resolve(aToB(a)))
);
};
};
We're using Js.Promise.then_
here to just wait for the promise of the 'a => 'b
function to resolve, then we wait for the promise of the 'a
value to
resolve, and then we finally resolve the chain with aToB(a)
to get our 'b
value. This chain might look like we're making this operation sequential, but
remember that both promises are already running when we get them so the inner
promise might finish before the outer, but it doesn't matter to us. We need
them both to resolve before we can get our 'b
value. If we were
constructing the inner promise here, it would matter, but the promise was
already constructed and running when we got it.
Another thing to note is that if either of the Js.Promises
fail, the
apply
function will fail. Again, we can only succeed if both inputs succeed
and give us our 'a => 'b
function and our 'a
value. With applicatives, we
actually have a few different options for handling errors, but here I'm going
to keep it simple and just fail fast using the default failure mechanism of
chained Promises
.
Example usage:
Promise.apply(Js.Promise.resolve(a => a * 2), Js.Promise.resolve(42))
> Js.Promise.then_(a => Js.Promise.resolve(Js.log(a)));
// 84
We could have also “cheated” and just used Js.Promise.all2
here, because
someone has already implemented for us:
let apply = (promiseAToB, promiseA) =>
Js.Promise.all2((promiseAToB, promiseA))
> Js.Promise.then_(((aToB, a)) => Js.Promise.resolve(aToB(a)));
This isn't actually cheating  it's perfectly fine to use functions that
already exist to implement typeclasses, assuming they follow the laws.
Speaking of all2
, we'll soon see how to implement this ourselves for any
applicative, not just Js.Promise
, and we'll also see that by implementing
apply
and pure
for a type, we can get a ton of other stuff for free!
As for pure
, we just need to take a value of type 'a
and get it into the
Js.Promise
context, so we can use Js.Promise.resolve
.
Here is the full Promise
module:
module Promise = {
type t('a) = Js.Promise.t('a);
let map = (aToB, promiseA) => {
promiseA > Js.Promise.then_(a => Js.Promise.resolve(aToB(a)));
};
let apply = (promiseAToB: Js.Promise.t('a => 'b), promiseA: Js.Promise.t('a)) => {
promiseAToB
> Js.Promise.then_(aToB =>
promiseA > Js.Promise.then_(a => Js.Promise.resolve(aToB(a)))
);
};
let pure = a => Js.Promise.resolve(a);
module Functor: FUNCTOR with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let map = map;
};
module Apply: APPLY with type t('a) = t('a) = {
include Functor;
let apply = apply;
};
module Applicative: APPLICATIVE with type t('a) = t('a) = {
include Apply;
let pure = pure;
};
};
The implementation of apply
for Js.Promise
gives us a first glimpse as to
why applicatives are associated with parallel processing  we get two
independent “effectful values” (the promises), and we wait for both of them
to finish independently before we use them to do our final computation. I'll
talk about what “effectful values” means a little later.
List/array/tree applicative
The types list('a)
, array('a)
, and other “multivalue data types” like
binary trees, etc. have applicative instances, but I'm going to skip these in
this article, as I don't personally think they are immediately helpful in
gaining an initial intuition about applicatives.
If you think about the signature of apply
for a list('a)
, you'd get the
following:
let apply = (list('a => 'b), list('a)) => list('b);
Basically, you have a list of functions and a list of values, and you need to
apply some or all of the functions to some or all the values. You can
probably imagine how you might end up with a cartesian product where all the
functions are applied to all the values to produce a new longer list of
unobvious utility. The same idea applies for array('a)
. For a binary tree,
imagine a binary tree of functions Tree.t('a => 'b)
applied to a binary
tree of values Tree.t('a)
. These applicatives can be implemented, but it's
not something you run into as much in daytoday use, so I'll not get into
these at this time, but you are welcome to give it a shot yourself!
Result applicative
Now let's try to implement APPLICATIVE
for Result.t('a, 'e)
. Since
FUNCTOR
/APPLY
/APPLICATIVE
want to operate on a type t('a)
, we'll use
the module functor trick (see the functors
article) again
to “lockin” our error type:
module type TYPE = {
type t;
};
module type FUNCTOR = {
type t('a);
let map: ('a => 'b, t('a)) => t('b);
};
module type APPLY = {
include FUNCTOR;
let apply: (t('a => 'b), t('a)) => t('b);
};
module type APPLICATIVE = {
include APPLY;
let pure: 'a => t('a);
};
module Result = {
type t('a, 'e) =  Ok('a)  Error('e);
let map = (aToB, resultA) => switch(resultA) {
 Ok(a) => Ok(aToB(a))
 Error(e) => Error(e)
};
let apply = (resultAToB, resultA) => switch((resultAToB, resultA)) {
 (Ok(aToB), Ok(a)) => Ok(aToB(a))
 (Error(e1), Ok(a)) => Error(e1)
 (Ok(aToB), Error(e2)) => Error(e2)
 (Error(e1), Error(e2)) => Error(e1) // !!!
};
let pure = a => Ok(a);
module WithError = (E: TYPE) => {
module Functor: FUNCTOR with type t('a) = t('a, E.t) = {
type nonrec t('a) = t('a, E.t);
let map = map;
};
module Apply: APPLY with type t('a) = t('a, E.t) = {
include Functor;
let apply = apply;
};
module Applicative: APPLICATIVE with type t('a) = t('a, E.t) = {
include Apply;
let pure = pure;
};
};
};
The implementation of pure
is obvious because there's only one way to turn
a value of type 'a
into a Result.t('a, 'e)
; however, the implementation
of apply
poses an interesting conundrum in the  (Error(e1), Error(e2))
case  we have two errors (of the same type 'e
), but we need to return a
single error value in the resulting Error('e)
constructor. We don't know
what the 'e
value is, so we don't have enough information to know how to
pick one error or the other, or combine them, so we'll just pick one side
(e1
in this case), and fail the function. Just like with option
and
Js.Promise
, the only way we can succeed is in the  (Ok(aToB), Ok(a))
case  we need both the function and the value in order to produce the result
of type 'b
with Ok(aToB(a))
.
We'll leave Result
like this for now, but we'll revisit this error handling
issue later in the section on “applicative validation”.
Function applicative
As we saw in the functor
article, we
could implement FUNCTOR
for the function of type 'x => 'a
. Let's see how
to implement APPLICATIVE
for this type:
module Function = {
type t('x, 'a) = 'x => 'a;
let map = (aToB: 'a => 'b, xToA: 'x => 'a): ('x => 'b) => {
x => aToB(xToA(x));
};
let apply = (xToAToB: 'x => ('a => 'b), xToA: 'x => 'a): ('x => 'b) => {
x => {
let aToB = xToAToB(x);
let a = xToA(x);
aToB(a);
};
};
let pure = (a: 'a): ('x => 'a) => {
_ => a;
};
module WithArgument = (X: TYPE) => {
module Functor: FUNCTOR with type t('a) = t(X.t, 'a) = {
type nonrec t('a) = t(X.t, 'a);
let map = map;
};
module Apply: APPLY with type t('a) = t(X.t, 'a) = {
include Functor;
let apply = apply;
};
module Applicative: APPLICATIVE with type t('a) = t(X.t, 'a) = {
include Apply;
let pure = pure;
};
};
};
The apply
function here is given a function 'x => ('a => 'b)
, and a
function 'x => 'a
, and needs to return a function 'x => 'b
. So in this
resulting function, we are given an 'x
, and need to return a 'b
. To do
this, we use the 'x
to get our 'a => 'b
function from the first argument,
and the same 'x
to get our 'a
value from the second argument, and just
apply the function. This is a great exercise in “following the types”  we
don't know upfront what we're doing, but we just look at the types and work
it out. If you do this in FP, you'll often find that there's only one valid
way to actually implement something, like in this case.
The pure
function is just given a value 'a
and needs to return a function 'x => 'a
,
but we already have our 'a
, so we just return a function that throws away the input
and returns our 'a
. This function is commonly called const
:
let const = (a, _) => a;
This looks funny in ReasonML syntax, because it looks like a function that
takes an a
argument and another ignored argument, and then just returns the
a
, but if you think about it in terms of partial application, if you supply
the a
argument, you now have a function _ => a
. This is useful for
creating a function that just produces a constant value regardless of the
input, hence the name const
. It's another one of those utilities like
identity: 'a => 'a
that is so trivially simple, but sometimes it's exactly
what you need.
Overall, the usefulness of this APPLICATIVE
instance for 'x => 'a
is not
immediately obvious, but we'll explore it more when we talk about the reader
monad.
Also, I wanted to show this as an example of an APPLICATIVE
that's not just
a static data value, to demonstrate that these concepts can apply to
functions too.
JSON decoder applicative
Let's show a more realworld example of an applicative: the JSON decoder function we saw in the functor article. I'll just jump right into it, and explain after, but I encourage you to try it yourself too.
module Decoder = {
module Error = {
type t =  ExpectedBool(Js.Json.t)  ExpectedString(Js.Json.t)  Other;
};
type t('a) =  Decode(Js.Json.t => Result.t('a, Error.t));
let map = (aToB, (Decode(jsonToResultA))) => {
Decode(json => jsonToResultA(json) > Result.map(aToB));
};
let apply = (Decode(jsonToResultAToB), Decode(jsonToResultA)) => {
Decode(json => {
let resultAToB: Result.t('a => 'b, Error.t) = jsonToResultAToB(json);
let resultA: Result.t('a, Error.t) = jsonToResultA(json);
Result.apply(resultAToB, resultA);
})
};
let pure = a => Decode(_ => Result.pure(a));
module Functor: FUNCTOR with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let map = map;
};
module Apply: APPLY with type t('a) = t('a) = {
include Functor;
let apply = apply;
};
module Applicative: APPLICATIVE with type t('a) = t('a) = {
include Apply;
let pure = pure;
};
};
The apply
function here feels a lot like the apply
function we saw for
'x => 'a
, and that's because it is very similar: Js.Json.t => Result.t('a, Error.t))
is of a similar form to 'x => 'a
, except the 'a
value is just
buried in another (applicative) type Result
. In Decoder.apply
we have a
decoder of a function, and a decoder of a value. Recall that a decoder
is just a function that takes a Js.Json.t
value and produces a Result
, so
we feed the input Js.Json.t
value into each decoder argument to get a
Result.t('a => 'b, Error.t)
and a Result.t('a, Error.t)
. Here we have a
function 'a => 'b
buried in a Result
and a value 'a
buried in a
Result
 so we can take advantage of the fact that Result
is also an
applicative functor, and just use Result.apply
to apply the wrapped
function to our wrapped value to get our Result.t('b, Error.t)
! Take note
that we've again not actually done any work at this point  no JSON has been
decoded  we've simply composed some functions to turn a decoder of 'a
into
a decoder of 'b
. Think about how you might do something like this in an
imperative or OO language  I'm pretty sure I would have done it in a much
less elegant way!
This example also serves to show that you don't always need to completely
understand what you're doing (or even the end result) to create the typeclass
instances for a type. As long as you can do it, the types line up, and your
implementation follows the laws, there's a good chance you did it correctly.
Sometimes all you need to do is just follow the types! We're kind of glossing
over the laws, and we haven't yet seen how to test the laws, but we'll
hopefully try that out in another blog post. The concept of a “decoder of a
function” 'a => 'b
doesn't make much intuitive sense, but we'll soon see
where this comes into play, and maybe it will become clear as to why we're
doing these things.
map in terms of apply and pure
We've now seen that APPLICATIVE
is an extension of FUNCTOR
 it adds the
functions apply
and pure
to the t('a)
and map
function we have from
FUNCTOR
. One interesting thing to note at this point is that you can actually
implement map
in terms of just apply
and pure
:
// (pseudocode)
let apply: (t('a => 'b), t('a)) => t('b) = ...;
let pure = 'a => t('a) = ...;
let map = (aToB, fa) => apply(pure(aToB), fa);
Here, our map
function is given a function 'a => 'b
and a value of type
t('a)
. We “lift” the function into our applicative context t
using
pure(aToB)
, which gives us a value of type t('a => 'b)
, then we apply
that “wrapped” function to our t('a)
value using apply
, because that's
what apply
does:
let apply: (t('a => 'b), t('a)) => t('b);
When we get into monads, we'll also see that you can implement apply
in
terms of the monadic function flatMap
(aka bind
or >>=
).
I just wanted to point this out, as sometimes it's actually easier or more convenient to implement one of the higherlevel functions for a type, and then just implement the lowerlevel functions in terms of the “more powerful” ones.
Reformulation of apply as tuple2
Let's now look at a reformulation of apply
that better demonstrates the
parallel nature of applicatives. If this section bends your brain too far,
I'd highly recommend just trying it for yourself  it can take some time and
practice for this to sink in, especially if you're not used to working with
functions with curried arguments.
Our goal will be to come up with a function of the following type, using only
the powers offered to us by APPLICATIVE
: t('a)
, map
, apply
, and
pure
, and nothing more. To be honest, we're actually just going to just use
APPLY
here  we don't actually need pure
for this.
let tuple2: (t('a), t('b)) => t(('a, 'b));
Given two “effectful” values t('a)
and t('b)
, let see if we can “run”
them both to get at the values 'a
and 'b
, then combine those values in a
“effectful” tuple. We're going to assume the result is also wrapped in our
applicative context t
, because given the types of map
, apply
, and
pure
, it doesn't look like we have any way to “get rid of” or “get out of”
our t
context  we can only put things into it using pure
, and operate on
things inside of our context using map
and apply
.
Let's start with a function that can create our tuple:
let makeTuple2 = (a, b) => (a, b);
Since we're in a curried language, another way to think about this function is like this:
let makeTuple2 = a => {
b => (a, b);
};
These are both the same thing, but the second form might help with the
intuition  given an 'a
, we can return a function 'b => ('a, 'b)
. The
first makeTuple2
can also do this just as well, but it's not as clear with
the ReasonML syntax of (a, b) => (a, b)
 this looks like a function of
arity 2 that just returns a tuple, but in reality, it's a curried function in
ReasonML  a function of a single argument that returns another function of a
single argument.
So back to our tuple2
challenge:
let tuple2: (t('a), t('b)) => t(('a, 'b));
We're given a t('a)
and a t('b)
, and we have a function makeTuple2: 'a => ('b => ('a, 'b))
. We have no way to get the 'a
value out of our
t('a)
, we can only operate on the value inside the context using map
and
apply
. We again have a value of the form t('a)
(and a t('b)
), but we
don't have a function of the form t('a => 'b)
so apply
doesn't seem
immediately applicable, but we do have our “pure” makeTuple2
function, so
let's just try the only thing that seems to make sense, map: ('a => 'b, t('a)) => t('b)
Let's quickly consider our function makeTuple2: 'a => ('b => ('a, 'b))
again.
If we write it like this, we can sort of fuzz the right hand side into an opaque
type like this:
'a => ('b => ('a, 'b));
'a => 'c
where 'c represents our 'b => ('a, 'b) function
Given this type 'c
, we can write our map
like this:
let map: ('a => 'c, t('a)) => t('c);
and if we substitute 'c
with what it actually is:
let map: ('a => 'c, , t('a)) => t('c );
let map: ('a => ('b => ('a, 'b)), t('a)) => t('b => ('a, 'b));
If it's not clear, I'm just trying to demonstrate what happens when you map a function of more than one argument over a functor whose type matches the type of the first argument of our function  you end up with a function inside your functor context. And this function has one fewer argument than what you started with.
Let's try mapping makeTuple2
over our fa
, which is a t('a)
:
let fa: t('a) = ...;
let fBToAB: t('b => ('a, 'b)) = map(makeTuple2, fa);
Our fBToAB
has the type t('b => ('a, 'b))
 we mapped a function of
multiple arguments over our functor and ended up with a function of one fewer
arguments inside our functor context. This ability to map
and apply
a
function of multiple arguments over a bunch of individual functor values is
the key to how applicatives work.
After this map
over the first value t('a)
, we now have a function of the
form t('x => 'y)
(actually t('b => ('a, 'b))
). map
can't deal with this
because map
wants a pure/nonwrapped function, but apply
knows what to do
with a wrapped function:
let fa: t('a) = ...;
let fb: t('b) = ...;
// map first  now we have a function inside a functor
let fBToAB: t('b => ('a, 'b)) = map(makeTuple2, fa);
// then apply this wrapped function to the wrapped 'b value
let fAB: t(('a, 'b)) = apply(fBToAB, fb);
We've now “filled” all the arguments of makeTuple2
, and we get our final
result  a tuple wrapped in our applicative context: t(('a, 'b))
.
To wrap up, we'll now write our tuple2
function in terms of map
and
apply
. (Stay tuned to see a cleaner, more intuitive way to do this
below):
let tuple2 = (fa: t('a), fb: t('b)) => {
let makeTuple2 = (a, b) => (a, b);
apply(map(makeTuple2, fa), fb);
};
Let's run through a quick demonstration of this using options to make it more concrete:
let makeTuple2 = (a, b) => (a, b);
let optionA = Some(42);
let optionB = Some("hi");
// Map the pure function on our optionA
// This results in a function 'b => ('a, 'b) **that is inside an option**
let optionBToTuple: option('b => (int, 'b)) = Option.map(makeTuple2, optionA);
// Now we have a function inside an option, and a value `b inside
// an option, and that's what apply deals with:
let optionTuple: option((int, string)) = Option.apply(optionBToTuple, optionB);
// optionTuple == Some((42, "hi"))
If you're not getting it just by looking, try working through it it yourself,
and look carefully at the types along the way. Basically, we're using map
to apply the pure function to our option('a)
for the first step, then using
apply
to apply a function (which is now inside an option
) to our
option('b)
.
One final note on this example: in the section where we implemented map
in
terms of apply
and pure
, we saw that if we lifted a function into the
applicative context using pure
, we could then just use apply
to apply the
function to our effectful value. We could also implement tuple2
in this
same style  we can lift the pure function into our context first, then just
apply
it, then apply
the resulting wrapped function to get the final
result:
let tuple2 = (fa: t('a), fb: t('b)) => {
let makeTuple2 = (a, b) => (a, b);
apply(apply(pure(makeTuple2), fa), fb);
};
Apply multiple times
Surprisingly, this pattern of map
and apply
works for any number of
arguments  we can just keep apply
ing the resulting functions to the next
values until we run out of arguments in our function and arrive at our final
result. That said, we will only succeed in getting the final result if each
step along the way is also successful. If any step fails, the whole
computation will fail, but we actually have some cool options for how we can
deal with errors.
If you think about how apply
is implemented, it gets a wrapped function,
which basically carries information about the previous computations, and a
wrapped value, which is the “current value” on which we want to operate. The
previous computations may have already failed, but apply
will still go
through the motion of considering each input value, regardless of what
happened in the past. That said, apply
doesn't give us the ability to
“recover” a computation that failed  we can't create a successful result
unless we have both the 'a => 'b
function and the 'a
value, so all we can
do is propagate the previous errors, or possibly add our own error to the mix
(which we'll see later). This inability to fork or recover a computation is
one of the reasons that applicatives are strictly less powerful than monads.
However, being less powerful sometimes has its advantages  for example,
because an applicative can't fork the flow of a computation (i.e. recover
from an error, or create some new processing branch), we are forced to feed
it all the information we want to process at once. Knowing the fullscope of
the problem upfront can sometimes unlock certain optimizations and allow us
to make certain assumptions about how the computation will occur.
Anyway, to demonstrate how to apply the pattern to functions of more
arguments, below is an example of the map
/apply
pattern being used with a
(curried) arity 3 function:
let makeTuple3 = (a, b, c) => (a, b, c);
let optionA = Some(42);
let optionB = Some("hi");
let optionC = Some(true);
let optionBToCToTuple: option(('b, 'c) => (int, 'b, 'c)) = Option.map(makeTuple3, optionA);
let optionCToTuple: option('c => (int, string, 'c)) = Option.apply(optionBToCToTuple, optionB);
let optionTuple: option((int, string, bool)) = Option.apply(optionCToTuple, optionC);
// optionTuple = Some((42, "hi", true))
More succinctly, we could write it like this:
let tuple3 = (fa: t('a), fb: t('b), fc: t('c)) => {
let f = (a, b, c) => (a, b, c);
apply(apply(map(f, fa), fb), fc);
};
// or using pure to immediately lift our function into the applicative context:
let tuple3 = (fa: t('a), fb: t('b), fc: t('c)) => {
let f = (a, b, c) => (a, b, c);
apply(apply(apply(pure(f), fa), fb), fc);
};
Again, this pattern works for any number of arguments  just keep apply
ing
until you “fill” all the arguments of your function. Try it yourself with
tuple4
and soon.
The tuple2: (t('a), t('b)) => t(('a, 'b))
function is sometimes called
product
, product2
or zip
, because it takes two effectful values and
combines them into the most basic product type  a tuple. I hope to write
about product and sum types in a later post. Looking at tuple2
it's likely
more clear as to why applicatives are associated with parallel operations 
we start with two independent “effectful” values t('a)
, and t('b)
, and we
produce a value t(('a, 'b))
that's a combination of our two inputs
(assuming they both “succeed”).
Well, that was my attempt at describing how applicatives work, but not sure
how successful it was. If you got lost somewhere along the way, I'd again
recommend trying it yourself with a concrete type like option
. It's hard to
explain, but once you slog through it enough times, you'll get it. Try
implementing these functions too, to see how the patterns expands to more
values:
let tuple4: (t('a), t('b), t('c), t('d)) => t(('a, 'b, 'c, 'd));
let tuple5: (t('a), t('b), t('c), t('d), t('e)) => t(('a, 'b, 'c, 'd, 'e));
// etc.
If you haven't recognized it yet, these look at lot like the
Js.Promise.allN
functions, but there's no specific mention of Js.Promise
here  we're just using the functions from our APPLY
abstraction.
map2, map3, etc.
Now let's generalize our tuple2
, tuple3
, etc. functions. Recall the defintion
we made for tuple2
:
let tuple2 = (fa: t('a), fb: t('b)) => {
let makeTuple2 = (a, b) => (a, b);
apply(map(makeTuple2, fa), fb);
};
The makeTuple2
here is just a function that basically takes two inputs, and
“combines” them into something else. We can allow the caller to pass in this
function, so they can combine the values however they want, rather than
having to deal with a tuple. We'll call this function map2
:
let map2 = (f: ('a, 'b) => 'c, fa: t('a), fb: t('b)): t('c) => {
apply(map(f, fa), fb);
};
We call this map2
because it looks a lot like map
 it applies a pure
function to some independent values that are each inside their own
functor context.
let map: ('a => 'b, t('a)) => t('b);
let map2: (('a, 'b) => 'c, t('a), t('b)) => t('c);
It's important to note that the result of both of these is still inside our
functor context t
 we don't have a way to get it out of that context. You
can also create map3
, map4
, etc. in the same way. These functions are
sometimes called liftA2
, liftA3
, etc., as they act to “lift” a pure
function ('a, 'b) => 'c
into the applicative context (t('a), t('b)) => t('c)
(using the power of the APPLY
typeclass).
For one final side note, we can also now implement our tuple2
function in terms of
map2
, and the same applies for map3
/tuple3
, etc.
let tuple2 = (fa: t('a), fb: t('b)): t(('a, 'b)) => map2((a, b) => (a, b), fa, fb);
These mapN
functions are useful for running a bunch of independent
applicative effects and combining the results however we please.
The map/ap/ap pattern
Let's get super fancy, and cast off our fear of weird operators, and do the above using some Haskellstyle infix operators. Once you grok the pattern (and memorize the operators), the infixbased approach actually becomes quite beautiful. The usual caveat with operators applies  they add a level of opacity and abstraction that can be quite hostile to newcomers, so it's best to introduce them with some handholding, and not just dump them on people without the prerequisite setup.
Let's first define an operator for map. We're going to use <$>
because it's
the conventional operator for map
in many other FP languages, and it's
useful to start to recognize it for what it is.
let map: ('a => 'b, t('a)) => t('b) = ...;
let (<$>) = map;
Remember that the function is on the left, and the effectful value on the right, so you'd use this like so:
let f = a => a * 2;
let fa = Some(42);
// The following are all the same:
let _ = f <$> fa;
let _ = map(f, fa);
let _ = fa > map(f);
Now let's define the operator <*>
as apply
, again the “function” part is
on the left (even though the function for apply
is wrapped), and the
effectful value is on the right. Again, we're using <*>
for apply
,
because that's what many other languages do, and it helps to get used to it.
let apply: (t('a => 'b), t('a)) => t('b) = ...;
let (<*>) = apply;
Now let's take a look back at our map3
function:
let map3 = (f: ('a, 'b, 'c) => 'd, fa: t('a), fb: t('b), fc: t('c)): t('d) => {
apply(apply(map(f, fa), fb), fc);
};
Infix operators let you move the name of a function between the two
arguments, so we could write this like below  just start from the innermost
function application (map
) and move the name of the function between the
args, and work your way out:
apply(apply(map(f, fa), fb), fc);
apply(apply(f `map` fa), fb, fc);
apply(f `map` fa `apply` fb, fc);
f `map` fa `apply` fb `apply` fc;
Now just replace map
with <$>
and apply
with <*>
:
let map3 = (f, fa, fb, fc) => f <$> fa <*> fb <*> fc;
If you read it lefttoright, we start with our function f
, and we map it
over our first effectful value fa: t('a)
, so now we have a wrapped
function, which we apply to our next effectful value fb: t('b)
, and so on.
The two things below are the same thing  one just uses infix operators, and
the other uses normal named functions:
let _ = f <$> fa <*> fb <*> fc;
let _ = apply(apply(map(f, fa), fb), fc);
The operators work for any number of arguments, so we could implement map4
,
map5
, etc. like this too:
let map4 = (f, fa, fb, fc, fd) => f <$> fa <*> fb <*> fc <*> fd;
let map5 = (f, fa, fb, fc, fd, fe) => f <$> fa <*> fb <*> fc <*> fd <*> fe;
// etc.
This approach of applying a pure function to a series of “effectful” values
is very common in FP languages like Haskell and PureScript, so you'll likely
run into this at some point. I like to call this this map/ap/ap pattern,
because you just start with a pure function, you map
it over the first
value, then you just ap
it over all the remaining values.
You may run into the alternate form of this pattern, like this:
let map3 = (f, fa, fb, fc) => pure(f) <*> fa <*> fb <*> fc;
We've seen before how we can lift a pure function into our applicative
context using pure
and then just apply
it, so this is just another way of
achieving the same thing as map/ap/ap. The difference is that we
immediately lift our function into the applicative context, so we can't use
map
(<$>
) to apply it the first time  we have to go straight to apply
(<*>
) for the first application, and for all the rest.
We'll see some more concrete uses of these operators below in “applicative validation”.
Applicative “effects”
Let's take a break from code to talk about the concept of “effects” and
“effectful” values. These concepts are a bit overloaded in functional
programming, so let's take a look at a few examples. Let's assume our
functions can't throw exceptions, and we're in a language that doesn't have
the concept of a null
value.
The function 'a => 'b
is an example of what appears to be a
pure/noneffectful function. Based on the types, there's no indication that
the function can fail to produce a value or fail for any other reason.
There's also no indication that the computation will be asynchronous  it
appears to just map an argument of type 'a
to a result value of type 'b
.
Let's now consider the function 'a => option('b)
. In this case, we are now
made aware (via the type system) that this function might either produce a
value of type 'b
(Some(b)
), or might fail to produce a value (None
). In
terms of “effects” we can say that this function has the “effect” of being
unable to produce a value in some, or possibly all cases. This isn't a “side
effect” like writing to STDOUT or reading the system time, but a behavior of
the function where we're no longer just mapping inputs to nice and clean
outputs. Just like option
, it turns out we can actually model realworld
side effects via the type system, but we'll not get into that now.
How about the function 'a => Result.t('b, 'e)
? Here we can observe the
“effect” of possible failure  the function can either succeed and produce a
value of type Ok(b)
, or fail and produce an error of type Error(e)
. This
effect of possible failure is represented by the type Result.t('a, 'e)
.
Let's quickly mention the function 'a => list('b)
. We've ignored
lists/arrays/trees/etc. to this point, but this type of function can also be
seen as an effect  the effect of indeterminate results. Here we could get no
results, a single result, or any number of results, and we can think about
this effect similarly to how we think about other effects.
Now consider the RationalJS/future
Future
type. The way this type is defined and implemented, it represents an
asynchronous computation that cannot fail. Being asynchronous, it's possible
that it will just never complete, but it has no way of representing a
computation that completed but failed. We can call this an “async effect.”
With this library, if you need to represent an asynchronous computation that
can fail, you're advised to use the type Future.t(Result.t('a, 'e))

here we're combining the effect of possible failure with a separate async
effect. This is a good demonstration of how effects can be “stacked” via the
type system, and it's also a good example to show that it's possible to “run”
or “remove” a single effect, while leaving other effects intact for separate
or later processing. For example, if you were to allow the Future
to
complete, you're given a value of type Result.t('a, 'e)
which still has the
effect of possible failure. This failure effect can be “removed” by
attempting to map
or flatMap
the value, and handling the case of Error
,
either by converting the error to a successful value, or handling it in some
other way.
The Js.Promise.t('a)
type also has two effects: an async effect and the
effect of possibile failure. With this type, the possibility of failure is
not directly observable by the type alone. This is okay  it just means that
the type has an implicit/hidden/nonpolymorphic way of representing the
failure condition, but there's nothing inherently wrong with that, you just
have to know that with a Js.Promise
, there is a possibility of failure. You
might run into code that uses Js.Promise.t(Result.t('a, 'e))
, but this is
problematic in that we now have two ways to represent failure  the “hidden”
Js.Promise
error type, and the 'e
type from the Result
. I hope to talk
about Js.Promise
more in another blog post, but I'll leave it at that for
now.
As a final example, if we look at the JSON decoder type like the
Decoder.t('a)
we defined above, we have a value that represents the effect
of parsing a JSON value into some type, and the effect of possible of failure
(which again is not represented by a polymorphic error type, but by a fixed
error type that we've defined with the decoder). This decoding effect is a
little different than the others in that it's more of a deferred computation,
but the same idea applies  the effectful value itself doesn't do anything
until we “run it” (by giving it a JSON value), and letting it produce a
Result
, which is how the effect of possible failure manifests itself, and
can be handled or “run” separately from the decoding effect.
Before we move on, let's look back at the “pure” function 'a => 'b
. In many
languages, including ReasonML, this type of function can actually have all
sorts of effects which do not manifest themselves as “effectful values,” but
just as plain old side effects. We can do I/O, read the system time, and even
launch the nukes, and nobody would be the wiser. It turns out you can
actually encapsulate these kinds of “side effects” in the type system using a
wide variety of different techniques, but we won't get into that now. The
topic of effect management is a very actively evolving and quite fascinating
topic in FP right now.
In summary, when we talk about applicatives (and monads), it's common to talk
about “effectful” values and “running” said effects. The concept of “running”
an effect sometimes can result in the “removal” of the effect, which
indicates that the effect has been handled, processed, or executed, but this
can mean different things for different types of effects. One key aspect of
functional programming is the separation of describing or encoding work to do
and the actual execution or interpretation of that work. If you're coming
from a more imperative language or style, where effects kind of just happen
when they happen, and are manifested by null
values, exceptions, or
spaghetti code that just does whatever it wants whenever it wants, the
functional approach will take a little getting used to, but it unlocks a
great deal of control and power.
Apply extensions
Now that we've seen some of the things that you can do with APPLICATIVE
,
let's try to capture those ideas so we can reuse them for every applicative.
In the functor article,
we saw that we could use a module functor to add “extensions” or “freebies”
for any instance of FUNCTOR
, and now we'll do the same thing for APPLY
.
Note that all of these extensions just need APPLY
and not APPLICATIVE
, because
these things only need map
and apply
, and not pure
.
module ApplyExtensions = (A: APPLY) => {
let applyFirst = (fa: A.t('a), fb: A.t('b)): A.t('a) => {
let f = (a, _) => a; // const
A.apply(A.map(f, fa), fb);
};
let applySecond = (fa: A.t('a), fb: A.t('b)): A.t('b) => {
let f = (_, b) => b; // const(id)
A.apply(A.map(f, fa), fb);
};
let map2 = (f: ('a, 'b) => 'c, fa: A.t('a), fb: A.t('b)): A.t('c) => {
A.apply(A.map(f, fa), fb);
};
let map3 = (f: ('a, 'b, 'c) => 'd, fa: A.t('a), fb: A.t('b), fc: A.t('c)): A.t('d) => {
A.apply(A.apply(A.map(f, fa), fb), fc);
};
// TODO: map4, map5, etc.
let mapTuple2 = (f: ('a, 'b) => 'c, (fa: A.t('a), fb: A.t('b))): A.t('c) => {
map2(f, fa, fb);
};
let mapTuple3 = (f: ('a, 'b, 'c) => 'd, (fa: A.t('a), fb: A.t('b), fc: A.t('c))): A.t('d) => {
map3(f, fa, fb, fc);
};
// TODO: mapTuple4, mapTuple5, etc.
let tuple2 = (fa: A.t('a), fb: A.t('b)) => map2((a, b) => (a, b), fa, fb);
let tuple3 = (fa: A.t('a), fb: A.t('b), A.t('c)) => map3((a, b, c) => (a, b, c), fa, fb, fc);
// TODO: tuple4, tuple5, etc.  as many as you want
};
module ApplyInfix = (A: APPLY) => {
module AE = ApplyExtensions(A);
let (<*>) = A.apply;
let (<*) = AE.applyFirst;
let (*>) = AE.applySecond;
};
There's a lot going on here, so let's break it down. We're creating a module
functor ApplyExtensions
that takes an instance of APPLY
and uses that
instance to define a bunch of extension functions. Because we have an APPLY
we are constrained to only having access to a type A.t('a)
, the A.map
function, and the A.apply
function, but we can do a lot with just these. Note
that you can see this in action in
Relude_Extensions_Apply.
We're first defining functions called applyFirst
and applySecond
. These
are interesting functions that take two effectful values, and runs them
both, but then only produces the result of the first or the second,
respectively. The key here is that we're actually running both effects, and
not just discarding the undesired side immediately  both effects must
succeed in order for us to get the result. If either or both effects fail, we
get an unsuccessful result, which can mean different things depending on
which APPLY
instance we're using. E.g for option
, the “unsuccessful
value” is None
, and for Result
the unsucessful value is Error(e)
, etc.
These two functions are more commonly seen in their operator forms: <*
and
*>
they kind of look like half of the apply
operator <*>
, and point at
the argument that we want to keep. See
ReludeParse for a realworld
nontrivial use case for these operators. You can also think of these
operators as similar to tuple2
where we first run and tuple the results,
then just take one side or the other.
Next we're defining a bunch of mapN
functions  here we only go up to
map2
and map3
, but in your own library, you could go as high as you
wanted. Note that if you go above 5 or so arguments, you might be better off
just using the more flexible map/ap/ap pattern with <$>
and <*>
operators. If you were curious, the reason we need all these numbered map and
other functions, rather than operating on lists or arrays of effectful
values, it's because lists and arrays are homogenous  they can only carry
values of the same type, but we want to operate on values of different types.
In a later blog post, we'll see a new abstraction called TRAVERSABLE
which
can help us deal with lists/arrays/options/etc. that containing effectful
values.
We then define an alternative version of mapN
called mapTupleN
 this
function simply takes a tuple of the effectful values, and then just
destructures the tuple and runs them through mapN
. Using a tuple can
sometimes be a more convenient way of collecting and chaining a series of
operations, e.g.
(Some(42), Some("hi"), Some(bool))
> Option.mapTuple3((i, s, b) => ...do something here...);
Finally we create a separate module functor for defining infix operators.
Having a separate module for infix operators can be handy for when you want
to do a local open and just get access to the operators for a few small
operations. Note that using the OCaml/ReasonML include
mechanism, you can
choose to include the infix operators into any organizational module you
want.
The pattern we use in Relude to incorporate these extensions is something like this:
module Option = {
type t('a) = option('a);
let map = ...;
let apply = ...;
let pure = ...;
module Functor: FUNCTOR = {
type nonrec t('a) = t('a); // alias
let map = map; // alias
};
include FunctorExtensions(Functor);
module Apply: APPLY = { ... };
include ApplyExtensions(Apply);
module Applicative: APPLICATIVE = { ... };
include ApplicativeExtensions(Applicative);
// ... other typeclasses and extension module functors
module Infix = {
include FunctorInfix(Functor);
include ApplyInfix(Functor);
// ... other infix module functors
};
};
This way, all the toplevel functions are exposed at the top level of
the module for convenience, then we have the typeclasses also at the top level
with their corresponding functions and types defined as just aliases, and we
immediately construct and include each of the typeclass extension modules.
This include puts the extension functions like map3
, tuple3
, etc. right
at the toplevel of the module, so you can do things like Option.map3(...)
.
Finally, we create a separate Infix
module, where we include all of the
Infix
extension modules. This puts all the operators in one common scope,
so you can do things like this:
let x = Option.Infix.(
f <$> Some(42) <*> Some("hi") <*> Some(true)
);
For types with more than one type parameter (like Result.t('a, 'e)
), we
unfortunately have to implement all of our typeclass instances and extensions
inside a module functor (like Result.WithError
), so we lose a little of the
convenience of the extensions. In order to get access to map3
, etc. for Result
,
you have to do something like this:
module ResultE = Result.WithError({ type t = myErrorType });
ResultE.map3(...);
Unfortunately, you can't inline module functor stuff with function invocations, so you can't do this, which is a big bummer:
// Can't do this :(
let x = Result.WithError({ type t = string }).map3(...);
Also, as we saw in the functor article, you can't pass modules that deal with higherkinded types via firstclass modules, so we're kind of stuck with the extra boilerplate of instantiating our module in one line, and using it where we need it.
That was a lot of discussion, but in case you missed it, we just gave
ourselves an implementation of mapN
, tupleN
, mapTupleN
, <*>
, <*
,
*>
for any module that has an APPLY
instance! I think that's pretty
awesome! Your option
, Result
, Js.Promise
, Future
, IO
, and every
other APPLY
gets a handy, consistent and powerful set of functions for
free, just because we took the time to implement map
and apply
and setup
some typeclass instances. Plus, we now have a layer of centralized
abstraction and extension where we can add more of these types of functions,
and we just get them all for free anywhere we're using ApplyExtensions
! If
you think this is cool, you should take a look at languages like Haskell or
PureScript and see how they do all this with much less ceremony, but it's
still pretty cool that we can achieve something very similar in ReasonML!
Applicative validation
We now have a bunch of useful tools for dealing with APPLY
and
APPLICATIVE
instances and their “effectful values,” so let's see what we
can do with them.
First of all, if you've used Promise.all
in JavaScript (or the
Js.Promise.allN
equivalents in ReasonML), you've already used an
applicativestyle API. We have our own versions of these in our extensions
like mapN
, tupleN
, mapTupleN
, etc. These are all just helper functions
that let us process a bunch of effectful values in parallel and combine the
results in different ways, which is quite handy all by itself.
If you have a situation where you have a bunch effectful values to deal with
(say N+1
values), and your mapN
functions only “go up to N
,” you can
deal with an arbitrary number of arguments using the map/ap/ap pattern
using <$>
and <*>
, which we'll see below.
Let's now look at one more very useful use case for applicatives: applicative validation, parsing, and decoding. But first, let's introduce a few additional concepts to help us out:
SEMIGROUP typeclass
Now that we've seen FUNCTOR
, APPLY
, and APPLICATIVE
, we know that
typeclasses are sort of just sets of functions that must conform to some
laws, so I think we can start to introduce more of these things without fear.
Just look at the types of the functions and the laws  you don't necessarily
need to understand why it exists or what it's for right away, just see it for
what it is: functions and laws.
The SEMIGROUP
typeclass is simply about the ability to combine values of the
same type:
module type SEMIGROUP = {
type t;
let append: (t, t) => t;
};
The append
function often has an operator version too  in Haskell, it's
<>
, but we'll ignore that for now (other than for noting the law). The name
append
is commonly used, but you could also think of this operation as
combine
.
The laws for this typeclass is that the append
function must be associative,
i.e.
// Associativity (<> is the operator for append)
(a <> b) <> c == a <> (b <> c)
If you remember from algebra class, it's the same law of associativity that
we learned for addition and multiplication. Basically, it requires that if
you have 3 values, you can combine the first two first, then combine the
third and that should be the same as combining the second two, then combining
that with the first. Note that this is not commutativity  the items have to
be combined in the order above, we're not saying that you can switch the
order of the arguments, we're just saying exactly what's described above.
There is another typeclass that requires commutativity, but SEMIGROUP
doesn't  only associativity.
// Commutativity  this is NOT a law required by SEMIGROUP
a <> b == b <> a
Let's see a few quick examples for int
addition, int
multiplication,
bool
AND and OR, string
, and list('a)
.
Integers actually have multiple semigroups. One semigroup is integer
addition, where the append
operation is just +
. Another different
semigroup is multiplication where the append
function is *
. You might
hear these notions described as something like “integers form a semigroup
under addition.”
module Integer = {
module Addition = {
module Semigroup: SEMIGROUP = {
type t = int;
let append = (a, b) => a + b;
};
};
module Multiplication = {
module Semigroup: SEMIGROUP = {
type t = int;
let append = (a, b) => a * b;
};
};
};
We define these as distinct instances, because they are distinct semigroups.
Note that integer subtraction and division are not SEMIGROUPS because
those operations are not associative (they don't conform to the semigroup
law, which requires that the append operation be associative). These
operations might be usable for other typeclasses, but not SEMIGROUP
.
(12)3 != 1(23)
(1/2)/3 != 1/(2/3)
Boolean has a few semigroups too: the AND
and OR
operators. These would
be implemented similarly to how I split up Addition
and Multiplication
for Integer
, and for booleans, you'll see words like Conjunctive
for
AND
and Disjunctive
for OR
.
Strings have a SEMIGROUP too, just string concatenation:
let append = (str1, str2) => str1 ++ str2;
Lists and arrays also form a semigroup, but since these are type constructors
of the form t('a)
, we need a variation of SEMIGROUP
that works with types
of that form. We'll call it SEMIGROUP_ANY
 that's what it's called in
bsabstract.
module type SEMIGROUP_ANY = {
type t('a);
let append: (t('a), t('a)) => t('a);
};
The semigroup for list
and array
is just concatenation, just like
string
. Here's a quick implementation for list('a)
module List = {
type t('a) = list('a);
let concat = (list1, list2) =>
Belt.List.concat(list1, list2); // I hope I got the order right for Belt.List...
module SemigroupAny: SEMIGROUP_ANY = {
type nonrec t('a) = t('a);
let append = concat;
};
};
I hope that's clear enough for now, and let's move on.
NonEmptyList data type
Let's now introduce a useful type for error handling  a NonEmptyList
.
Sometimes when you're dealing with collections of values, it can be really
helpful to guarantee that there's at least one value. Imagine a validation
function in JavaScript, where maybe it gives you back an array of errors 
what happens if it returns you an empty array? That seems odd and
undesirable, so let's not allow ambiguous things like that by using better
types. Just for expedience, I'm going to go ahead and implement this now,
including a SemigroupAny
instance  below is how we typically implement
new data types in Relude:
module NonEmptyList = {
type t('a) =  Nel('a, list('a));
let map = ...;
let apply = ...;
let pure = a => Nel(a, []);
let concat = (Nel(h1, t1), Nel(h2, t2)) => {
Nel(h1, Belt.List.concat(t1, [h2, ...t2]));
};
module SemigroupAny: SEMIGROUP_ANY = {
type nonrec t('a) = t('a);
let append = concat;
};
include SemigroupAnyExtensions(SemigroupAny);
module Functor: FUNCTOR with type t('a) = t('a) = ...
include FunctorExtensions(Functor);
module Apply: APPLY with type t('a) = t('a) = ...
include ApplyExtensions(Apply);
module Applicative: APPLY with type t('a) = t('a) = ...
include ApplicativeExtensions(Applicative);
};
The NonEmptyList
is basically just a data structure with a constructor
Nel
that has one guaranteed value 'a
, and a “tail” list('a)
, which can
be empty. We can implement map
, apply
, pure
, concat
and many other
useful things for this type, and all the corresponding typeclass instances.
There's nothing magic about any of this  it's simply a type that is exactly what it says it is: a nonempty list. We're going to use this below for “collecting errors” in our applicative validation, so that we can be sure that if an applicative operation fails, we'll get at least one error value out.
Validation data type
If you recall above when we implemented apply
for Result.t('a, 'e)
, we
ran into a conundrum when faced with the “both sides failed” case:
let apply = (ff: Result.t('a => 'b, 'e), fa: Result.t('a, 'e)) => switch(ff, fa) {
 (Ok(aToB), Ok(a)) => Ok(aToB(a))
 (OK(aToB), Error(e2)) => Error(e2)
 (Error(e1), Ok(a)) => Error(e1)
 (Error(e1), Error(e2)) => Error(e1) // !!!
}
In this Error/Error
case, we have no way to combine the errors, and no
real way to decide which one to return, so we'll just arbitrarily return the
left side's error.
Let's introduce a new data type called Validation.t('a, 'e)
to deal with
this case specifically. We're going to add a little extra spice to our
apply
implementation to deal with these two errors. The type itself is
going to look almost exactly the same as Result.t('a, 'e)
, but it will
have different semantics for the applicative behavior.
module Validation = {
type t('a, 'e) =  VOk('a)  VError('e);
let map = (aToB, fa) => switch(fa) {
 VOk(a) => VOk(aToB(a))
 VError(e) => VError(e)
};
let applyWithCombineErrors = (combineErrors: ('e, 'e) => 'e, ff: t('a => 'b, 'e), fa: t('a, 'e)): t('b, 'e) => {
switch(ff, fa) {
 (VOk(aToB), VOk(a)) => VOk(aToB(a))
 (VError(e1), VOk(a)) => VError(e1)
 (VOk(aToB), VError(e2)) => VError(e2)
 (VError(e1), VError(e2)) => VError(combineErrors(e1, e2)) // 🎉
};
};
let pure = a => VOk(a);
};
To implement our typeclass instances like Functor
, etc. we'll use the
WithError
module functor trick to “lock down” our error type. However, when
we get to APPLY
, where is this combineErrors
function going to come from?
We don't know what the error type is (well, we'll know once we lock it down
using WithError
), and moreover, we don't know how to combine these errors.
Oftentimes, when you need to do something, and you don't know how to do it,
you can just make someone pass it to you. In this case, we need a mechanism
for combining values of the same type, and for this, we have the SEMIGROUP
abstraction! Let's see how this works:
module Validation = {
type t('a, 'e) =  VOk('a)  VError('e);
let map = (aToB, fa) => switch(fa) {
 VOk(a) => VOk(aToB(a))
 VError(e) => VError(e)
};
let applyWithCombineErrors = (combineErrors: ('e, 'e) => 'e, ff: t('a => 'b, 'e), fa: t('a, 'e)): t('b, 'e) => {
switch(ff, fa) {
 (VOk(aToB), VOk(a)) => VOk(aToB(a))
 (VError(e1), VOk(a)) => VError(e1)
 (VOk(aToB), VError(e2)) => VError(e2)
 (VError(e1), VError(e2)) => VError(combineErrors(e1, e2)) // 🎉
};
};
let pure = a => VOk(a);
module WithErrors = (Errors: SEMIGROUP_ANY, Error: TYPE) => {
module Functor: FUNCTOR with type t('a) = t('a, Errors.t(Error.t)) = {
type nonrec t('a) = t('a, Errors.t(Error.t));
let map = map;
};
include FunctorExtensions(Functor);
module Apply: APPLY with type t('a) = t('a, Errors.t(Error.t)) = {
include Functor;
let apply = (ff, fa) => applyWithCombineErrors(Errors.append, ff, fa);
};
include ApplyExtensions(Apply);
module Applicative: APPLICATIVE with type t('a) = t('a, Errors.t(Error.t)) = {
include Apply;
let pure = pure;
};
include ApplicativeExtensions(Applicative);
module Infix = {
include FunctorInfix(Functor);
include ApplyInfix(Apply);
};
};
};
There are multiple ways of doing this, but I'm just going to require a
SEMIGROUP_ANY
instance to collect errors, and a TYPE
instance to lock
down my error type. You could also probably use a plain SEMIGROUP
, but that
requires some different techniques for dealing with the error type.
Now in order to use the applicative stuff with this type, I just need to give
it a SEMIGROUP_ANY
and an error TYPE
, like so:
module ValidationNel = Validation.WithErrors(NonEmptyList.SemigroupAny, { type t = string });
ValidationNel.tuple3(VOk(42), VOk("hi"), VOk(true)) // VOk((42, "hi", true))
ValidationNel.tuple3(VOk(42), VError("oops"), VError("darn")) // VError(Nel("oops", ["darn"]))
With this type and the SEMIGROUP_ANY
semantics, we have the ability to
collect the error for each individual failure in the parallel operation,
rather than just exploding at the first error we encounter! To understand how
this works, just think about how apply
is implemented for Validation
,
especially in the case of both sides failing. We can use a NonEmptyList
to
collect our errors, because there's never a scenario where this can fail and
we don't have an error value to deal with. We could also use a plain old
list('a)
, but then you lose the guarantee at the callsite of there being at
least one error  the caller has to deal with the empty list case, when we
know that will never actually happen. This is all about making impossible states
impossible.
Applicative validation with error collection
We now have a type Validation.t('a, 'e)
that looks a lot like a Result.t('a, 'e)
,
but has slightly different semantics in its APPLICATIVE
instance. The Result.t('a, 'e)
apply
function “fails fast” in that it just propagates the first error it encounters
all the way to the final result, whereas the Validation
has “error collecting” semantics
where it will collect and combine each error it encounters using a SEMIGROUP
or in our
case a SEMIGROUP_ANY
instance, which is just a semigroup for types of the form t('a)
.
Note that when I say “Result fails fast” in its applicative, I don't mean it
instantly explodes or earlyreturns out of the function. I just mean that if
some individual operation fails, the apply
function will just propagate
that error along, and ignore any further errors, until it reaches the final
result  just think about how apply
is implemented for Result
.
We can use applicatives for a wide variety of validation, decoding, parsing,
and data normalization tasks. Some real world use cases for applicative
validation might include CSV parsing, JSON decoding, web form validation,
string parsing, database record parsing, parallel async data fetching, and
countless more. Let's see a few examples, but for all of these remember that
you can use any type that has an applicative instance to do the same thing
but with different error handling semantics, and in fact, you can abstract
the APPLICATIVE
instance away using a module functor to make a validation
module that can just work with any applicative right off the bat. When I say
“desired semantics” I just mean that you can choose the type of error
handling that fits your needs  if you just want to know whether the
validation succeeded or failed, you could use option('a)
, if you wanted to
know whether it failed, and what the first error was, you might use
Result.t('a, 'e)
(which doesn't know how to combine the errors in apply
),
or if you wanted to capture all the errors, you might use a Validation.t('a, 'e)
. If your validation requires async computations, you might need to use
something like Promise.t('a)
or Future.t(Result.t('a, 'e))
, or even
Future.t(Validation.t('a, 'e))
.
Anyway, let's see a quick example. Let's say we have a model like the following:
module User = {
type t = {
name: string,
email: string,
age: int,
isAdmin: bool
};
let make = (
name: string,
email: string,
age: int,
isAdmin: bool
) => {name, email, age, isAdmin};
We have a type User.t
which has information about users, and a pure make
function which
simply acts as a nonvalidating constructor for the type User.t
. Let's say we're getting a
JSON value representing a user, and we want to decode it into a value of type User.t
, so that
we can be confident that we know what data we're working with.
Let's build a tiny JSON decoder library based on Validation.t('a, 'e)
, and then see how we
can implement a basic JSON decoder using Validation, and the JSON decoder type we saw above.
Here is the full example, and I'll explain more below.
module type TYPE = {type t;};
module type SEMIGROUP_ANY = {
type t('a);
let append: (t('a), t('a)) => t('a);
};
module type FUNCTOR = {
type t('a);
let map: ('a => 'b, t('a)) => t('b);
};
module FunctorExtensions = (F: FUNCTOR) => {
let voidLeft = (b: 'b, fa: F.t('a)) => F.map(_ => b, fa);
let voidRight = (fa: F.t('a), b: 'b) => F.map(_ => b, fa);
// etc.
};
module FunctorInfix = (F: FUNCTOR) => {
module FE = FunctorExtensions(F);
let (<$>) = F.map;
let (<$) = FE.voidLeft;
let ($>) = FE.voidRight;
// etc.
};
module type APPLY = {
include FUNCTOR;
let apply: (t('a => 'b), t('a)) => t('b);
};
module ApplyExtensions = (A: APPLY) => {
let map2 = (f, fa, fb) => A.apply(A.map(f, fa), fb);
let map3 = (f, fa, fb, fc) => A.apply(A.apply(A.map(f, fa), fb), fc);
let map4 = (f, fa, fb, fc, fd) =>
A.apply(A.apply(A.apply(A.map(f, fa), fb), fc), fd);
let map5 = (f, fa, fb, fc, fd, fe) =>
A.apply(A.apply(A.apply(A.apply(A.map(f, fa), fb), fc), fd), fe);
// etc.
};
module ApplyInfix = (A: APPLY) => {
let (<*>) = A.apply;
// etc.
};
module type APPLICATIVE = {
include APPLY;
let pure: 'a => t('a);
};
module NonEmptyList = {
type t('a) =
 Nel('a, list('a));
let pure = a => Nel(a, []);
let concat = (Nel(h1, t1), Nel(h2, t2)) => {
Nel(h1, List.concat([t1, [h2], t2]));
};
let toList = (Nel(h, t)) => [h, ...t];
let toArray = nonEmptyList => nonEmptyList > toList > Belt.List.toArray;
module SemigroupAny: SEMIGROUP_ANY with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let append = concat;
};
};
module Validation = {
type t('a, 'e) =
 VOk('a)
 VError('e);
let ok = a => VOk(a);
let error = e => VError(e);
let errorNel = e => VError(NonEmptyList.pure(e));
let map = (f, fa) =>
switch (fa) {
 VOk(a) => VOk(f(a))
 VError(e) => VError(e)
};
let applyWithAppendErrors =
(appendErrors: ('e, 'e) => 'e, ff: t('a => 'b, 'e), fa: t('a, 'e)) =>
switch (ff, fa) {
 (VOk(f), VOk(a)) => VOk(f(a))
 (VError(e1), VOk(_)) => VError(e1)
 (VOk(_), VError(e2)) => VError(e2)
 (VError(e1), VError(e2)) => VError(appendErrors(e1, e2))
};
let pure = ok;
module WithErrors = (S: SEMIGROUP_ANY, E: TYPE) => {
type nonrec t('a) = t('a, S.t(E.t));
module Functor: FUNCTOR with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let map = map;
};
let map = map;
include FunctorExtensions(Functor);
module Apply: APPLY with type t('a) = t('a) = {
include Functor;
let apply = (ff, fa) => applyWithAppendErrors(S.append, ff, fa);
};
let apply = Apply.apply;
include ApplyExtensions(Apply);
module Applicative: APPLICATIVE with type t('a) = t('a) = {
include Apply;
let pure = pure;
};
let pure = pure;
module Infix = {
include FunctorInfix(Functor);
include ApplyInfix(Apply);
};
};
};
module Decode = {
module Error = {
type t =
 ExpectedString(Js.Json.t)
 ExpectedBool(Js.Json.t)
 ExpectedInt(Js.Json.t)
 ExpectedObject(Js.Json.t)
 ExpectedObjectWithKey(Js.Json.t, string);
let show = e =>
switch (e) {
 ExpectedString(json) =>
"Expected string, got: " ++ Js.Json.stringify(json)
 ExpectedBool(json) =>
"Expected bool, got: " ++ Js.Json.stringify(json)
 ExpectedInt(json) => "Expected int, got: " ++ Js.Json.stringify(json)
 ExpectedObject(json) =>
"Expected object, got: " ++ Js.Json.stringify(json)
 ExpectedObjectWithKey(json, key) =>
"Expected object with key "
++ key
++ ", got: "
++ Js.Json.stringify(json)
};
};
// For this example, we're going to use `Validation.t('a, NonEmptyList.t(Error.t))`
// as our applicative "effect" type for the decoder. This will give us errorcollecting
// semantics in the decode operation.
module ValidationE =
Validation.WithErrors(NonEmptyList.SemigroupAny, Error);
// Our decoder type is a function Js.Json.t => A.t('a)
// Think of `A` being option, Result, Validation, Promise, etc. but here we're just
// specialized to using ValidationE
type t('a) =
 Decode(Js.Json.t => ValidationE.t('a));
// define map/Functor
let map = (f, Decode(jsonToFA)) =>
Decode(json => jsonToFA(json) > ValidationE.map(f));
module Functor: FUNCTOR with type t('a) = t('a) = {
type nonrec t('a) = t('a);
let map = map;
};
include FunctorExtensions(Functor);
// define apply/Apply
let apply = (Decode(jsonToFAToB), Decode(jsonToFA)) =>
Decode(
json => {
let fAToB = jsonToFAToB(json);
let fA = jsonToFA(json);
ValidationE.apply(fAToB, fA);
},
);
module Apply: APPLY with type t('a) = t('a) = {
include Functor;
let apply = apply;
};
include ApplyExtensions(Apply);
// define pure/Applicative
let pure = a => Decode(_ => ValidationE.pure(a));
module Applicative: APPLICATIVE with type t('a) = t('a) = {
include Apply;
let pure = pure;
};
// Add an infix operator module
module Infix = {
include FunctorInfix(Functor);
include ApplyInfix(Apply);
};
// Now define some specific types of decoders
let string: t(string) =
Decode(
json =>
switch (Js.Json.classify(json)) {
 JSONString(str) => ValidationE.pure(str)
 _ => Validation.errorNel(Error.ExpectedString(json))
},
);
let bool: t(bool) =
Decode(
json =>
switch (Js.Json.classify(json)) {
 JSONTrue => VOk(true)
 JSONFalse => VOk(false)
 _ => Validation.errorNel(Error.ExpectedBool(json))
},
);
let int: t(int) =
Decode(
json =>
switch (Js.Json.classify(json)) {
 JSONNumber(f) => VOk(Js.Math.floor(f))
 _ => Validation.errorNel(Error.ExpectedInt(json))
},
);
let field = (key: string, decode: t('a)): t('a) =>
Decode(
json =>
switch (Js.Json.classify(json)) {
 JSONObject(dict) =>
switch (Js.Dict.get(dict, key)) {
 Some(value) =>
let Decode(decodeA) = decode;
decodeA(value);
 None =>
Validation.errorNel(Error.ExpectedObjectWithKey(json, key))
}
 _ => Validation.errorNel(Error.ExpectedObject(json))
},
);
let run = (json: Js.Json.t, Decode(f)) => f(json);
};
module User = {
type t = {
name: string,
email: string,
age: int,
isAdmin: bool,
};
let make = (name, email, age, isAdmin) => {name, email, age, isAdmin};
let decode =
Decode.Infix.(
make
<$> Decode.field("name", Decode.string)
<*> Decode.field("email", Decode.string)
<*> Decode.field("age", Decode.int)
<*> Decode.field("isAdmin", Decode.bool)
);
// Note: could also use map4 rather than <$>/<*>
};
let json1 =
Js.Json.object_(
Js.Dict.fromList([
("name", Js.Json.string("Andy")),
("email", Js.Json.string("test@example.com")),
("age", Js.Json.number(55.0)),
("isAdmin", Js.Json.boolean(true)),
]),
);
let validatedUser1 = User.decode > Decode.run(json1);
switch (validatedUser1) {
 VOk(user) => Js.log2("Success!", user)
 VError(errors) => Js.log2("Failure!", errors > NonEmptyList.toArray)
};
// Success! [ 'Andy', 'test@example.com', 55, true ]
let json2 =
Js.Json.object_(
Js.Dict.fromList([
("name", Js.Json.string("Andy")),
("email", Js.Json.number(42.0)),
("age", Js.Json.number(55.0)),
("isAdmin", Js.Json.string("supposed to be a boolean")),
]),
);
let validatedUser2 = User.decode > Decode.run(json2);
switch (validatedUser2) {
 VOk(user) => Js.log2("Success!", user)
 VError(errors) =>
Js.log2(
"Failure!",
errors > NonEmptyList.toArray > Array.map(Decode.Error.show),
)
};
// Failure! [ 'Expected string, got: 42',
// 'Expected bool, got: "supposed to be a boolean"' ]
So that was a long code sample, but most of it was just setting up all the core pieces that would normally come from a library like Relude  I'm just showing it here to have a complete and somewhat realistic example for demonstrating all the moving parts.
 At the top of the file, we have all the core typeclass definitions
TYPE
SEMIGROUP_ANY
FUNCTOR
 etc.
 We also have a couple
Extensions
andInfix
modules for a few of these, which we'll use below.  Next we define out
NonEmptyList
type, which we use in theValidation
to collect the errors in theapply
function  Then we have the
Validation
module, with the core type, functions, then the typeclass instances wrapped intheWithErrors
module functor, where we require aSEMIGROUP_ANY
instance to collect our errors, and aTYPE
instance to lock down the error type.  Lastly for the “library” pieces, we define a
Decoder
module, which is essentially a functionJs.Json.t => Validation.t('a, NonEmptyList.t(Error.t))
, and all the supporting functions and typeclass instances.
As for the actual demonstration, I have a User
module which has:
 a basic record type with 4 fields
 a pure, nonvalidating
make
function which acts as a constructor  a decoder, which uses the applicative map/ap/ap pattern to map the
pure
make
function over a series of decoders for the individual fields. Note that instead of
<$>
/<*>
we could use theDecode.map4
function which we got “for free” from ourDecode.Apply
and theinclude ApplyExtensions(Apply)
 Note that instead of
The Decoder
itself has an applicative instance, which is backed by the
applicative for the Validation
module, which has error collecting
semantics. If you want to try an exercise, see if you can make the decoder
a module functor like module WithApplicative => (A: APPLICATIVE, ???) =>
and replace all usages of Validation
/ValidationE
with just A
references.
Note: you'll need some way to deal with the errors, like the places where
I have Validation.errorNel(...)
.
Anyway, that was a long example, with possibly insufficient explanation, but I hope that by providing a complete, working example, anyone who is interested can dig in more.
If you've used Elm, you might have recognized this type of pattern from the elmdecodepipeline library. That library is more builtout, but it is fundamentally built around the idea of applicatives and parallel decoding of values.
You can also see this same pattern in ReasonML in the bsdecode library. This library is also more robust than our simple example here, but the applicatives are there at the core!
The applicative laws
As I mentioned in the functor article, the typeclass laws are quite important, but given that this article is already quite long, I'm just going to link to the laws on a couple different sites  I encourage you to check these out on your own:
Conclusion
I hope you enjoyed my article on applicative functors  let me know if you find any problems with the code samples, or if you have any questions or suggestions.