Introduction

Recently I went through this fascinating article Professor Frisby’s mostly adequate guide to Functional Programming and would like to summarise my understanding in this post.

f(x)

In imperative programming, you get things done by giving the computer a sequence of tasks and then it executes them. While executing them, it can change state.  In purely functional programming you don’t tell the computer what to do as such but rather you tell it what stuff is.

// Imperative

var original = [1,2,3,4,5]
var squared = []

for(var i = 0; i < original.length; i++) {  
  var squaredNumber = original[i] * original[i]  
  squared.push(squaredNumber) 
} 

console.log(squared) //=> [1, 4, 9, 16, 25]

// Functional

var original = [1,2,3,4,5]

var squared = original.map(n => n * n);

console.log(squared) //=> [1, 4, 9, 16, 25]

Functions are first-class

In Javascript, functions are “first class”, we mean they are just like everyone else for example, like a Number, String etc.,

Functions are better if they are pure

A pure function is a function that, given the same input, 
will always return the same output and does not have any 
observable side effect.

Side effects may include but not limited to,

Changing the file system
Inserting/Updating/Deleting a record into a database
Making a http call
Mutations
Printing to the screen/logging
Obtaining User Input
Querying the DOM
Accessing system state

8th grade math

From mathisfun.com:

A function is a special relationship between values: 
Each of its input values gives back exactly one output value.

Take a function which calculates square

function square(x) {
  return x * x;
}

If function is just a special relationship between values as mentioned above, we can visualize the function as a simple table:

There’s no need for implementation details if the input dictates the output. Pure functions are mathematical functions and they’re what functional programming is all about.

A pure function is:

• Cacheable
• Self Documenting
• Testable
• Reasonable (Referential transparent: A spot of code is referentially transparent when it can be substituted for its evaluated value without changing the behavior of the program.)
• Can be made to execute in parallel

Curried Functions

The concept of curried functions is, you can call a function with fewer arguments than it expects. It returns a function that takes the remaining arguments.

For eg., a curried add function can be,

function add = a => b => {
    return a+b;
}

add(1)(2) // Result: 3

As explained earlier, a function is a special relationship between values: Each of its input values gives back exactly one output value. Using curried functions, we can make new, useful functions on the fly simply by passing in a few arguments and as a bonus, we’ve retained the mathematical function definition of each input mapping to exactly one output despite multiple arguments. This is the reason, why every function in Haskell officially only takes only one parameter. All the functions that accepted several parameters will be curried functions.

Composition – Holy Grail

compose function can be defined as:

var compose = function(f, g) {
  return function(x) {
    return f(g(x));
  };
};

f and g are functions and x is the value being piped through them.

The most important building block in FP is functions. Composition allows us to combine these smaller building blocks to build larger programs. If you look at the function definition, compose function takes 2 functions f and g and creates a brand new function which takes a value x calls g and then passes the result to function fand returns the result.

Functional composition is associative.

// associativity

var associative = 
compose(f, compose(g, h)) == compose(compose(f, g), h);

// true

This means that it doesn’t matter how we group our calls to compose, the result will be the same. This allows us to write a variadic compose (which is implemented in libraries like lodash, ramda etc.,).

The best analogy to think about the power of composition is UNIX pipes. 2 of the important points in Unix  philosophy are:

1. Write programs that do one thing and do it well.
2. Write programs to work together.

For eg;

find - walk a file hierarchy
cat - concatenate and print files
grep - file pattern searcher
xargs - construct argument list(s) and execute utility

Each program mentioned above exactly does one job very well. 
But when we combine/compose them together you get powerful programs.

find . -name "fp.txt" | xargs cat | grep "FP"

Find a file called "fp.txt" searching from the current directory
and pipe the contents to cat through xargs and then pipe the 
contents to grep to search for the text "FP"

In FP, we write pure functions that do one thing and do it well. We use composition to combine/compose functions so that they work together.

Hindley-Milner

In functional world, it won’t be long before we find ourself knee deep in type signatures. Types are the meta language that enables people from all different backgrounds to communicate succinctly and effectively. For the most part, they’re written with a system called “Hindley-Milner”.

Type signatures for the below functions are as follows:

:t capitalize
capitalize :: String -> String

:t strLength
strLength :: String -> Number

:t head
head :: [a] -> a

:t reverse
reverse :: [a] -> [a]

:t sort
sort :: Ord a => [a] -> [a]

:t foldl
foldl :: Foldable t => (b -> a -> b) -> b -> t a -> b

Once a type variable is introduced, there emerges a curious property called parametricity. This property states that a function will act on all types in a uniform manner which makes the possible behavior of the function massively narrowed due to its polymorphic type.

Container/Box/Computational Context

We have seen how to apply functions to a value.

function addOne(x) {
  return x + 1;
}

var r = addOne(10) // r = 11

We will extend this idea by saying that any value can be in a context. For now we can think of this as a CONTAINER or BOX or COMPUTATIONAL CONTEXT. 

You can create this type with the following definition:

var Container = function(x) {
  this.__value = x;
}

Container.of = function(x) { return new Container(x); };

We will take a simple type called Maybewhich can be in 2 different states: One state containing a value and the other Nothing. In Haskell, this type is represented as

data Maybe a = Nothing | Just a

In Scala, this type is called Option, which will be in 2 different states: Some and None.

In Javascript, in the below example we will just define a  Maybe class which encapsulates both the states. In practice, we should mirror the Haskell or Scala or some other language have separate type constructors for each of the states.

var Maybe = function(x) {
  this.__value = x;
};

Maybe.of = function(x) {
  return new Maybe(x);
};

Maybe.prototype.isNothing = function() {
  return (this.__value === null || this.__value === undefined);
};

Maybe.of("JS rocks!!"); // State containing a value
Maybe.of(null); // State containing null

Functor

The definition of functor:

map :: Functor f => (a -> b) -> f a -> f b

From the signature, we can clearly see that Functor defines a map method and its implementors must provide an implementation to the map function. From the signature, we can see that map method allows us to do a data transformationof the value that is contained in the computational context. Another very important property of map function is it preserves structure.

A Functor is a type that implements map and obeys some laws

Maybe is a Functorand it provides a map implementation.

Maybe.prototype.map = function(f) {
  return this.isNothing() ? Maybe.of(null) : Maybe.of(f(this.__value));
};

Maybe.of('Mayakumar').map(match(/a/ig));
// => Maybe['a', 'a', 'a']

Functor Laws

// identity
map(id) === id;

// composition
compose(map(f), map(g)) === map(compose(f, g));

Diagrammatically, what happens under the hood of Functor's map method can be shown as below:

Monad

Pointy Functor Factory

of method is to place values in what’s called a default minimal context part of an important interface called Pointed.

A pointed functor is a functor with an of method

Maybe.of(100).map(add(1));
// Maybe(101)

When the map function has a signature a => M[B] we will get nested container types. If we had to get to the value, we have to map as many times  as the wrapped container types which is not great from the caller perspective.

We somehow need to remove the extra container/box/computation context, M[M[a] to M[a]

var mmo = Maybe.of(Maybe.of('value'));
// Maybe(Maybe('value'))

mmo.join();
// Maybe('value')

Monads are pointed functors that can flatten

Any functor which defines a join method, has an of method, and obeys a few laws is a monad. Defining join is not too difficult so let’s do so for Maybe:

Maybe.prototype.join = function() {
  return this.isNothing() ? Maybe.of(null) : this.__value;
}

We can call join right after map which can be abstracted in a function called chain.

//  chain :: Monad m => (a -> m b) -> m a -> m b
var chain = curry(function(f, m){
  return m.map(f).join(); // or compose(join, map(f))(m)
});

For Maybe:

Maybe.prototype.chain = function(f) {
  return this.isNothing() ? Maybe.of(null) : this.map(f).join();
}

chain nests effects and we can capture both sequence and variable assignment in a purely functional way.

For eg:

var result = Maybe.of(3).chain(a => {
   return Maybe.of(30).chain(b => {
      return Maybe.of(300).chain(c => {
        return Maybe.of(3000).map(d => a + b + c + d);
      });
   })
});

// => 3333

Monad Laws

// associativity
compose(join, map(join)) == compose(join, join);

// identity for all (M a)
compose(join, of) === compose(join, map(of)) === id

var mcompose = function(f, g) {
  return compose(chain(f), g);
};

// left identity
mcompose(M, f) == f;

// right identity
mcompose(f, M) == f;

// associativity
mcompose(mcompose(f, g), h) === mcompose(f, mcompose(g, h));

Diagrammatically, what happens under the hood of Monad's chain method  can be shown as below:

function even(x) {
   if(x % 2 === 0) {
      return Maybe.of(x/2);
    } else {
      return Maybe.of(null);
    }
}

Applicatives

Applicatives provide the ability to apply functors to each other.  We will take an example to understand better:

var add = a => b => {
 return a + b;
}

add(Maybe.of(2), Maybe.of(3));
// Not possible

var maybe_of_add_2 = map(add, Maybe.of(2));
// Maybe(add(2))

We have ourselves a Maybe with a partially applied function inside. More specifically, we have a Maybe(add(2)) and we’d like to apply its add(2) to the 3 in Maybe(3) to complete the call. In other words, we’d like to apply one functor to another.  We can achieve this using chain and map functions as defined below:

Maybe.of(2).chain(function(two) {
 return Maybe.of(3).map(add(two));
});

The issue here is that we are stuck in the sequential world of monads. We have ourselves two strong, independent values and we should think it unnecessary to delay the creation of Maybe(3) merely to satisfy the monad’s sequential demands. So applicatives provides the functionality where we need to apply functions within a computational context to values in a computational context but the values are independent and hence there is no need to sequence them. A good example to think of the use of Applicatives is when we do parallel independent asynchronous computations and apply the results of each to a function contained in a functor.

Maybe.prototype.ap = function(other_container) {
 return this.isNothing() ? this : other_container.map(this.__value);
}

Maybe.of(add).ap(Maybe.of(2)).ap(Maybe.of(3));
// Maybe 5

An applicative functor is a pointed functor with an ap method

Applicative Laws

// identity
A.of(id).ap(v) == v

// homomorphism
A.of(f).ap(A.of(x)) == A.of(f(x))

// interchange
v.ap(A.of(x)) == A.of(function(f) { return f(x) }).ap(v)

// composition
A.of(compose).ap(u).ap(v).ap(w) == u.ap(v.ap(w));

So a good use case for applicatives is when one has multiple functor arguments. They give us the ability to apply functions to arguments all within the functor world. Though we could already do so with monads, we should prefer applicative functors when we aren’t in need of monadic specific functionality.

Diagrammatically, what happens under the hood of Applicative's ap method  can be shown as below:

map/of/chain/ap

We have explained the concepts behind Functor, Monad, Applicatives. There are so many structures that obey/satisfy the properties of being a functor, monad and applicative. From the above example we can see that Maybe is a Functor, Monad and Applicative. There are many structures that satisfy functor, monad, applicatives which include:

List
Maybe
IO
Task (https://github.com/folktale/data.task)
etc.,

If a type is a Monad it has to be both an Appliacative and a Functor. If a type is an Applicative it has to be a Functor.

Summary

Professor Frisby’s mostly adequate guide to Functional Programming is one of the best articles that I have read recently. From the article, the author wonderfully shows how powerful and deep functional programming constructs are and also explains how they can all be implemented with Javascript language.

References

1. https://www.gitbook.com/book/drboolean/mostly-adequate-guide/details
2. http://learnyouahaskell.com/
3. http://adit.io/posts/2013-04-17-functors,_applicatives,_and_monads_in_pictures.html (The diagrammatic representation for Functor, Monad, Applicatives are inspired from this post).