1st Edition

Lambda calculus is a formal system for expressing computation introduced in the 1930s by Alonzo Church.

It is based on function abstraction λx.(x + 2), application λx.(x + 2)(3), variable binding [x := 3] and substitution (3 + 2) → 5.

An anonymous function that accepts exactly one argument is a lambda expression, or lambda for short.

"Functions that operate on other functions, either by taking them as arguments or by returning them"

— Marijn Haverbeke, Eloquent JavaScript

const numbers = [1, 2, 3, 4];

numbers.filter(n => n > 2); // → [6, 8]
numbers.find(n => n === 2); // → 2

Functional programming is a declarative programming paradigm.

It expresses the logic of a computation without describing its control flow.

// imperative code
if (CONDITION) {
    TRUE PATH
} else {
    FALSE PATH
}

// declarative code
ifElse(CONDITION, TRUE PATH, FALSE PATH);

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.

— Brian Lonsdorf, Mostly Adequate Guide to Functional Programming

const numbers = [1, 2, 3, 4];

// pure
numbers.slice(0, 2); // → [1, 2, 3]
numbers.slice(0, 2); // → [1, 2, 3]
numbers.slice(0, 2); // → [1, 2, 3]

// impure
numbers.splice(0, 2); // → [1, 2]
numbers.splice(0, 2); // → [3, 4]
numbers.splice(0, 2); // → []

In functional programming, immutability is an important quality of data that makes reasoning about the logic of a program much easier.

// mutable
let counter = 0;

function increment() {
    return ++counter;
}

// immutable
function increment(n) {
    return n + 1;
}

As of ES5, there exists a convenient method for making JavaScript objects immutable.

// mutable
const user = { name: 'John', lastName: 'Doe' };

// immutable
const frozenUser = Object.freeze(user);

"The process of reducing functions of more than one argument to functions of one argument"

— Haskell B. Curry

// uncurried function
const add = (a, b) => a + b;

add(2, 5); // → 7

// curried function
const add = a => b => a + b;

add(2)(5); // → 7

Applying a function to one argument at a time is partial application.

A function has to be curried in order to be partially applied.

const add = a => b => a + b;

const add2 = add(2); // → b => 2 + b

const sum = add2(5); // → 7

Function composition is the application of one function to the result of another.

In other words, the output of one function feeds the input of another.

The following example can be read as "bar after foo", "bar of foo", etc.

bar(foo(x)) = (bar ∘ foo )(x)

The compose function is right-associative — it works right-to-left.

const addThenDiv = compose(div(2), add(5));

addThenDiv(3); // → 4

The pipe function is left-associative — it works left-to-right.

const addThenDiv = pipe(add(5), div(2));

addThenDiv(3); // → 4

Point-free style is concerned with functions that do not reference the arguments (points) on which they operate. Instead, the definitions merely compose other functions.

// pointful
const toKebabCase = text => text.toLowerCase().replace(/\s+/ig, '-');

// point-free
const toKebabCase = compose(replace(/\s+/ig, '_'), toLowerCase);

Such a composition is possible only with curried, data-last functions.

const toLowerCase = string => string.toLowerCase();

const replace = curry((pattern, replacement, string) => string.replace(pattern, replacement));

Hindley-Milner is a type system for lambda calculus with parametric polymorphism. It is based on two papers: The principal type scheme of an object in combinatory logic (Roger J. Hindley, 1969) and A Theory of Type Polymorphism (Robin Milner, 1978).

Writing function signatures in Hindley-Milner notation is a common practice among functional programmers.

// toLowerCase :: String -> String
const toLowerCase = s => s.toLowerCase();

// add :: Number -> Number -> Number
const add = curry((a, b) => a + b);

// length :: String -> Number
const length = s => s.length;

// equals :: a -> b -> Boolean
const equals = curry((a, b) => a === b);

// head :: [a] -> a
const head = xs => xs[0];

// filter :: (a -> Boolean) -> [a] -> [a]
const filter = curry((fn, xs) => xs.filter(fn));

• Ramda
• Rambda

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 Unported License.

The materials herein are all © 2020 Jakub Barczyk.