Functional Programming Jargon

Functional programming (FP) provides many advantages, and its popularity has been increasing as a result. However, each programming paradigm comes with its own unique jargon and FP is no exception. By providing a glossary, we hope to make learning FP easier.

Examples are presented in JavaScript (ES2015). Why JavaScript?

Where applicable, this document uses terms defined in the Fantasy Land spec.

Translations

Table of Contents

Arity

The number of arguments a function takes. From words like unary, binary, ternary, etc.

const sum = (a, b) => a + b
// The arity of sum is 2 (binary)
const inc = a => a + 1
// The arity of inc is 1 (unary)
const zero = () => 0
// The arity of zero is 0 (nullary)

Further reading

Arity on Wikipedia

Higher-Order Functions (HOF)

A function which takes a function as an argument and/or returns a function.

const filter = (predicate, xs) => xs.filter(predicate)

const is = (type) => (x) => Object(x) instanceof type

filter(is(Number), [0, '1', 2, null]) // [0, 2]

Closure

A closure is a scope which captures local variables of a function for access even after the execution has moved out of the block in which it is defined. This allows the values in the closure to be accessed by returned functions.

const addTo = x => y => x + y
const addToFive = addTo(5)
addToFive(3) // => 8

In this case the x is retained in addToFive's closure with the value 5. addToFive can then be called with the y to get back the sum.

Further reading/Sources

Partial Application

Partially applying a function means creating a new function by pre-filling some of the arguments to the original function.

// Helper to create partially applied functions
// Takes a function and some arguments
const partial = (f, ...args) =>
  // returns a function that takes the rest of the arguments
  (...moreArgs) =>
    // and calls the original function with all of them
    f(...args, ...moreArgs)

// Something to apply
const add3 = (a, b, c) => a + b + c

// Partially applying `2` and `3` to `add3` gives you a one-argument function
const fivePlus = partial(add3, 2, 3) // (c) => 2 + 3 + c

fivePlus(4) // 9

You can also use Function.prototype.bind to partially apply a function in JS:

const add1More = add3.bind(null, 2, 3) // (c) => 2 + 3 + c

Partial application helps create simpler functions from more complex ones by baking in data when you have it. Curried functions are automatically partially applied.

Currying

The process of converting a function that takes multiple arguments into a function that takes them one at a time.

Each time the function is called it only accepts one argument and returns a function that takes one argument until all arguments are passed.

const sum = (a, b) => a + b

const curriedSum = (a) => (b) => a + b

curriedSum(40)(2) // 42.

const add2 = curriedSum(2) // (b) => 2 + b

add2(10) // 12

Auto Currying

Transforming a function that takes multiple arguments into one that if given less than its correct number of arguments returns a function that takes the rest. When the function gets the correct number of arguments it is then evaluated.

Lodash & Ramda have a curry function that works this way.

const add = (x, y) => x + y

const curriedAdd = _.curry(add)
curriedAdd(1, 2) // 3
curriedAdd(1) // (y) => 1 + y
curriedAdd(1)(2) // 3

Further reading

Function Composition

The act of putting two functions together to form a third function where the output of one function is the input of the other. This is one of the most important ideas of functional programming.

const compose = (f, g) => (a) => f(g(a)) // Definition
const floorAndToString = compose((val) => val.toString(), Math.floor) // Usage
floorAndToString(121.212121) // '121'

Continuation

At any given point in a program, the part of the code that's yet to be executed is known as a continuation.

const printAsString = (num) => console.log(`Given ${num}`)

const addOneAndContinue = (num, cc) => {
  const result = num + 1
  cc(result)
}

addOneAndContinue(2, printAsString) // 'Given 3'

Continuations are often seen in asynchronous programming when the program needs to wait to receive data before it can continue. The response is often passed off to the rest of the program, which is the continuation, once it's been received.

const continueProgramWith = (data) => {
  // Continues program with data
}

readFileAsync('path/to/file', (err, response) => {
  if (err) {
    // handle error
    return
  }
  continueProgramWith(response)
})

Pure Function

A function is pure if the return value is only determined by its input values, and does not produce side effects. The function must always return the same result when given the same input.

const greet = (name) => `Hi, ${name}`

greet('Brianne') // 'Hi, Brianne'

As opposed to each of the following:

window.name = 'Brianne'

const greet = () => `Hi, ${window.name}`

greet() // "Hi, Brianne"

The above example's output is based on data stored outside of the function...

let greeting

const greet = (name) => {
  greeting = `Hi, ${name}`
}

greet('Brianne')
greeting // "Hi, Brianne"

... and this one modifies state outside of the function.

Side effects

A function or expression is said to have a side effect if apart from returning a value, it interacts with (reads from or writes to) external mutable state.

const differentEveryTime = new Date()

console.log('IO is a side effect!')

Idempotence

A function is idempotent if reapplying it to its result does not produce a different result.

Math.abs(Math.abs(10))

sort(sort(sort([2, 1])))

Point-Free Style

Writing functions where the definition does not explicitly identify the arguments used. This style usually requires currying or other Higher-Order functions. A.K.A Tacit programming.

// Given
const map = (fn) => (list) => list.map(fn)
const add = (a) => (b) => a + b

// Then

// Not point-free - `numbers` is an explicit argument
const incrementAll = (numbers) => map(add(1))(numbers)

// Point-free - The list is an implicit argument
const incrementAll2 = map(add(1))

Point-free function definitions look just like normal assignments without function or =>. It's worth mentioning that point-free functions are not necessarily better than their counterparts, as they can be more difficult to understand when complex.

Predicate

A predicate is a function that returns true or false for a given value. A common use of a predicate is as the callback for array filter.

const predicate = (a) => a > 2

;[1, 2, 3, 4].filter(predicate) // [3, 4]

Contracts

A contract specifies the obligations and guarantees of the behavior from a function or expression at runtime. This acts as a set of rules that are expected from the input and output of a function or expression, and errors are generally reported whenever a contract is violated.

// Define our contract : int -> boolean
const contract = (input) => {
  if (typeof input === 'number') return true
  throw new Error('Contract violated: expected int -> boolean')
}

const addOne = (num) => contract(num) && num + 1

addOne(2) // 3
addOne('some string') // Contract violated: expected int -> boolean

Value

Anything that can be assigned to a variable.

5
Object.freeze({ name: 'John', age: 30 }) // The `freeze` function enforces immutability.
;(a) => a
;[1]
undefined

Constant

A variable that cannot be reassigned once defined.

const five = 5
const john = Object.freeze({ name: 'John', age: 30 })

Constants are referentially transparent. That is, they can be replaced with the values that they represent without affecting the result.

With the above two constants the following expression will always return true.

john.age + five === ({ name: 'John', age: 30 }).age + 5

Constant Function

A curried function that ignores its second argument:

const constant = a => () => a

;[1, 2].map(constant(0)) // => [0, 0]

Constant Functor

Object whose map doesn't transform the contents. See Functor.

Constant(1).map(n => n + 1) // => Constant(1)

Constant Monad

Object whose chain doesn't transform the contents. See Monad.

Constant(1).chain(n => Constant(n + 1)) // => Constant(1)

Functor

An object that implements a map function that takes a function which is run on the contents of that object. A functor must adhere to two rules:

Preserves identity

object.map(x => x)

is equivalent to just object.

Composable

object.map(x => g(f(x)))

is equivalent to

object.map(f).map(g)

(f, g are arbitrary composable functions)

The reference implementation of Option is a functor as it satisfies the rules:

Some(1).map(x => x) // = Some(1)

and

const f = x => x + 1
const g = x => x * 2

Some(1).map(x => g(f(x))) // = Some(4)
Some(1).map(f).map(g) // = Some(4)

Pointed Functor

An object with an of function that puts any single value into it.

ES2015 adds Array.of making arrays a pointed functor.

Array.of(1) // [1]

Lift

Lifting is when you take a value and put it into an object like a functor. If you lift a function into an Applicative Functor then you can make it work on values that are also in that functor.

Some implementations have a function called lift, or liftA2 to make it easier to run functions on functors.

const liftA2 = (f) => (a, b) => a.map(f).ap(b) // note it's `ap` and not `map`.

const mult = a => b => a * b

const liftedMult = liftA2(mult) // this function now works on functors like array

liftedMult([1, 2], [3]) // [3, 6]
liftA2(a => b => a + b)([1, 2], [30, 40]) // [31, 41, 32, 42]

Lifting a one-argument function and applying it does the same thing as map.

const increment = (x) => x + 1

lift(increment)([2]) // [3]
;[2].map(increment) // [3]

Lifting simple values can be simply creating the object.

Array.of(1) // => [1]

Referential Transparency

An expression that can be replaced with its value without changing the behavior of the program is said to be referentially transparent.

Given the function greet:

const greet = () => 'Hello World!'

Any invocation of greet() can be replaced with Hello World! hence greet is referentially transparent. This would be broken if greet depended on external state like configuration or a database call. See also Pure Function and Equational Reasoning.

Equational Reasoning

When an application is composed of expressions and devoid of side effects, truths about the system can be derived from the parts. You can also be confident about details of your system without having to go through every function.

const grainToDogs = compose(chickenIntoDogs, grainIntoChicken)
const grainToCats = compose(dogsIntoCats, grainToDogs)

In the example above, if you know that chickenIntoDogs and grainIntoChicken are pure then you know that the composition is pure. This can be taken further when more is known about the functions (associative, commutative, idempotent, etc...).

Lambda

An anonymous function that can be treated like a value.

;(function (a) {
  return a + 1
})

;(a) => a + 1

Lambdas are often passed as arguments to Higher-Order functions:

;[1, 2].map((a) => a + 1) // [2, 3]

You can assign a lambda to a variable:

const add1 = (a) => a + 1

Lambda Calculus

A branch of mathematics that uses functions to create a universal model of computation.

Functional Combinator

A higher-order function, usually curried, which returns a new function changed in some way. Functional combinators are often used in Point-Free Style to write especially terse programs.

// The "C" combinator takes a curried two-argument function and returns one which calls the original function with the arguments reversed.
const C = (f) => (a) => (b) => f(b)(a)

const divide = (a) => (b) => a / b

const divideBy = C(divide)

const divBy10 = divideBy(10)

divBy10(30) // => 3

See also List of Functional Combinators in JavaScript which includes links to more references.

Lazy evaluation

Lazy evaluation is a call-by-need evaluation mechanism that delays the evaluation of an expression until its value is needed. In functional languages, this allows for structures like infinite lists, which would not normally be available in an imperative language where the sequencing of commands is significant.

const rand = function * () {
  while (1 < 2) {
    yield Math.random()
  }
}

const randIter = rand()
randIter.next() // Each execution gives a random value, expression is evaluated on need.

Monoid

An object with a function that "combines" that object with another of the same type (semigroup) which has an "identity" value.

One simple monoid is the addition of numbers:

1 + 1 // 2

In this case number is the object and + is the function.

When any value is combined with the "identity" value the result must be the original value. The identity must also be commutative.

The identity value for addition is 0.

1 + 0 // 1
0 + 1 // 1
1 + 0 === 0 + 1

It's also required that the grouping of operations will not affect the result (associativity):

1 + (2 + 3) === (1 + 2) + 3 // true

Array concatenation also forms a monoid:

;[1, 2].concat([3, 4]) // [1, 2, 3, 4]

The identity value is empty array []:

;[1, 2].concat([]) // [1, 2]

As a counterexample, subtraction does not form a monoid because there is no commutative identity value:

0 - 4 === 4 - 0 // false

Monad

A monad is an object with of and chain functions. chain is like map except it un-nests the resulting nested object.

// Implementation
Array.prototype.chain = function (f) {
  return this.reduce((acc, it) => acc.concat(f(it)), [])
}

// Usage
Array.of('cat,dog', 'fish,bird').chain((a) => a.split(',')) // ['cat', 'dog', 'fish', 'bird']

// Contrast to map
Array.of('cat,dog', 'fish,bird').map((a) => a.split(',')) // [['cat', 'dog'], ['fish', 'bird']]

of is also known as return in other functional languages. chain is also known as flatmap and bind in other languages.

Comonad

An object that has extract and extend functions.

const CoIdentity = (v) => ({
  val: v,
  extract () {
    return this.val
  },
  extend (f) {
    return CoIdentity(f(this))
  }
})

extract takes a value out of a functor:

CoIdentity(1).extract() // 1

extend runs a function on the comonad. The function should return the same type as the comonad:

CoIdentity(1).extend((co) => co.extract() + 1) // CoIdentity(2)

Kleisli Composition

An operation for composing two monad-returning functions (Kleisli Arrows) where they have compatible types. In Haskell this is the >=> operator.

Using Option:

// safeParseNum :: String -> Option Number
const safeParseNum = (b) => {
  const n = parseNumber(b)
  return isNaN(n) ? None() : Some(n)
}

// validatePositive :: Number -> Option Number
const validatePositive = (a) => a > 0 ? Some(a) : None()

// kleisliCompose :: Monad M => ((b -> M c), (a -> M b)) -> a -> M c
const kleisliCompose = (g, f) => (x) => f(x).chain(g)

// parseAndValidate :: String -> Option Number
const parseAndValidate = kleisliCompose(validatePositive, safeParseNum)

parseAndValidate('1') // => Some(1)
parseAndValidate('asdf') // => None
parseAndValidate('999') // => Some(999)

This works because:

option is a monad,
both validatePositive and safeParseNum return the same kind of monad (Option),
the type of validatePositive's argument matches safeParseNum's unwrapped return.

Applicative Functor

An applicative functor is an object with an ap function. ap applies a function in the object to a value in another object of the same type.

// Implementation
Array.prototype.ap = function (xs) {
  return this.reduce((acc, f) => acc.concat(xs.map(f)), [])
}

// Example usage
;[(a) => a + 1].ap([1]) // [2]

This is useful if you have two objects and you want to apply a binary function to their contents.

// Arrays that you want to combine
const arg1 = [1, 3]
const arg2 = [4, 5]

// combining function - must be curried for this to work
const add = (x) => (y) => x + y

const partiallyAppliedAdds = [add].ap(arg1) // [(y) => 1 + y, (y) => 3 + y]

This gives you an array of functions that you can call ap on to get the result:

partiallyAppliedAdds.ap(arg2) // [5, 6, 7, 8]

Morphism

A relationship between objects within a category. In the context of functional programming all functions are morphisms.

Homomorphism

A function where there is a structural property that is the same in the input as well as the output.

For example, in a Monoid homomorphism both the input and the output are monoids even if their types are different.

// toList :: [number] -> string
const toList = (a) => a.join(', ')

toList is a homomorphism because:

array is a monoid - has a concat operation and an identity value ([]),
string is a monoid - has a concat operation and an identity value ('').

In this way, a homomorphism relates to whatever property you care about in the input and output of a transformation.

Endomorphisms and Isomorphisms are examples of homomorphisms.

Further Reading

Homomorphism | Learning Functional Programming in Go

Endomorphism

A function where the input type is the same as the output. Since the types are identical, endomorphisms are also homomorphisms.

// uppercase :: String -> String
const uppercase = (str) => str.toUpperCase()

// decrement :: Number -> Number
const decrement = (x) => x - 1

Isomorphism

A morphism made of a pair of transformations between 2 types of objects that is structural in nature and no data is lost.

For example, 2D coordinates could be stored as an array [2,3] or object {x: 2, y: 3}.

// Providing functions to convert in both directions makes the 2D coordinate structures isomorphic.
const pairToCoords = (pair) => ({ x: pair[0], y: pair[1] })

const coordsToPair = (coords) => [coords.x, coords.y]

coordsToPair(pairToCoords([1, 2])) // [1, 2]

pairToCoords(coordsToPair({ x: 1, y: 2 })) // {x: 1, y: 2}

Isomorphisms are an interesting example of morphism because more than single function is necessary for it to be satisfied. Isomorphisms are also homomorphisms since both input and output types share the property of being reversible.

Catamorphism

A function which deconstructs a structure into a single value. reduceRight is an example of a catamorphism for array structures.

// sum is a catamorphism from [Number] -> Number
const sum = xs => xs.reduceRight((acc, x) => acc + x, 0)

sum([1, 2, 3, 4, 5]) // 15

Anamorphism

A function that builds up a structure by repeatedly applying a function to its argument. unfold is an example which generates an array from a function and a seed value. This is the opposite of a catamorphism. You can think of this as an anamorphism builds up a structure and catamorphism breaks it down.

const unfold = (f, seed) => {
  function go (f, seed, acc) {
    const res = f(seed)
    return res ? go(f, res[1], acc.concat([res[0]])) : acc
  }
  return go(f, seed, [])
}

const countDown = n => unfold((n) => {
  return n <= 0 ? undefined : [n, n - 1]
}, n)

countDown(5) // [5, 4, 3, 2, 1]

Hylomorphism

The function which composes an anamorphism followed by a catamorphism.

const sumUpToX = (x) => sum(countDown(x))
sumUpToX(5) // 15

Paramorphism

A function just like reduceRight. However, there's a difference:

In paramorphism, your reducer's arguments are the current value, the reduction of all previous values, and the list of values that formed that reduction.

// Obviously not safe for lists containing `undefined`,
// but good enough to make the point.
const para = (reducer, accumulator, elements) => {
  if (elements.length === 0) { return accumulator }

  const head = elements[0]
  const tail = elements.slice(1)

  return reducer(head, tail, para(reducer, accumulator, tail))
}

const suffixes = list => para(
  (x, xs, suffxs) => [xs, ...suffxs],
  [],
  list
)

suffixes([1, 2, 3, 4, 5]) // [[2, 3, 4, 5], [3, 4, 5], [4, 5], [5], []]

The third parameter in the reducer (in the above example, [x, ... xs]) is kind of like having a history of what got you to your current acc value.

Apomorphism

The opposite of paramorphism, just as anamorphism is the opposite of catamorphism. With paramorphism, you retain access to the accumulator and what has been accumulated, apomorphism lets you unfold with the potential to return early.

Setoid

An object that has an equals function which can be used to compare other objects of the same type.

Make array a setoid:

Array.prototype.equals = function (arr) {
  const len = this.length
  if (len !== arr.length) {
    return false
  }
  for (let i = 0; i < len; i++) {
    if (this[i] !== arr[i]) {
      return false
    }
  }
  return true
}

;[1, 2].equals([1, 2]) // true
;[1, 2].equals([0]) // false

Semigroup

An object that has a concat function that combines it with another object of the same type.

;[1].concat([2]) // [1, 2]

Foldable

An object that has a reduce function that applies a function against an accumulator and each element in the array (from left to right) to reduce it to a single value.

const sum = (list) => list.reduce((acc, val) => acc + val, 0)
sum([1, 2, 3]) // 6

Lens

A lens is a structure (often an object or function) that pairs a getter and a non-mutating setter for some other data structure.

// Using [Ramda's lens](http://ramdajs.com/docs/#lens)
const nameLens = R.lens(
  // getter for name property on an object
  (obj) => obj.name,
  // setter for name property
  (val, obj) => Object.assign({}, obj, { name: val })
)

Having the pair of get and set for a given data structure enables a few key features.

const person = { name: 'Gertrude Blanch' }

// invoke the getter
R.view(nameLens, person) // 'Gertrude Blanch'

// invoke the setter
R.set(nameLens, 'Shafi Goldwasser', person) // {name: 'Shafi Goldwasser'}

// run a function on the value in the structure
R.over(nameLens, uppercase, person) // {name: 'GERTRUDE BLANCH'}

Lenses are also composable. This allows easy immutable updates to deeply nested data.

// This lens focuses on the first item in a non-empty array
const firstLens = R.lens(
  // get first item in array
  xs => xs[0],
  // non-mutating setter for first item in array
  (val, [__, ...xs]) => [val, ...xs]
)

const people = [{ name: 'Gertrude Blanch' }, { name: 'Shafi Goldwasser' }]

// Despite what you may assume, lenses compose left-to-right.
R.over(compose(firstLens, nameLens), uppercase, people) // [{'name': 'GERTRUDE BLANCH'}, {'name': 'Shafi Goldwasser'}]

Other implementations:

partial.lenses - Tasty syntax sugar and a lot of powerful features
nanoscope - Fluent-interface

Type Signatures

Often functions in JavaScript will include comments that indicate the types of their arguments and return values.

There's quite a bit of variance across the community, but they often follow the following patterns:

// functionName :: firstArgType -> secondArgType -> returnType

// add :: Number -> Number -> Number
const add = (x) => (y) => x + y

// increment :: Number -> Number
const increment = (x) => x + 1

If a function accepts another function as an argument it is wrapped in parentheses.

// call :: (a -> b) -> a -> b
const call = (f) => (x) => f(x)

The letters a, b, c, d are used to signify that the argument can be of any type. The following version of map takes a function that transforms a value of some type a into another type b, an array of values of type a, and returns an array of values of type b.

// map :: (a -> b) -> [a] -> [b]
const map = (f) => (list) => list.map(f)

Further reading

Ramda's type signatures
Mostly Adequate Guide
What is Hindley-Milner? on Stack Overflow

Algebraic data type

A composite type made from putting other types together. Two common classes of algebraic types are sum and product.

Sum type

A Sum type is the combination of two types together into another one. It is called sum because the number of possible values in the result type is the sum of the input types.

JavaScript doesn't have types like this, but we can use Sets to pretend:

// imagine that rather than sets here we have types that can only have these values
const bools = new Set([true, false])
const halfTrue = new Set(['half-true'])

// The weakLogic type contains the sum of the values from bools and halfTrue
const weakLogicValues = new Set([...bools, ...halfTrue])

Sum types are sometimes called union types, discriminated unions, or tagged unions.

There's a couple libraries in JS which help with defining and using union types.

Flow includes union types and TypeScript has Enums to serve the same role.

Product type

A product type combines types together in a way you're probably more familiar with:

// point :: (Number, Number) -> {x: Number, y: Number}
const point = (x, y) => ({ x, y })

It's called a product because the total possible values of the data structure is the product of the different values. Many languages have a tuple type which is the simplest formulation of a product type.

Further reading

Set theory on Wikipedia

Option

Option is a sum type with two cases often called Some and None.

Option is useful for composing functions that might not return a value.

// Naive definition

const Some = (v) => ({
  val: v,
  map (f) {
    return Some(f(this.val))
  },
  chain (f) {
    return f(this.val)
  }
})

const None = () => ({
  map (f) {
    return this
  },
  chain (f) {
    return this
  }
})

// maybeProp :: (String, {a}) -> Option a
const maybeProp = (key, obj) => typeof obj[key] === 'undefined' ? None() : Some(obj[key])

Use chain to sequence functions that return Options:

// getItem :: Cart -> Option CartItem
const getItem = (cart) => maybeProp('item', cart)

// getPrice :: Item -> Option Number
const getPrice = (item) => maybeProp('price', item)

// getNestedPrice :: cart -> Option a
const getNestedPrice = (cart) => getItem(cart).chain(getPrice)

getNestedPrice({}) // None()
getNestedPrice({ item: { foo: 1 } }) // None()
getNestedPrice({ item: { price: 9.99 } }) // Some(9.99)

Option is also known as Maybe. Some is sometimes called Just. None is sometimes called Nothing.

Function

A function f :: A => B is an expression - often called arrow or lambda expression - with exactly one (immutable) parameter of type A and exactly one return value of type B. That value depends entirely on the argument, making functions context-independent, or referentially transparent. What is implied here is that a function must not produce any hidden side effects - a function is always pure, by definition. These properties make functions pleasant to work with: they are entirely deterministic and therefore predictable. Functions enable working with code as data, abstracting over behaviour:

// times2 :: Number -> Number
const times2 = n => n * 2

;[1, 2, 3].map(times2) // [2, 4, 6]

Partial function

A partial function is a function which is not defined for all arguments - it might return an unexpected result or may never terminate. Partial functions add cognitive overhead, they are harder to reason about and can lead to runtime errors. Some examples:

// example 1: sum of the list
// sum :: [Number] -> Number
const sum = arr => arr.reduce((a, b) => a + b)
sum([1, 2, 3]) // 6
sum([]) // TypeError: Reduce of empty array with no initial value

// example 2: get the first item in list
// first :: [A] -> A
const first = a => a[0]
first([42]) // 42
first([]) // undefined
// or even worse:
first([[42]])[0] // 42
first([])[0] // Uncaught TypeError: Cannot read property '0' of undefined

// example 3: repeat function N times
// times :: Number -> (Number -> Number) -> Number
const times = n => fn => n && (fn(n), times(n - 1)(fn))
times(3)(console.log)
// 3
// 2
// 1
times(-1)(console.log)
// RangeError: Maximum call stack size exceeded

Dealing with partial functions

Partial functions are dangerous as they need to be treated with great caution. You might get an unexpected (wrong) result or run into runtime errors. Sometimes a partial function might not return at all. Being aware of and treating all these edge cases accordingly can become very tedious. Fortunately a partial function can be converted to a regular (or total) one. We can provide default values or use guards to deal with inputs for which the (previously) partial function is undefined. Utilizing the Option type, we can yield either Some(value) or None where we would otherwise have behaved unexpectedly:

// example 1: sum of the list
// we can provide default value so it will always return result
// sum :: [Number] -> Number
const sum = arr => arr.reduce((a, b) => a + b, 0)
sum([1, 2, 3]) // 6
sum([]) // 0

// example 2: get the first item in list
// change result to Option
// first :: [A] -> Option A
const first = a => a.length ? Some(a[0]) : None()
first([42]).map(a => console.log(a)) // 42
first([]).map(a => console.log(a)) // console.log won't execute at all
// our previous worst case
first([[42]]).map(a => console.log(a[0])) // 42
first([]).map(a => console.log(a[0])) // won't execute, so we won't have error here
// more of that, you will know by function return type (Option)
// that you should use `.map` method to access the data and you will never forget
// to check your input because such check become built-in into the function

// example 3: repeat function N times
// we should make function always terminate by changing conditions:
// times :: Number -> (Number -> Number) -> Number
const times = n => fn => n > 0 && (fn(n), times(n - 1)(fn))
times(3)(console.log)
// 3
// 2
// 1
times(-1)(console.log)
// won't execute anything

Making your partial functions total ones, these kinds of runtime errors can be prevented. Always returning a value will also make for code that is both easier to maintain and to reason about.

Total Function

A function which returns a valid result for all inputs defined in its type. This is as opposed to Partial Functions which may throw an error, return an unexpected result, or fail to terminate.

Functional Programming Libraries in JavaScript

P.S: This repo is successful due to the wonderful contributions!

Functional Programming Jargon

Examples are presented in JavaScript (ES2015). Why JavaScript?

Where applicable, this document uses terms defined in the Fantasy Land spec.

Translations

Table of Contents

Arity

The number of arguments a function takes. From words like unary, binary, ternary, etc.

const sum = (a, b) => a + b
// The arity of sum is 2 (binary)
const inc = a => a + 1
// The arity of inc is 1 (unary)
const zero = () => 0
// The arity of zero is 0 (nullary)

Further reading

Arity on Wikipedia

Higher-Order Functions (HOF)

A function which takes a function as an argument and/or returns a function.

const filter = (predicate, xs) => xs.filter(predicate)

const is = (type) => (x) => Object(x) instanceof type

filter(is(Number), [0, '1', 2, null]) // [0, 2]

Closure

const addTo = x => y => x + y
const addToFive = addTo(5)
addToFive(3) // => 8

In this case the x is retained in addToFive's closure with the value 5. addToFive can then be called with the y to get back the sum.

Further reading/Sources

Partial Application

Partially applying a function means creating a new function by pre-filling some of the arguments to the original function.

// Helper to create partially applied functions
// Takes a function and some arguments
const partial = (f, ...args) =>
  // returns a function that takes the rest of the arguments
  (...moreArgs) =>
    // and calls the original function with all of them
    f(...args, ...moreArgs)

// Something to apply
const add3 = (a, b, c) => a + b + c

// Partially applying `2` and `3` to `add3` gives you a one-argument function
const fivePlus = partial(add3, 2, 3) // (c) => 2 + 3 + c

fivePlus(4) // 9

You can also use Function.prototype.bind to partially apply a function in JS:

const add1More = add3.bind(null, 2, 3) // (c) => 2 + 3 + c

Partial application helps create simpler functions from more complex ones by baking in data when you have it. Curried functions are automatically partially applied.

Currying

The process of converting a function that takes multiple arguments into a function that takes them one at a time.

Each time the function is called it only accepts one argument and returns a function that takes one argument until all arguments are passed.

const sum = (a, b) => a + b

const curriedSum = (a) => (b) => a + b

curriedSum(40)(2) // 42.

const add2 = curriedSum(2) // (b) => 2 + b

add2(10) // 12

Auto Currying

Lodash & Ramda have a curry function that works this way.

const add = (x, y) => x + y

const curriedAdd = _.curry(add)
curriedAdd(1, 2) // 3
curriedAdd(1) // (y) => 1 + y
curriedAdd(1)(2) // 3

Further reading

Function Composition

The act of putting two functions together to form a third function where the output of one function is the input of the other. This is one of the most important ideas of functional programming.

const compose = (f, g) => (a) => f(g(a)) // Definition
const floorAndToString = compose((val) => val.toString(), Math.floor) // Usage
floorAndToString(121.212121) // '121'

Continuation

At any given point in a program, the part of the code that's yet to be executed is known as a continuation.

const printAsString = (num) => console.log(`Given ${num}`)

const addOneAndContinue = (num, cc) => {
  const result = num + 1
  cc(result)
}

addOneAndContinue(2, printAsString) // 'Given 3'

const continueProgramWith = (data) => {
  // Continues program with data
}

readFileAsync('path/to/file', (err, response) => {
  if (err) {
    // handle error
    return
  }
  continueProgramWith(response)
})

Pure Function

A function is pure if the return value is only determined by its input values, and does not produce side effects. The function must always return the same result when given the same input.

const greet = (name) => `Hi, ${name}`

greet('Brianne') // 'Hi, Brianne'

As opposed to each of the following:

window.name = 'Brianne'

const greet = () => `Hi, ${window.name}`

greet() // "Hi, Brianne"

The above example's output is based on data stored outside of the function...

let greeting

const greet = (name) => {
  greeting = `Hi, ${name}`
}

greet('Brianne')
greeting // "Hi, Brianne"

... and this one modifies state outside of the function.

Side effects

A function or expression is said to have a side effect if apart from returning a value, it interacts with (reads from or writes to) external mutable state.

const differentEveryTime = new Date()

console.log('IO is a side effect!')

Idempotence

A function is idempotent if reapplying it to its result does not produce a different result.

Math.abs(Math.abs(10))

sort(sort(sort([2, 1])))

Point-Free Style

Writing functions where the definition does not explicitly identify the arguments used. This style usually requires currying or other Higher-Order functions. A.K.A Tacit programming.

// Given
const map = (fn) => (list) => list.map(fn)
const add = (a) => (b) => a + b

// Then

// Not point-free - `numbers` is an explicit argument
const incrementAll = (numbers) => map(add(1))(numbers)

// Point-free - The list is an implicit argument
const incrementAll2 = map(add(1))

Predicate

A predicate is a function that returns true or false for a given value. A common use of a predicate is as the callback for array filter.

const predicate = (a) => a > 2

;[1, 2, 3, 4].filter(predicate) // [3, 4]

Contracts

// Define our contract : int -> boolean
const contract = (input) => {
  if (typeof input === 'number') return true
  throw new Error('Contract violated: expected int -> boolean')
}

const addOne = (num) => contract(num) && num + 1

addOne(2) // 3
addOne('some string') // Contract violated: expected int -> boolean

Value

Anything that can be assigned to a variable.

5
Object.freeze({ name: 'John', age: 30 }) // The `freeze` function enforces immutability.
;(a) => a
;[1]
undefined

Constant

A variable that cannot be reassigned once defined.

const five = 5
const john = Object.freeze({ name: 'John', age: 30 })

Constants are referentially transparent. That is, they can be replaced with the values that they represent without affecting the result.

With the above two constants the following expression will always return true.

john.age + five === ({ name: 'John', age: 30 }).age + 5

Constant Function

A curried function that ignores its second argument:

const constant = a => () => a

;[1, 2].map(constant(0)) // => [0, 0]

Constant Functor

Object whose map doesn't transform the contents. See Functor.

Constant(1).map(n => n + 1) // => Constant(1)

Constant Monad

Object whose chain doesn't transform the contents. See Monad.

Constant(1).chain(n => Constant(n + 1)) // => Constant(1)

Functor

An object that implements a map function that takes a function which is run on the contents of that object. A functor must adhere to two rules:

Preserves identity

object.map(x => x)

is equivalent to just object.

Composable

object.map(x => g(f(x)))

is equivalent to

object.map(f).map(g)

(f, g are arbitrary composable functions)

The reference implementation of Option is a functor as it satisfies the rules:

Some(1).map(x => x) // = Some(1)

and

const f = x => x + 1
const g = x => x * 2

Some(1).map(x => g(f(x))) // = Some(4)
Some(1).map(f).map(g) // = Some(4)

Pointed Functor

An object with an of function that puts any single value into it.

ES2015 adds Array.of making arrays a pointed functor.

Array.of(1) // [1]

Lift

Lifting is when you take a value and put it into an object like a functor. If you lift a function into an Applicative Functor then you can make it work on values that are also in that functor.

Some implementations have a function called lift, or liftA2 to make it easier to run functions on functors.

const liftA2 = (f) => (a, b) => a.map(f).ap(b) // note it's `ap` and not `map`.

const mult = a => b => a * b

const liftedMult = liftA2(mult) // this function now works on functors like array

liftedMult([1, 2], [3]) // [3, 6]
liftA2(a => b => a + b)([1, 2], [30, 40]) // [31, 41, 32, 42]

Lifting a one-argument function and applying it does the same thing as map.

const increment = (x) => x + 1

lift(increment)([2]) // [3]
;[2].map(increment) // [3]

Lifting simple values can be simply creating the object.

Array.of(1) // => [1]

Referential Transparency

An expression that can be replaced with its value without changing the behavior of the program is said to be referentially transparent.

Given the function greet:

const greet = () => 'Hello World!'

Equational Reasoning

const grainToDogs = compose(chickenIntoDogs, grainIntoChicken)
const grainToCats = compose(dogsIntoCats, grainToDogs)

Lambda

An anonymous function that can be treated like a value.

;(function (a) {
  return a + 1
})

;(a) => a + 1

Lambdas are often passed as arguments to Higher-Order functions:

;[1, 2].map((a) => a + 1) // [2, 3]

You can assign a lambda to a variable:

const add1 = (a) => a + 1

Lambda Calculus

A branch of mathematics that uses functions to create a universal model of computation.

Functional Combinator

A higher-order function, usually curried, which returns a new function changed in some way. Functional combinators are often used in Point-Free Style to write especially terse programs.

// The "C" combinator takes a curried two-argument function and returns one which calls the original function with the arguments reversed.
const C = (f) => (a) => (b) => f(b)(a)

const divide = (a) => (b) => a / b

const divideBy = C(divide)

const divBy10 = divideBy(10)

divBy10(30) // => 3

See also List of Functional Combinators in JavaScript which includes links to more references.

Lazy evaluation

const rand = function * () {
  while (1 < 2) {
    yield Math.random()
  }
}

const randIter = rand()
randIter.next() // Each execution gives a random value, expression is evaluated on need.

Monoid

An object with a function that "combines" that object with another of the same type (semigroup) which has an "identity" value.

One simple monoid is the addition of numbers:

1 + 1 // 2

In this case number is the object and + is the function.

When any value is combined with the "identity" value the result must be the original value. The identity must also be commutative.

The identity value for addition is 0.

1 + 0 // 1
0 + 1 // 1
1 + 0 === 0 + 1

It's also required that the grouping of operations will not affect the result (associativity):

1 + (2 + 3) === (1 + 2) + 3 // true

Array concatenation also forms a monoid:

;[1, 2].concat([3, 4]) // [1, 2, 3, 4]

The identity value is empty array []:

;[1, 2].concat([]) // [1, 2]

As a counterexample, subtraction does not form a monoid because there is no commutative identity value:

0 - 4 === 4 - 0 // false

Monad

A monad is an object with of and chain functions. chain is like map except it un-nests the resulting nested object.

// Implementation
Array.prototype.chain = function (f) {
  return this.reduce((acc, it) => acc.concat(f(it)), [])
}

// Usage
Array.of('cat,dog', 'fish,bird').chain((a) => a.split(',')) // ['cat', 'dog', 'fish', 'bird']

// Contrast to map
Array.of('cat,dog', 'fish,bird').map((a) => a.split(',')) // [['cat', 'dog'], ['fish', 'bird']]

of is also known as return in other functional languages. chain is also known as flatmap and bind in other languages.

Comonad

An object that has extract and extend functions.

const CoIdentity = (v) => ({
  val: v,
  extract () {
    return this.val
  },
  extend (f) {
    return CoIdentity(f(this))
  }
})

extract takes a value out of a functor:

CoIdentity(1).extract() // 1

extend runs a function on the comonad. The function should return the same type as the comonad:

CoIdentity(1).extend((co) => co.extract() + 1) // CoIdentity(2)

Kleisli Composition

An operation for composing two monad-returning functions (Kleisli Arrows) where they have compatible types. In Haskell this is the >=> operator.

Using Option:

// safeParseNum :: String -> Option Number
const safeParseNum = (b) => {
  const n = parseNumber(b)
  return isNaN(n) ? None() : Some(n)
}

// validatePositive :: Number -> Option Number
const validatePositive = (a) => a > 0 ? Some(a) : None()

// kleisliCompose :: Monad M => ((b -> M c), (a -> M b)) -> a -> M c
const kleisliCompose = (g, f) => (x) => f(x).chain(g)

// parseAndValidate :: String -> Option Number
const parseAndValidate = kleisliCompose(validatePositive, safeParseNum)

parseAndValidate('1') // => Some(1)
parseAndValidate('asdf') // => None
parseAndValidate('999') // => Some(999)

This works because:

option is a monad,
both validatePositive and safeParseNum return the same kind of monad (Option),
the type of validatePositive's argument matches safeParseNum's unwrapped return.

Applicative Functor

An applicative functor is an object with an ap function. ap applies a function in the object to a value in another object of the same type.

// Implementation
Array.prototype.ap = function (xs) {
  return this.reduce((acc, f) => acc.concat(xs.map(f)), [])
}

// Example usage
;[(a) => a + 1].ap([1]) // [2]

This is useful if you have two objects and you want to apply a binary function to their contents.

// Arrays that you want to combine
const arg1 = [1, 3]
const arg2 = [4, 5]

// combining function - must be curried for this to work
const add = (x) => (y) => x + y

const partiallyAppliedAdds = [add].ap(arg1) // [(y) => 1 + y, (y) => 3 + y]

This gives you an array of functions that you can call ap on to get the result:

partiallyAppliedAdds.ap(arg2) // [5, 6, 7, 8]

Morphism

A relationship between objects within a category. In the context of functional programming all functions are morphisms.

Homomorphism

A function where there is a structural property that is the same in the input as well as the output.

For example, in a Monoid homomorphism both the input and the output are monoids even if their types are different.

// toList :: [number] -> string
const toList = (a) => a.join(', ')

toList is a homomorphism because:

array is a monoid - has a concat operation and an identity value ([]),
string is a monoid - has a concat operation and an identity value ('').

In this way, a homomorphism relates to whatever property you care about in the input and output of a transformation.

Endomorphisms and Isomorphisms are examples of homomorphisms.

Further Reading

Homomorphism | Learning Functional Programming in Go

Endomorphism

A function where the input type is the same as the output. Since the types are identical, endomorphisms are also homomorphisms.

// uppercase :: String -> String
const uppercase = (str) => str.toUpperCase()

// decrement :: Number -> Number
const decrement = (x) => x - 1

Isomorphism

A morphism made of a pair of transformations between 2 types of objects that is structural in nature and no data is lost.

For example, 2D coordinates could be stored as an array [2,3] or object {x: 2, y: 3}.

// Providing functions to convert in both directions makes the 2D coordinate structures isomorphic.
const pairToCoords = (pair) => ({ x: pair[0], y: pair[1] })

const coordsToPair = (coords) => [coords.x, coords.y]

coordsToPair(pairToCoords([1, 2])) // [1, 2]

pairToCoords(coordsToPair({ x: 1, y: 2 })) // {x: 1, y: 2}

Catamorphism

A function which deconstructs a structure into a single value. reduceRight is an example of a catamorphism for array structures.

// sum is a catamorphism from [Number] -> Number
const sum = xs => xs.reduceRight((acc, x) => acc + x, 0)

sum([1, 2, 3, 4, 5]) // 15

Anamorphism

const unfold = (f, seed) => {
  function go (f, seed, acc) {
    const res = f(seed)
    return res ? go(f, res[1], acc.concat([res[0]])) : acc
  }
  return go(f, seed, [])
}

const countDown = n => unfold((n) => {
  return n <= 0 ? undefined : [n, n - 1]
}, n)

countDown(5) // [5, 4, 3, 2, 1]

Hylomorphism

The function which composes an anamorphism followed by a catamorphism.

const sumUpToX = (x) => sum(countDown(x))
sumUpToX(5) // 15

Paramorphism

A function just like reduceRight. However, there's a difference:

In paramorphism, your reducer's arguments are the current value, the reduction of all previous values, and the list of values that formed that reduction.

// Obviously not safe for lists containing `undefined`,
// but good enough to make the point.
const para = (reducer, accumulator, elements) => {
  if (elements.length === 0) { return accumulator }

  const head = elements[0]
  const tail = elements.slice(1)

  return reducer(head, tail, para(reducer, accumulator, tail))
}

const suffixes = list => para(
  (x, xs, suffxs) => [xs, ...suffxs],
  [],
  list
)

suffixes([1, 2, 3, 4, 5]) // [[2, 3, 4, 5], [3, 4, 5], [4, 5], [5], []]

The third parameter in the reducer (in the above example, [x, ... xs]) is kind of like having a history of what got you to your current acc value.

Apomorphism

Setoid

An object that has an equals function which can be used to compare other objects of the same type.

Make array a setoid:

Array.prototype.equals = function (arr) {
  const len = this.length
  if (len !== arr.length) {
    return false
  }
  for (let i = 0; i < len; i++) {
    if (this[i] !== arr[i]) {
      return false
    }
  }
  return true
}

;[1, 2].equals([1, 2]) // true
;[1, 2].equals([0]) // false

Semigroup

An object that has a concat function that combines it with another object of the same type.

;[1].concat([2]) // [1, 2]

Foldable

An object that has a reduce function that applies a function against an accumulator and each element in the array (from left to right) to reduce it to a single value.

const sum = (list) => list.reduce((acc, val) => acc + val, 0)
sum([1, 2, 3]) // 6

Lens

A lens is a structure (often an object or function) that pairs a getter and a non-mutating setter for some other data structure.

// Using [Ramda's lens](http://ramdajs.com/docs/#lens)
const nameLens = R.lens(
  // getter for name property on an object
  (obj) => obj.name,
  // setter for name property
  (val, obj) => Object.assign({}, obj, { name: val })
)

Having the pair of get and set for a given data structure enables a few key features.

const person = { name: 'Gertrude Blanch' }

// invoke the getter
R.view(nameLens, person) // 'Gertrude Blanch'

// invoke the setter
R.set(nameLens, 'Shafi Goldwasser', person) // {name: 'Shafi Goldwasser'}

// run a function on the value in the structure
R.over(nameLens, uppercase, person) // {name: 'GERTRUDE BLANCH'}

Lenses are also composable. This allows easy immutable updates to deeply nested data.

// This lens focuses on the first item in a non-empty array
const firstLens = R.lens(
  // get first item in array
  xs => xs[0],
  // non-mutating setter for first item in array
  (val, [__, ...xs]) => [val, ...xs]
)

const people = [{ name: 'Gertrude Blanch' }, { name: 'Shafi Goldwasser' }]

// Despite what you may assume, lenses compose left-to-right.
R.over(compose(firstLens, nameLens), uppercase, people) // [{'name': 'GERTRUDE BLANCH'}, {'name': 'Shafi Goldwasser'}]

Other implementations:

partial.lenses - Tasty syntax sugar and a lot of powerful features
nanoscope - Fluent-interface

Type Signatures

Often functions in JavaScript will include comments that indicate the types of their arguments and return values.

There's quite a bit of variance across the community, but they often follow the following patterns:

// functionName :: firstArgType -> secondArgType -> returnType

// add :: Number -> Number -> Number
const add = (x) => (y) => x + y

// increment :: Number -> Number
const increment = (x) => x + 1

If a function accepts another function as an argument it is wrapped in parentheses.

// call :: (a -> b) -> a -> b
const call = (f) => (x) => f(x)

// map :: (a -> b) -> [a] -> [b]
const map = (f) => (list) => list.map(f)

Further reading

Ramda's type signatures
Mostly Adequate Guide
What is Hindley-Milner? on Stack Overflow

Algebraic data type

A composite type made from putting other types together. Two common classes of algebraic types are sum and product.

Sum type

A Sum type is the combination of two types together into another one. It is called sum because the number of possible values in the result type is the sum of the input types.

JavaScript doesn't have types like this, but we can use Sets to pretend:

// imagine that rather than sets here we have types that can only have these values
const bools = new Set([true, false])
const halfTrue = new Set(['half-true'])

// The weakLogic type contains the sum of the values from bools and halfTrue
const weakLogicValues = new Set([...bools, ...halfTrue])

Sum types are sometimes called union types, discriminated unions, or tagged unions.

There's a couple libraries in JS which help with defining and using union types.

Flow includes union types and TypeScript has Enums to serve the same role.

Product type

A product type combines types together in a way you're probably more familiar with:

// point :: (Number, Number) -> {x: Number, y: Number}
const point = (x, y) => ({ x, y })

Further reading

Set theory on Wikipedia

Option

Option is a sum type with two cases often called Some and None.

Option is useful for composing functions that might not return a value.

// Naive definition

const Some = (v) => ({
  val: v,
  map (f) {
    return Some(f(this.val))
  },
  chain (f) {
    return f(this.val)
  }
})

const None = () => ({
  map (f) {
    return this
  },
  chain (f) {
    return this
  }
})

// maybeProp :: (String, {a}) -> Option a
const maybeProp = (key, obj) => typeof obj[key] === 'undefined' ? None() : Some(obj[key])

Use chain to sequence functions that return Options:

// getItem :: Cart -> Option CartItem
const getItem = (cart) => maybeProp('item', cart)

// getPrice :: Item -> Option Number
const getPrice = (item) => maybeProp('price', item)

// getNestedPrice :: cart -> Option a
const getNestedPrice = (cart) => getItem(cart).chain(getPrice)

getNestedPrice({}) // None()
getNestedPrice({ item: { foo: 1 } }) // None()
getNestedPrice({ item: { price: 9.99 } }) // Some(9.99)

Option is also known as Maybe. Some is sometimes called Just. None is sometimes called Nothing.

Function

// times2 :: Number -> Number
const times2 = n => n * 2

;[1, 2, 3].map(times2) // [2, 4, 6]

Partial function

// example 1: sum of the list
// sum :: [Number] -> Number
const sum = arr => arr.reduce((a, b) => a + b)
sum([1, 2, 3]) // 6
sum([]) // TypeError: Reduce of empty array with no initial value

// example 2: get the first item in list
// first :: [A] -> A
const first = a => a[0]
first([42]) // 42
first([]) // undefined
// or even worse:
first([[42]])[0] // 42
first([])[0] // Uncaught TypeError: Cannot read property '0' of undefined

// example 3: repeat function N times
// times :: Number -> (Number -> Number) -> Number
const times = n => fn => n && (fn(n), times(n - 1)(fn))
times(3)(console.log)
// 3
// 2
// 1
times(-1)(console.log)
// RangeError: Maximum call stack size exceeded

Dealing with partial functions

// example 1: sum of the list
// we can provide default value so it will always return result
// sum :: [Number] -> Number
const sum = arr => arr.reduce((a, b) => a + b, 0)
sum([1, 2, 3]) // 6
sum([]) // 0

// example 2: get the first item in list
// change result to Option
// first :: [A] -> Option A
const first = a => a.length ? Some(a[0]) : None()
first([42]).map(a => console.log(a)) // 42
first([]).map(a => console.log(a)) // console.log won't execute at all
// our previous worst case
first([[42]]).map(a => console.log(a[0])) // 42
first([]).map(a => console.log(a[0])) // won't execute, so we won't have error here
// more of that, you will know by function return type (Option)
// that you should use `.map` method to access the data and you will never forget
// to check your input because such check become built-in into the function

// example 3: repeat function N times
// we should make function always terminate by changing conditions:
// times :: Number -> (Number -> Number) -> Number
const times = n => fn => n > 0 && (fn(n), times(n - 1)(fn))
times(3)(console.log)
// 3
// 2
// 1
times(-1)(console.log)
// won't execute anything

Making your partial functions total ones, these kinds of runtime errors can be prevented. Always returning a value will also make for code that is both easier to maintain and to reason about.

Total Function

A function which returns a valid result for all inputs defined in its type. This is as opposed to Partial Functions which may throw an error, return an unexpected result, or fail to terminate.

Functional Programming Libraries in JavaScript

P.S: This repo is successful due to the wonderful contributions!