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


    Table of Contents


    The number of arguments a function takes. From words like unary, binary, ternary, etc. This word has the distinction of being composed of two suffixes, "-ary" and "-ity." Addition, for example, takes two arguments, and so it is defined as a binary function or a function with an arity of two. Such a function may sometimes be called "dyadic" by people who prefer Greek roots to Latin. Likewise, a function that takes a variable number of arguments is called "variadic," whereas a binary function must be given two and only two arguments, currying and partial application notwithstanding (see below).

    const sum = (a, b) => a + b
    const arity = sum.length
    console.log(arity) // 2
    // The arity of sum is 2

    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]


    A closure is a way of accessing a variable outside its scope. Formally, a closure is a technique for implementing lexically scoped named binding. It is a way of storing a function with an environment.

    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. ie. they allow referencing a scope after the block in which the variables were declared has finished executing.

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

    The function addTo() returns a function(internally called add()), lets store it in a variable called addToFive with a curried call having parameter 5.

    Ideally, when the function addTo finishes execution, its scope, with local variables add, x, y should not be accessible. But, it returns 8 on calling addToFive(). This means that the state of the function addTo is saved even after the block of code has finished executing, otherwise there is no way of knowing that addTo was called as addTo(5) and the value of x was set to 5.

    Lexical scoping is the reason why it is able to find the values of x and add - the private variables of the parent which has finished executing. This value is called a Closure.

    The stack along with the lexical scope of the function is stored in form of reference to the parent. This prevents the closure and the underlying variables from being garbage collected(since there is at least one live reference to it).

    Lambda Vs Closure: A lambda is essentially a function that is defined inline rather than the standard method of declaring functions. Lambdas can frequently be passed around as objects.

    A closure is a function that encloses its surrounding state by referencing fields external to its body. The enclosed state remains across invocations of the closure.

    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.


    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.

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


    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
    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


    A function is pure if the return value is only determined by its input values, and does not produce side effects.

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

    As opposed to each of the following: = 'Brianne'
    const greet = () => `Hi, ${}`
    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}`
    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!')


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

    f(f(x)) ≍ f(x)
    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) =>
    const add = (a) => (b) => a + b
    // Then
    // Not points-free - `numbers` is an explicit argument
    const incrementAll = (numbers) => map(add(1))(numbers)
    // Points-free - The list is an implicit argument
    const incrementAll2 = map(add(1))

    incrementAll identifies and uses the parameter numbers, so it is not points-free. incrementAll2 is written just by combining functions and values, making no mention of its arguments. It is points-free.

    Points-free function definitions look just like normal assignments without function or =>.


    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]


    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


    A category in category theory is a collection of objects and morphisms between them. In programming, typically types act as the objects and functions as morphisms.

    To be a valid category 3 rules must be met:

    1. There must be an identity morphism that maps an object to itself. Where a is an object in some category, there must be a function from a -> a.
    2. Morphisms must compose. Where a, b, and c are objects in some category, and f is a morphism from a -> b, and g is a morphism from b -> c; g(f(x)) must be equivalent to (g • f)(x).
    3. Composition must be associative f • (g • h) is the same as (f • g) • h

    Since these rules govern composition at very abstract level, category theory is great at uncovering new ways of composing things.

    Further reading


    Anything that can be assigned to a variable.

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


    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)


    An object that implements a map function which, while running over each value in the object to produce a new object, adheres to two rules:

    Preserves identity => x) ≍ object

    Composable, g)) ≍

    (f, g are arbitrary functions)

    A common functor in JavaScript is Array since it abides to the two functor rules:

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


    const f = x => x + 1
    const g = x => x * 2
    ;[1, 2, 3].map(x => f(g(x))) // = [3, 5, 7]
    ;[1, 2, 3].map(g).map(f)     // = [3, 5, 7]

    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]


    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) => // 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], [3, 4]) // [4, 5, 5, 6]

    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]

    Referential Transparency

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

    Say we have function greet:

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

    Any invocation of greet() can be replaced with Hello World! hence greet is referentially transparent.

    Equational Reasoning

    When an application is composed of expressions and devoid of side effects, truths about the system can be derived from the parts.


    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.

    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() // Each execution gives a random value, expression is evaluated on need.


    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


    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.


    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)

    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(, [])
    // 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]


    A transformation function.


    A function where the input type is the same as the output.

    // uppercase :: String -> String
    const uppercase = (str) => str.toUpperCase()
    // decrement :: Number -> Number
    const decrement = (x) => x - 1


    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 them 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}


    A homomorphism is just a structure preserving map. In fact, a functor is just a homomorphism between categories as it preserves the original category's structure under the mapping.

    A.of(f).ap(A.of(x)) == A.of(f(x))
    Either.of(_.toUpper).ap(Either.of("oreos")) == Either.of(_.toUpper("oreos"))


    A reduceRight function that applies a function against an accumulator and each value of the array (from right-to-left) to reduce it to a single value.

    const sum = xs => xs.reduceRight((acc, x) => acc + x, 0)
    sum([1, 2, 3, 4, 5]) // 15


    An unfold function. An unfold is the opposite of fold (reduce). It generates a list from a single value.

    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]


    The combination of anamorphism and catamorphism.


    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],
    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.


    it's the opposite of paramorphism, just as anamorphism is the opposite of catamorphism. Whereas with paramorphism, you combine with access to the accumulator and what has been accumulated, apomorphism lets you unfold with the potential to return early.


    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


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

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


    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


    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](
    const nameLens = R.lens(
      // getter for name property on an object
      (obj) =>,
      // 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:

    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) =>

    Further reading

    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.

    See also Set theory.


    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.


    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-independant, 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))
    // 3
    // 2
    // 1
    // 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 execte, 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))
    // 3
    // 2
    // 1
    // 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 as well as to reason about.

    Functional Programming Libraries in JavaScript

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


    🚀 Github 镜像仓库 🚀




    贡献者 82