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 translated from original JavaScript (ES2015) to Elixir. Why JavaScript? Why Elixir? Just because.

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

Functional Programming Jargon

Arity

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).

def sum(a, b), do: a + b

# The function is referred to as sum/2, because Elixir differentiates functions
# by name AND arity.

arity = :erlang.fun_info(&sum/2)[:arity] IO.puts(arity) # => 2

Higher-Order Functions (HOF)

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

def filter(predicate, xs), do: Enum.filter(xs, predicate)

filter(is_number, [0, '1', 2, nil]) # => [0, 2]

Closure

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.

def add_to(x), do: fn y -> x + y end

add_to_five = add_to(5)
add_to_five(3) # => 8

The function add_to() returns a function, lets store it in a variable called add_to_five with a curried call having parameter 5.

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

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.

Partial Application

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

def add_three(a, b, c), do: a + b + c

def five_plus(x), do: add_three(2, 3, x)

five_plus(4) # => 9

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.

def sum(a, b), do: a + b

def curried_sum(a), do: fn b -> a + b end

curried_sum(40).(2) # => 42

add_two = curried_sum(2)

add_two.(10) # 12

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.

def compose(f, g), do: fn a -> a |> g |> f end

def floor_and_to_string(val), do: compose(Integer.to_string/1, floor/1)

floor_and_to_string(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.

def print_as_string(num), do: IO.puts("Given #{num}")

def add_one_and_continue(num, cc) do
  result = num + 1
  cc.(result)
end

add_one_and_continue(2, print_as_string/1) # => "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.

def continue_program_with(data) do
  # Continues program with data
end

case File.read("/path/to/file") do
  {:ok, response} -> continue_program_with(response)
  {:error, reason} => # handle error
end

Purity

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

def greet(name), do: "Hi, #{name}"

greet("Brianne") # => "Hi, Brianne"

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.

different_every_time = NaiveDateTime.utc_now()

IO.puts("IO is a side effect!")

Idempotent

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

f(f(x)) ≍ f(x)

10 |> abs() |> abs()

[2, 1] |> Enum.sort() |> Enum.sort()

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.

def predicate(a), do: a > 2

[1, 2, 3, 4] |> Enum.filter(&predicate/1) # => [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
def contract(input) do
  if is_integer(input) do
    true
  else
    raise ArgumentError, "Contract violated: expected int -> boolean"
  end
end

def add_one(num), do: contract(num) and num +_1

add_one(2) # => 3
add_one("some string") # => Contract violated: expected int -> boolean

Value

Anything that can be assigned to a variable.

5
%{name: "John", age: 30}
fn a -> a end
[1]
nil

Constant

A variable that cannot be reassigned once defined.

defmodule Constants do
  @five 5
  @john %{name: "John", age: 30}
end

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

Functor

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

Functor.map(object, fn x -> x end) == object

Composable

Functor.map(object, fn x -> f(g(x)) end) ==
  object |> Enum.map(&g/1) |> Enum.map(&f/1)

(f, g are arbitrary functions)

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], [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:

def greet(), do: "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.

Lambda

An anonymous function that can be treated like a value.

fn a -> a + 1 end

&(&1 + 1)

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

Enum.map([1, 2], &(&1 + 1)) # => [2, 3]

You can assign a lambda to a variable.

add_one = fn a -> a + 1 end

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.

Monoid

An object with a function that “combines” that object with another of the same type.

One simple monoid is the addition of numbers:

1 + 1 == 2

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

An “identity” value must also exist that when combined with a value doesn’t change it.

The identity value for addition is 0.

1 + 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:

Enum.concat([1, 2], [3, 4]) # => [1, 2, 3, 4]

The identity value is empty array []

Enum.concat([1, 2], []) # => [1, 2]

If identity and compose functions are provided, functions themselves form a monoid:

def identity(a), do: a

def compose(f, g), do: fn x -> f(g(x)) end

foo is any function that takes one argument.

compose(&foo/1, &identity/1) ≍ compose(&identity/1, &foo/1) ≍ &foo/1

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)

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 transformation function.

Endomorphism

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

Isomorphism

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}

Homomorphism

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"))

Catamorphism

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

Anamorphism

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]

Hylomorphism

The combination of anamorphism and catamorphism.

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

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.

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 TODO * Ramda’s type signatures * Mostly Adequate Guide * TODO 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 =Set=s 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 TODO 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.

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.

Use chain to sequence functions that return Options

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

Function

A function TODO f :: A => B is an expression - often called arrow or lambda expression - with exactly one (immutable) TODO parameter of type A and exactly one TODO 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 * TODO 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 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))
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 as well as to reason about.

Br1ght0ne/functional-programming-jargon.ex