Functional Programming Jargon

This document is a port of the original Functional Programming Jargon to the Kotlin programming language. Feel free to read the original if you are looking for a Javascript version.

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.

This document improves upon the original one with real world code examples which will make the concepts much easier to understand.

Table of Contents

• Arity
• Higher-Order Functions (HOF)
• Closure
• Partial Application
• Currying
• Auto Currying
• Function Composition
• Continuation
• Purity
• Side effects
• Idempotent
• Point-Free Style
• Predicate
• Contracts
• Category
• Value
• Constant
• Functor
• Pointed Functor
• Lift
• Referential Transparency
• Lambda
• Lambda Calculus
• Lazy evaluation
• Monoid
• Monad
• Comonad
• Applicative Functor
• Morphism
  • Endomorphism
  • Isomorphism
  • Homomorphism
  • Catamorphism
• Semigroup
• Foldable
• Lens
• Algebraic data type
  • Sum type
  • Product type
• Option
• Function
• Partial function
• Functional Programming Libraries in JavaScript

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
val times2 = { n: Int -> n * 2 }

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

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).

val sum = { x: Int, y: Int ->
    x + y
}

val arity = sum.reflect()!!.parameters.size

println(arity) // 2

// The arity of sum is 2

Higher-Order Functions

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

val filter = { predicate: (Any) -> Boolean, xs: Array<out Any> ->
    xs.filter(predicate)
}

val isA = { type: KClass<out Any> ->
    { x: Any ->
        type.isSuperclassOf(x::class)
    }
}

filter(isA(Int::class), arrayOf(0, "1", 2)) // [0, 2]

Closure

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

In other words, 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.

val addTo = { x: Int ->
    fun add(y: Int): Int {
        return x + y
    }
    ::add
}

val addToFive = addTo(5)

addToFive(3) // 8

Note that we could have defined addTo with just lambdas. add here demonstrates how local function declaration works in Kotlin.

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

Ideally, when the function addTo finishes execution, its scope, with local variables add, x and 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

• Lambda Vs Closure
• Closures in Kotlin

Partial Application

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

Kotlin doesn't support partial application out of the box, but it is easy to write extension functions to enable doing this:

fun <A, B, C> Function2<A, B, C>.partial(a: A): (B) -> C {
    return { b -> invoke(a, b) }
}

// Something to apply
val add2 = { a: Int, b: Int ->
    a + b
}

// Partially applying `5` to `add2` gives us a one-argument function
val fivePlus = add2.partial(5)

fivePlus(4).print() // 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. Creating partially applied functions is also a good example of higher order functions.

Currying

The process of converting a function that takes multiple arguments into a chain of higher order functions that take 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.

Currying is not supported in Kotlin out of the box, but we can implement it using extension functions just like we did with partially applied functions.

fun <A, B, Z> ((A, B) -> Z).curry(): (A) -> (B) -> Z = { a: A ->
    { b: B ->
        invoke(a, b)
    }
}

val sum = { a: Int, b: Int ->
    a + b
}

val curriedSum = sum.curry()

curriedSum(40)(2) // 42.

val add2 = curriedSum(2) // (b: Int) -> 2 + b

add2(10).print() // 12

When using Kotlin you can add Arrow to your project dependencies which comes with built-in support for currying.

Further reading

• Favoring Curry

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.

typealias Function<T, R> = (T) -> R

fun <T, U, R> compose(f: Function<T, U>, g: Function<U, R>): Function<T, R> {
    return { param: T ->
        g(f(param))
    }
}

val floorAndToString = compose(Double::roundToInt) { it.toString() }

floorAndToString(121.212121) // "121"

Note that the Kotlin compiler can infer the generic type parameters so we don't have to pass them here.

Continuation

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

val printAsString = { num: Int ->
    println("Given $num")
}

val addOneAndContinue = { num: Int, cc: (Int) -> Unit ->
    val 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.

val continueProgramWith = { data: Any ->
    // continues program with data
}

try {
    continueProgramWith(File("path/to/file").readBytes())
} catch (e: Exception) {
    // handle error
}

Purity

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

val greet = { name: String ->
    "Hi, $name"
}

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

As opposed to each of the following:

var name = "Brianne"

val greet = {
    "Hi, $name"
}

greet() // "Hi, Brianne"

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

var greeting = ""

val greet = { name: String ->
    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.

val differentEveryTime = {
    Date()
}

println("IO is a side effect!")

Idempotence

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

f(f(x)) ≍ f(x)

abs(abs(10))

listOf(2, 1).sorted().sorted().sorted()

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
val map = { fn: (Int) -> Int ->
    { list: List<Int> ->
        list.map(fn)
    }
}

val add = { a: Int ->
    { b: Int ->
        a + b
    }
}

// Then

// Not points-free - `numbers` is an explicit argument
val incrementAll = { numbers: List<Int> ->
    map(add(1))(numbers)
}

// Points-free - The list is an implicit argument
val 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 numbers: List<Int> ->

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 filters.

val predicate = { a: Int ->
    a > 2
}

listOf(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.

val contract = { input: Any ->
    if (input is Int) {
        input
    } else throw RuntimeException("Contract violated: expected an Int")
}

val addOne = { num: Any ->
    contract(num) + 1
}

addOne(2) // 3

addOne("some string") // Contract violated: expected an Int

Note that Kotlin has its own implementation of Contracts but it is still an experimental feature

Category

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

Note that • is the compose operator. In Kotlin we can overload the + (plus) operator to enable function composition with infix syntax.

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

*Further reading

• Category Theory for Programmers

Value

Anything that can be assigned to a variable.

5
mapOf("name" to "John", "age" to 30) // a Map is immutable
{ a: Any ->
    a
}
listOf(1)
null

Constant

A variable that cannot be reassigned once defined.

val five = 5
val twoAndThree = listOf(2, 3)

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.

twoAndThree + five == listOf(2, 3) + 5

Functor

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

Preserves identity

object.map { x -> x } ≍ object

Composable

object.map(compose(f, g)) ≍ object.map(g).map(f)

(f, g are arbitrary functions)

A common functor in Kotlin is List since it adheres to the two functor rules:

listOf(1, 2, 3).map { it } // [1, 2, 3]

and

val f = { x: Int -> x + 1 }
val g = { x: Int -> x * 2 }

listOf(1, 2, 3).map { f(g(it)) } // [3, 5, 7]
listOf(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.

Kotlin provides *of functions for List for example:

listOf(1)

In other languages of might be called just.

Lift

TODO: this is a mess, fix it

Lifting is when you take a value and put it into an object like with a pointed 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:

val greet = { "Hello World!" }

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

Lambda

An anonymous function that can be treated like a value.

{ a: Int -> a + 1 }

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

listOf(1, 2).map { a: Int -> a + 1 } // [2, 3]

You can assign a lambda to a variable.

val add1 = { a: Int -> a + 1 }

Lambda Calculus

A branch of mathematics that uses functions to create a universal model of computation. This article explains it in depth and it is also very easy to understand.

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.

Note that in Kotlin we use Sequences for generating a potentially infinite sequence of values, like in the example below and lazy for lazy initialization.

val rand = {
    sequence {
        while (true) {
            yield(Random.nextInt())
        }
    }
}

rand().take(10).toList()

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  

Adding Lists together also forms a monoid:

listOf(1, 2) + listOf(3, 4) // [1, 2, 3, 4]

The identity value is an empty List: listOf<Int>()

listOf(1, 2) + listOf() // [1, 2]

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

val identity = { x: Any -> x }

typealias Function<T, R> = (T) -> R

// we have to call this plus instead of combine to overload the + operator
operator fun <T, U, R> Function<T, U>.plus(fn: Function<U, R>): Function<T, R> {
    return { t ->
        fn(this(t))
    }
}

foo is any function that takes one argument.

foo + identity ≍ identity + foo ≍ foo

Monad

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

// Implementation
fun <T, R> Iterable<T>.flatMap(f: (T) -> Iterable<R>): Iterable<R> {
    return this.fold(listOf()) { acc, next ->
        acc + f(next)
    }
}

// Usage
listOf("cat,dog", "fish,bird").flatMap { 
    it.split(",")
} // [cat, dog, fish, bird]

// Contrast to map
listOf("cat,dog", "fish,bird").map {
    it.split(",")
} // [[cat, dog], [fish, bird]]

of is also known as just in other languages. flatMap is also known as bind in other languages.

Comonad

An object that has extract and extend functions.

class CoIdentity<T>(val value: T) {

    fun extract() = value

    fun extend(f: (T) -> T) = CoIdentity(f(value))
}

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 { it + 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
fun <T, R> Iterable<Function<T, R>>.ap(elements: Iterable<T>): Iterable<R> {
    return this.fold(listOf()) { acc, next ->
        acc + elements.map(next)
    }
}

// Example usage
listOf({a: Int -> a + 1}).ap(listOf(1)) // [2]

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

// Lists that you want to combine
val arg0 = listOf(1, 3)
val arg1 = listOf(4, 5)

// combining function - must be curried for this to work

val add = { x: Int -> { y: Int -> x + y } }

val partiallyAppliedAdds = listOf(add).ap(arg0) // [(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(arg1) // [5, 6, 7, 8]

Morphism

A transformation function.

Endomorphism

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

// (String) -> String
val uppercase = { str: String -> str.toUpperCase() }

// (Int) -> Int
val decrement = { x: Int -> 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.
val pairToCoords = { (x, y): Pair<Int, Int> -> Coords(x, y) }

val coordsToPair = { (x, y): Coords -> x to y }

coordsToPair(pairToCoords(1 to 2)) // (1, 2)

pairToCoords(coordsToPair(Coords(1, 2))) // Coords(x=1, y=2)

Homomorphism

A homomorphism is 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))

listOf(uppercase).ap(listOf("oreos")) == listOf(uppercase("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.

val sum = { values: List<Int> -> values.reduceRight{ x, acc -> acc + x}}

sum(listOf(1, 2, 3, 4, 5)) // 15

Semigroup

An object that has a concat function that combines it with another object of the same type. Note that a semigroup is different from a monoid because it doesn't require an identity function.

listOf(1) + listOf(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

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

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.

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.

Functional Programming Libraries for Kotlin

Right now the de facto tool for functional programming in Kotlin is Arrow. It has everything you might need and comes with batteries included.