Monads

In functional programming, a monad is a design pattern that provides a way to encapsulate common patterns of computation. Specifically, a monad is a type constructor that provides a way to chain computations together in a composable and modular way.

One of the primary benefits of monads is that they allow for a separation of concerns between the specific details of a computation and the way that computation is combined with other computations. This can lead to code that is easier to understand, easier to modify, and less prone to errors.

There are many different types of monads, each with their own specific properties and use cases. Some common examples include the Maybe monad, the List monad, and the Reader monad.

The formal definition of monads

Given any well-defined, basic types T, U, a monad consists of three parts:

• A type constructor M that builds up a monadic type M
• A type converter, often called unit or return, that embeds a value in the monad:
  unit: T → M<T>
• A combinator, typically called bind and represented with an infix operator >>= or a method called flatMap, that unwraps a monadic variable, then inserts it into a monadic function, resulting in a new monadic value:
  (>>=): (M<T>, T → M<U>) → M<U>

These three components work together to define the behavior of a monad, allowing it to be used in a wide range of contexts and computations.

The monads' laws

The laws of monads ensure that the behavior of the monad is consistent and predictable, making it a powerful tool for working with abstract computations. The laws are:

• Left identity: The unit function acts as a left-identity for the bind function, ensuring that the monadic structure is preserved while applying a function to a value inside the monad unit(x) >>= f ↔ f(x)
• Right identity: The unit function also acts as a right-identity for the bind function, ensuring that the monadic structure is preserved while extracting a value from the monad. m >>= unit ↔ m
• Associativity: The bind function is associative, ensuring that the order in which monadic operations are performed does not affect the final result. m >>= ((x) → f(x) >>= g) ↔ (m >>= f) >>= g

These laws are essential for monads to be useful in practice, as they ensure that monads behave consistently and can be composed in predictable ways.

Example: The Maybe monad

In languages like TypeScript, monads can be implemented using techniques like higher-order functions and currying.

type Maybe<T> = {
  map: <R>(fn: (value: T) => Nullable<R>) => Maybe<R>;
  flatMap: <R>(fn: (value: T) => Maybe<R>) => Maybe<R>;
  value: Nullable<T>;
};

const Just = <T>(value: T): Maybe<T> => ({
  map: (fn) => Maybe(fn(value)),
  flatMap: (fn) => fn(value),
  value,
});

const Nothing = <T>(): Maybe<T> => ({
  map: () => Nothing(),
  flatMap: () => Nothing(),
  value: null,
});

const Maybe = <T>(value: Nullable<T>): Maybe<T> =>
  value != null ? Just(value) : Nothing();

// Example usage
const maybeValue = Maybe(5)
  .map((value) => value * 2)
  .flatMap((value) => Maybe(value + 1))
  .map((value) => value.toString());

console.log(maybeValue.value); // Outputs "11"

The left identity law states that calling the unit function on a value a, and then calling flatMap with a function f, should be equivalent to calling f directly on a.

const a = 5;
const f = (x: number) => Just(x * 2);

const leftIdentity1 = Just(a).flatMap(f); // Just(10)
const leftIdentity2 = f(a); // Just(10)

The right identity law states that calling flatMap on a monad with the unit function should return the same monad.

const m = Just(5);

const rightIdentity1 = m.flatMap(Just); // Just(5)
const rightIdentity2 = m; // Just(5)

The associativity law states that calling flatMap twice, with two different functions f and g, is equivalent to calling flatMap once with a function that returns another function that calls flatMap with g on the result of f.

const m = Just(5);
const f = (x: number) => Just(x * 2);
const g = (x: number) => Just(x + 1);

const associativity1 = m.flatMap(f).flatMap(g); // Just(11)
const associativity2 = m.flatMap((x) => f(x).flatMap(g)); // Just(11)

In these examples, we can see that the monad laws hold true for the Maybe monad, as they do for all monads.