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Fixed-Point Combinators in JavaScript


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An introduction to fixed-point combinators and lambda calculus with real-world JavaScript examples showing their power and beauty.


  • Lambda calculus can give you a deeper level of understanding of how functions work and how they relate.
  • Fixed-point combinators can help you to implement recursion.
  • Learning new stuff makes you stronger and better as a software engineer.
  • Scroll to the bottom for practical applications.

Fixed-Point Combinators

Fixed-point combinators are higher-order functions such that:

y f = f (y f)

for all f where y is a combinator, f is function and space is function application. This expands as follows:

y f = f (y f)
= f (f (y f))
= f (f (f (y f)))
= ...

This expansion is recursion.

With the use of combinators we can define self-referencing anonymous functions avoiding the use of variables. Let’s see the most simple one!

The U Combinator

The U combinator is the most simple fixed-point combinator.

U g := g gU = λg.(g g)

where λg. is lambda abstraction, which is the definition of an anonymous function that takes an argument g and substitutes it into its expression.

Example in JavaScript

Define the U combinator in JavaScript ES6 with arrow function.

U = g => g(g)

In practice we want our recursions to halt so we can calculate something. With currying we can introduce a halting condition.

Let’s calculate factorial with the U combinator.

fact = U(
g =>
// g for self-referencing
x => // currying is for passing the halting condition
(x === 0) // halting condition
? 1 // halt
: x * g(g)(x - 1) // recursion
)fact(5) === 120 // true

The Y Combinator

The Y Combinator in Lambda Calculus

Haskell B. Curry defined the Y combinator as follows.

Y := λf.(λx.f (x x)) (λx.f (x x))

The Y combinator could be used even in this form but we can simplify it.

Lets do β-reduction.

Y g := (λf.(λx.f (x x)) (λx.f (x x))) g
= (λx.g (x x)) (λx.g (x x))
= g ((λx.g (x x)) (λx.g (x x)))
= g (Y g)

The reduced form reveals that this is a fixed point combinator:

Y g = g (Y g)Y = λg.(g (Y g))

The Y Combinator in JavaScript

y is this meme here?

The Y combinator works only in lazy languages such as Haskell. In non-lazy languages like JavaScript the intuitive implementation expands until stack overflow or never halts in case of tail call optimization.

Y = g => g(Y(g)) // intuitive implementation
Y( a => a ) // call the Y combinator with the unit function
Uncaught RangeError: Maximum call stack size exceeded

Solution: let’s define laziness in JavaScript: wrap Y(g) in a zero-arity function.

Y =
g => g( () => Y(g) )

This way we can avoid automatic expansion. Let’s calculate factorial with Y.

fact = Y(
g => // g for self-referencing
x => // this curryed function is returned by g()
(x === 0)
// halting condition
? 1
// halt
: x * g()(x - 1)
// recursion
)fact(5) === 120 // true

The difference to the previous implementation is that there is no need to pass g to g(), it is already bound.

The Z Combinator

The Z Combinator in Lambda Calculus

Z := λg.(λx.g (λv.((x x) v))) (λx.g (λv.((x x) v)))

There is an extra v argument and an extra function application which is not present in the case of Y combinator definition.

After β-reduction we got:

Z g v = g (Z g) vZ g = λ.v(g (Z g) v)Z = λv.(λg.(g (Z g) v))

The Z Combinator in JavaScript

An extra v parameter and function application to the Y combinator gives us the Z combinator.

Because of the extra parameter v we do not need to construct a lazy recursion in JavaScript.

Z =
g => v => g(Z(g))(v)

Let’s define a function that sums up the numbers from a given whole number to an other one.

example series: 5 6 7 8
sum: 26

Define the sum function with the Z combinator:

// sum(from, to)
sum = Z(
g => // g for self-referencing
from =>
to =>
(from === to) // halting condition
? to // halt
: from + g(from + 1)(to) // step one and recurse
)sum(5)(8) === 26 // true

We can see that there is no extra laziness implemented because the function application is lazy in itself.

The Connection Between U, Y and Z

Let us consider the JavaScript implementations.

U = g => g(g) // recursion is strict, must curry
Y = g => g( () => Y(g) ) // we made the recursion 'lazy'
Z = g => v => g(Z(g))(v) // explicit currying makes it 'lazy'

Practical Examples in JavaScript

Here are some real-world examples using fixed point combinators.

Accept Terms

We would like users to accept terms. Ask them until they accept it.

It can be done without naming a function for the recursion.

Y(ask =>
(prompt('Accept terms?') !== 'yes')
? ask()
: alert('Thanks!')

We can even count how many times a user did not accept terms.

Y(ask => count =>
(prompt('Accept terms?') !== 'yes')
? ask()(count + 1)
: alert('At last! It took ' + count + ' times.')

We can notice that counting is done via currying.

You can read more on Currying in JavaScript ES6.

Fetch Retry

We can retry unsuccessful fetch() or XHR operations with a maximum number of attempts with manual operation on the UI.

retry =>
attempts =>
res => res.ok
? alert('Loaded!')
: (
(attempts > 1 && confirm('Reload?')) // retry on UI
? retry()(attempts - 1)
: alert('Could not load. ' + res.status)


I believe that lambda calculus and functional programming can help software engineers organize their thoughts in a consistent and rigorous way that helps write good quality code.

Enjoy the world of combinators and functional JavaScript.

See Also

Recursion with Combinators in JavaScript

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