# Decoding the Y Combinator: A Deep Dive into Recursion

The Y Combinator, an intriguing concept in functional programming, is a powerful tool that facilitates recursion even in programming languages that don't natively support it.

## What is a Y Combinator?

The Y Combinator can be represented as follows:

`Y = λf.(λx.f (x x)) (λx.f (x x))`

In this expression, `(x x)` means that we apply x to itself.

Sounds a bit abstract? Let's break it down!

## Inside the Lambda Function

The lambda function consists of three main components:

1. `f(r') -> r`: This is a recursive function generator, where f is a function that builds a recursive function by taking a reference to a semantically equivalent recursive function, and r is the recursive function that's returned.
2. `λx.f (x x)`: This is a function which uses a recursive function generator f, and itself x!
3. `(x x)` applies x to itself, creating a new recursive function generator that will receive a clone of x.

Essentially, we're creating new copies of our recursive function as we go further into the recursion.

## The Y Combinator in Action: A Python Example

To better illustrate how this works, let's explore a simple Python version:

``````# Example with function names

def y_combinator(f):
# the y combinator is a generator of new f references

def apply_x(x):
# x is the self duplicating function
# y is the argument we wanna pass to the recursive function
# f is the recursive function generator
return f(lambda y: x(x)(y))

def another_apply_x(x):
# same
return f(lambda y: x(x)(y))

# the trick can be found here!
# We get to avoid using a recursive reference through simply duplicating code.
return apply_x(another_apply_x)

# What does this look like?
# y=y_combinator and x = apply_x and x' is apply_other_x
# y(f)(n) = x(x')(n) = f(x(x'))(n)
# notice how evaluating x(x'), produces f(x(x')), a copy for the next iteration
``````

Let's imagine Python doesn't natively support recursion. We could still implement a recursive factorial function using our Y Combinator:

``````# Example usage assuming python doesn't have recursion yet
def factorial_func(f):
# even though we don't have recursion let's act like f is a recursive factorial function.
# even though the implementation below isn't recursive, it'd still work.
def factorial(n):
return 1 if n == 0 else n * f(n - 1)
return factorial

# factoral func is really just a factorial function builder.
# calling the function will look like this: factoral_func(*factorial)(number)
factorial = y_combinator(factorial_func)

print(factorial(5))  # Output: 120

# What does this look like?
# factorial = y(fac_gen) = factorial_func(x(x'))
# factorial(5) = 5 * f(n-1)
# remeber: f=x(x') = factorial_func(x(x'))
# factorial(5) = 5 * factorial_func(x(x'))(4) = 5 * y(fac_gen)(4)
# recursion!``````

This example shows how the Y Combinator creates new instances of the factorial function as required, facilitating recursion even when the language doesn't inherently support it.

## Wrapping Up

The Y Combinator is an cool solution to a complex problem, and it demonstrates the power of functional programming as a paradigm.