Fixed-point combinator
A fixed-point combinator is like a magical tool that lets us create a function that works on itself. Imagine you have a toy that you can put inside itself and it will keep making copies of itself over and over again. That's kind of what a fixed-point combinator does, but with functions.
Let's say you have a function that takes a number and adds 1 to it. Normally, if you want to use this function on a number, you would call it like this:
addOne(2) = 3
But what if you want the function to work on itself? You might try to do something like this:
addOne(addOne(2)) = addOne(3) = 4
That works, but it's not very flexible. If you have a different function, you would need to write a different expression every time you wanted to use it on itself. This is where the fixed-point combinator comes in.
The fixed-point combinator is like a little formula that you can use to turn any function into a function that works on itself. It looks like this:
Y(f) = f(Y(f))
Don't worry if this looks a bit scary! Basically, all it's saying is that we're defining a new function Y that takes another function f as an argument, and "wraps" it in a way that lets it work on itself.
Let's see how this works in practice. Let's take our addOne function and apply the fixed-point combinator to it:
Y(addOne)(2) = addOne(Y(addOne))(2) = addOne(addOne(Y(addOne))(2))
= addOne(addOne(addOne(Y(addOne))(2)))
Do you see what happened there? The Y(addOne) part means that we're using the fixed-point combinator to turn addOne into a function that can work on itself. Then we apply that function to the number 2, and we get a result. But because the function can work on itself, we can apply it again and again, as many times as we like!
This might not seem very useful at first, but it's actually incredibly powerful. It allows us to define recursive functions (functions that call themselves) in a very elegant and expressive way.
So next time you hear someone talk about fixed-point combinators, just remember: it's a magical toy that lets functions play with themselves!
Let's say you have a function that takes a number and adds 1 to it. Normally, if you want to use this function on a number, you would call it like this:
addOne(2) = 3
But what if you want the function to work on itself? You might try to do something like this:
addOne(addOne(2)) = addOne(3) = 4
That works, but it's not very flexible. If you have a different function, you would need to write a different expression every time you wanted to use it on itself. This is where the fixed-point combinator comes in.
The fixed-point combinator is like a little formula that you can use to turn any function into a function that works on itself. It looks like this:
Y(f) = f(Y(f))
Don't worry if this looks a bit scary! Basically, all it's saying is that we're defining a new function Y that takes another function f as an argument, and "wraps" it in a way that lets it work on itself.
Let's see how this works in practice. Let's take our addOne function and apply the fixed-point combinator to it:
Y(addOne)(2) = addOne(Y(addOne))(2) = addOne(addOne(Y(addOne))(2))
= addOne(addOne(addOne(Y(addOne))(2)))
Do you see what happened there? The Y(addOne) part means that we're using the fixed-point combinator to turn addOne into a function that can work on itself. Then we apply that function to the number 2, and we get a result. But because the function can work on itself, we can apply it again and again, as many times as we like!
This might not seem very useful at first, but it's actually incredibly powerful. It allows us to define recursive functions (functions that call themselves) in a very elegant and expressive way.
So next time you hear someone talk about fixed-point combinators, just remember: it's a magical toy that lets functions play with themselves!
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