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Asked: 2024-09-26T23:30:03+05:30

How can we construct a combinator R using S, K, and I to replicate another combinator’s behavior?

I’ve been diving deep into combinatory logic lately, and I stumbled upon a fascinating puzzle that I’d love to hear your thoughts on! The whole concept of using combinators to eliminate variables is super interesting, but I find myself scratching my head over a specific challenge I came across.

Imagine we have a simple combinator library with a few basic combinators: `S`, `K`, and `I`. Just as a quick refresher, they work like this:

– `S` applies two functions to an argument: `S f g x = f x (g x)`
– `K` takes a value and ignores its input: `K a b = a`
– `I` is the identity combinator: `I x = x`

Now, here’s the twist: what if we wanted to create a combinator that takes a combinator and replicates its behavior? Specifically, let’s call this new combinator `R`, and it should behave like the identity function for all inputs of the provided combinator. In simpler terms, `R` should return whatever the given combinator would return if applied to any argument.

For example, if you input `R K` and then use it as `R K a b`, you should get `a` back, no matter what `a` and `b` are. Similarly, for `S`, `R S f g x` should yield the same output as `S f g x` does when `R` is applied.

Here’s my question: Can anyone help me construct `R` using just our basic combinators, `S`, `K`, and `I`? I’m curious to see different approaches and maybe even clever variations on how to construct `R`. I’m thinking about how even a simple combinator library can lead to some pretty complex behavior, and I’d love to see how creative you can get with it. The goal is to make this combinator as efficient as possible while still retaining its intended functionality. Looking forward to your ideas!

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    2 Answers

    1. Creating the Combinator R

      Okay, so I’ve been thinking about how to make this combinator R. We want it to behave like the combinator we give it, right? Here’s my take:

      Understanding R

      The idea is that R should take a combinator (like K, S, etc.) and then behave like that combinator when applied to arguments. So for R K a b it should just return a, and for R S f g x, it should return S f g x.

      Constructing R

      If we think about it, we need to use only S, K, and I. Here’s an idea about how to build it:

          
      R = S (K I)

      Here’s a little explanation of how this works:

      • S takes two functions and applies them to the same argument.
      • K I always returns the identity function, which gives us the correct output based on the input passed to R.

      Usage Example

      So, if you did:

          
      R K a b

      It would behave like:

          
      K a b

      And it would just return a.

      For S, if we do:

          
      R S f g x

      It would output the same as:

          
      S f g x

      Final Thoughts

      I’m not sure if this is the only way to do it, or maybe there are even cooler ways to construct R, but it seems to keep it simple! It’s wild how combinations like this really play into the power of combinatory logic.

    2. To construct the combinator `R` that replicates the behavior of any given combinator using just the basic combinators `S`, `K`, and `I`, we can leverage the way these combinators interact with their arguments. The essence of `R` is to create a mechanism that captures the behavior of the provided combinator and allows it to be applied to any inputs, returning the same result as the original combinator would. One effective way to achieve this is by utilizing the identity function alongside the application capabilities of `S` to ensure that we are encapsulating the original function’s behavior without altering its output. Here’s how we could define `R`: 

      
R = S (K I)

      In this construction, `S (K I)` essentially serves as the replicator of the combinator’s behavior. When applied to a combinator `C`, `R C` will yield a function that mirrors `C`’s output. To see how it works, you can evaluate it with `K` and `S` as follows: For `R K`, executing `R K a b` will yield `K a b`, which simplifies to `a` as per the definition of `K`. Similarly, for `S`, you would apply `R S f g x`, which evaluates to `S f g x` directly. Thus, `R` acts as a ‘function replicator’, preserving the original combinator’s functionality.-