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Comment by __MatrixMan__ | Hacker News Reader

> it tells you a lot about what the author thinks is great about the language

I did not get so far as to understand that it was a language. I thought maybe it was some kind of higher order function to be implemented in a language of your closing. Like a map/reduce sort of thing.

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It's an example of Combinatory Logic. For another example check out SKI combinators. SKI is actually kind of close to this but using three combinators instead of one it's a bit simpler to understand (IMO).

https://en.wikipedia.org/wiki/Combinatory_logic

https://en.wikipedia.org/wiki/SKI_combinator_calculus

Note that the I combinator is redundant since `I = SK`, so all we really need is `SK`. There are alternatives like Iota with a single combinator, but they're essentially an obfuscated mix of S and K; so it's usually clearer to stick with SK.

Note that the first two rules of this Tree Calculus are precisely those of K and S.

> I = SK

Wrong. I = S K M where M can be anything:

    I x = S K M x = K x (M x) = x

K and S K represent True and False, since K x y = x and S K x y = y.

> essentially an obfuscated mix of S and K

Most alternatives are tuples of S and K like λz. z S K or λz. z K S K. But the arguably shortest one is indeed a proper mix:

    A = λx λy λz. x z (y (λ. z)) = λx λy λz. x z (y (K z))

Oops, sorry; I got part way through writing I = S K K, but went off to another tab to double-check, and forgot to update that expression when I came back!

You're absolutely right.

You know, it's entirely possible that it isn't a language and I just assumed that into existence. It's a very confusing landing page.

Edit: no, it's definitely a language of some sort:

> The syntax and semantics of Tree Calculus are minimal and self-contained, relying on no platform-specific concepts or abstractions. As a result, it is trivial to write interpreters on any platform or in any programming language (demo). ...

> The ability to bootstrap the full power of Tree Calculus anywhere makes it an excellent configuration as code language in a heterogeneous system. Developers could write Tree Calculus programs that generate, say, JSON, and use an interpreter ...

Thanks for the feedback! Tree Calculus is a calculus/logic, see Specification page or the book by Barry Jay (linked on that page) for way way better and detailed verbose explanations. It only defines what I chose to call "t" on the website, Barry uses "Δ" in his book and papers.

So without anything else, we'd have to talk about programs in terms of "(t t) t ..." or binary trees, which gets unwieldy quickly. The first natural step, for practical purposes, is to allow definitions "foo = ...", then some syntactic sugar for lists, functions, etc. Ooops and now we have a "language". If you open the "Playground" page there's a summary of what exactly is TC and what is syntactic sugar (and really nothing more!) on top of it.

I like to think that the line is so blurry precisely because TC needs nothing but a bit of syntactic sugar to feel like a usable PL haha.

Right, it's a programming language the way the lambda calculus or pi calculus or whatever are programming languages—I did understand that much!

I love the idea of having a website like this to introduce people to one of the less popular calculi, and the playground is a great touch. It might be helpful to have an introductory paragraph that explains exactly what the tree calculus is, starting from what a "calculus" is—your target audience seems to be people who aren't otherwise going to go out of their way to read Barry's papers, which means you can't assume as much background as you currently do. As a reference, I'm a casual PL nerd who actually has read academic papers related to some of the less well-known calculi with an eye towards implementing them, so I'm on the more informed end of the spectrum of your target audience.

Have you considered making this site open source? No pressure if not, but if so I'd be happy to contribute to polishing up the landing page. I'm very interested in learning more about this anyway, and I'm more than willing to help!

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