by Breck Yunits

May 31, 2024 โ Yesterday, on a plane, I found an equation I sought for a decade.

P = {C^R}^I

PCRI describes a symbolic notation.

PCRI says the number of possible programs **P** is equal to the number of columns **C** (aka characters, words, or bits), raised by the number of rows **R** (aka lines), raised by the number of indentation levels **I**.

If you view the source code of this post, you will see **C, R, and I** in action.

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PCRI explains the simplicity and power of Scroll Notation^{[1]}. Three syntax rules support a vast universe of concise programs. Another notation(s) might be found with superior metrics, but I have yet to see it.

PCRI also explains why *nearly* flat structures work so well: a little nesting goes a long way. If you set C and R to 8, changing I from 1 to 2 increases the amount of possible programs from 16 million to 281 trillion.

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59 days ago I announced the decade long Scroll Notation research endeavor over with a negative result.

It looks like I was wrong again. There was something special about Scroll Notation, and PCRI explains what that is.

*It shouldn't be possible to represent all programs in all programming language with such a minimal set of rules! Scroll Notation doesn't even have parentheses!*

And yet, the experimental evidence kept piling up.

The evidence hinted at some important natural formula, and now we have a name for it: PCRI.

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[1] Scroll Notation was originally named Tree Notation.

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It may help to explain from another angle. Imagine your boss comes up to you and says "We are giving you a special assignment. We want you to come up with the best possible syntax for all programming languages." What do you come up with?

You can present them with Scroll Notation, and explain how it supports a vast universe of programs, P = {C^R}^I, with just three syntax rules:

- lines split into columns via bits, characters or words
- groups of lines joined to each other via newlines
- scopes established via the indent trick (aka the "Offiside Rule", popularized by Python, HAML, et cetera)

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- Does PCRI also describe S-Expressions? S-Expression Notation works. Scroll Notation
*barely*works. - As always, all mistakes are on me, and credit goes to the people who have supported this effort with me.
- I think this equation is pretty interesting, so I really hope a lot of people on the Internet tell me how stupid it is and that it was discovered 100 years ago.
- Now, back to my vacation.