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Comment by VyseofArcadia | Hacker News Reader

A friend of mine once made the argument to me that software patents are nonsense because if pi is normal[0], then your source code, suitably encoded, appears somewhere in pi.

I don't know that that's a good argument, but it's a fun one.

[0] and while we don't know for sure signs point to yes

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It's a fun argument, but using it in earnest is also committing a category error. IP laws care not just about the bits, but also about their provenance (the famous "colour of your bits"). That is, the code in question may exist somewhere in pi, but it's not where you actually got it from.

(Also, even if the code exists somewhere in pi, finding it there requires more effort than coming up with it in the first place. There are interesting mathematical/philosophical consequences of this reasoning.)

Your coding process is different from mine

> finding it there requires more effort than coming up with it in the first place

What if you have access to [enter arbitrary number] computers?

[arbitrary number] is currently limited to somewhere in the low billions, which is unlikely to be of much help, even for finding the ASCII representation of "Hello, World!".

I meant in a fundamental sense, per information theory, i.e. physics.

[Arbitrary number] is big. You just won't believe how vastly, hugely, mind-bogglingly big it is. I mean, you may think it's a long way down the road, but that's just peanuts to [arbitrary number].

Assuming the binary digits of π are randomly distributed[0], any given n-bit binary sequence is going to come up with a probability of 1-in-2^n at any given point in the whole thing. A 1 kB file of any kind, app or data, is already one in 1-in-2^(8*(2^10))).

The Hitch Hiker's Guide to the Galaxy says that if you hold a lungful of air you can survive in the total vacuum of space for about thirty seconds. However it goes on to say that what with space being the mind boggling size it is the chances of getting picked up by another ship within those thirty seconds are two to the power of two hundred and sixty-seven thousand seven hundred and nine to one against.

This probability corresponds to a file just fractionally smaller than 33464 bytes.

For scale: The universe can have performed no more than 10^120 ops on 10^90 bits[1] (~2^399 and ~2^299 respectively), meaning that no IP-protected content over 299 bits in length is likely to be found even when the entire universe is dedicated to this task, and even if your algorithm for finding extra digits of π is efficient enough that you're not limited by it.

[0] is this proven for any value of "random"? I genuinely don't know.

[1] https://arxiv.org/abs/quant-ph/0110141

In a similar vein, the corecursive podcast[0] covered the Sloot Digital Coding System[1] in which it is hypothesized that he attempted something very similar but for video. Unfortunately, it seemed like overestimated his ability to produce this and resorted to smoke and mirrors for demos. It’s a facinating story that reads like a tale of industrial espionage.

[0]: https://corecursive.com/sloot-digital-coding-system/

[1]: https://en.m.wikipedia.org/wiki/Sloot_Digital_Coding_System

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

It’s a common logical error. Infinity doesn’t mean all possibilities. An infinite universe doesn’t have infinite copies of someone. Pi doesn’t have everyone’s source code. There an infinite number of infinite random digit series that don’t contain anyone’s source code.

Since pi never repeats it should actually contain any sequence of numbers possible though, right? So if you encode your source code as number thet argument should hold up.

Since pi never repeats it should actually contain any sequence of numbers possible though, right?

Simple counterexample: 0.101001000100001... which is 1 followed by one 0, then 1 followed by two 0s, then 1 followed by three 0s, and so on. It never repeats, but "11" will never appear.

"Almost all" real numbers do contain all finite sequences, but it hasn't been proven for pi (https://en.wikipedia.org/wiki/Normal_number).

But if my source code has length n in binary, and I had an infinite number of monkeys typing on an infinite number of terminals, they could find it in pi or any similar digit sequence within (((n * n) * n) * n)! years. Finding a jpeg of the Titanic sinking autographed by the captain might take a little longer, but Fox News is working on that.

GP didn't say "infinite," they said "normal" [1]. If Pi is normal, as conjectured, then it does contain every possible sequence.

1. https://en.wikipedia.org/wiki/Normal_number

wonder if e is normal

> Pi doesn’t have everyone’s source code.

This isn't actually proven though, as far as I know.

There are plenty of infinite non-repeating sequences that we know that don't contain a given sequence. (E.g. 0.1101001000100001000001...)

But pi is not among them. It might very well have everyone's source code. It also might not.

> In mathematics, a real number is said to be simply normal in an integer base b[1] if its infinite sequence of digits is *distributed uniformly* in the sense that each of the b digit values has the same natural density 1/b. A number is said to be normal in base b if, for every positive integer n, all possible strings n digits long have density b^−n.

> While √2, π, ln(2), and e are strongly conjectured to be normal, it is still not known whether they are normal or not.

> An infinite universe doesn’t have infinite copies of someone.

Depends on the uniformity of matter and energy and physical laws in the universe. If matter and energy are uniformly distributed with no large scale variation, and the laws of physics are constant then eventually somewhere in the universe you’d be able to find an Earth arbitrarily close to ours.

I wonder if there's an irrational number, not necessarily pi, whose computation is efficient enough that modern computers could find relatively short pieces of protected data in its digits, for example the AACS encryption key.

https://en.m.wikipedia.org/wiki/Champernowne_constant

That comes with a bonus ability to easily calculate a position of any particular digit sequence too. Nice.

It basically comes down to the million monkeys on a million typewriters argument. Any work of art or creation could be created randomly, given an infinity. Pi is just a sort of random number generator.

Although every PRNG must eventually repeat, while pi is mathematically proven to never repeat. And real-life RNGs are fundamentally limited by hardware, no? How long would it take to pull multiple terabytes off one? They're not meant for it.

Must it? What if you use the seed as an offset into pi, and then just return digits of pi from that point onwards?

(of course that will eventually run into a problem with the state of the RNG exceeding available memory, so it will ultimately fail on resource exhaustion, but it won't repeat)

I think the problem is calculating pi must be done in series, so you can't really get much further than anyone else. contrast that to a hypothetical situation where you could effortlessly start calculating at the nth position of pi. Then you'd have a rock-solid RNG where your real-world entropy source could repesent each digit of the index, so i.e. you get a 500 digit number, then start reciting pi from that position, now you're cooking with gas. Meaning your search space to crack the RNG would be about as difficult as cracking AES-256. but anyway I'm talking above my paygrade here, so no guarantees :)

> I think the problem is calculating pi must be done in series, so you can't really get much further than anyone else. contrast that to a hypothetical situation where you could effortlessly start calculating at the nth position of pi.

There is this: https://en.m.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80... , so you can calculate later digits of pi without the earlier ones.

Coincidentally, does your friend know about the Library of Babel?

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