This is just something I've been testing around for a few hours and thought was pretty neat so I decided to share it here. Please don't take it too seriously :)
What is able to express "any program"
Fortunately Alan Turing already solved this problem for us, and defined the term "Turing complete" to qualify anything that can express any possible Turing machine (thus, being able to express any algorithm/function/computable thing). One quite famous Turing-complete programming language is Brainfuck, an esoteric programming language that has no practical use (much like this own experiment) but was made to be challenge programmers and to be as simple as possible, having the smallest compiler. It only uses 8 symbols: +, -, >, <, [, ], ., and ,. After reading a bit about this language I thought about a Veritasium video I saw not long ago about Gödel numbering, and how it could be applied to also turn any program written in Brainfuck into an integer!
From Brainfuck to a number
Gödel's encoding works by assigning a number to every symbol used, and then when at the i-th symbol $$$s_i$$$ multiply to the answer $$$p_i^{s_i}$$$, with $$$p_i$$$ being the i-th prime. And that way, using prime factorization, we can decode the number and turn it into normal Brainfuck again. The code to do that is pretty simple, the only caveat is the need to use the numbers as strings because they can get quite big. I did all of this at dawn so I didn't really want to code from scratch, and just got the boring parts from Geeks for Geeks. Here is the encoding tool in C++ if you want to check it out.
From a number to running code
The same system works here, just get the prime factors of the number and use them to form the Brainfuck code. I also added a really cool Brainfuck interpreter that I copied from here, so just by giving a number it will run the code by itself. The complete runner code is here if you want to check it out.
Final thoughts
I tested it with a more complex program written by mitxela: a complete tic-tac-toe AI on BF, that actually works! While this may not be much, I still think it's really cool that you can give a single integer as an input to a relatively simple tool and run any conceivable program, even playable AIs. The "Gödel number" for the tic-tac-toe AI is available here, but you can generate it for any BF code in a few seconds using the encoding tool provided above. Thanks for reading this blog!
I guess it's cool, but here's a suggestion: why use Goedel's encoding with multiplying prime powers, when you can just regard a Brainfuck string of code as a huge base-8 number and convert it by summing powers of 8? The resulting number would still be huge (but grow less quickly), and you would get the benefit of looking cool by using binary-looking things (8 = 2^3).
Dang, that's true, didn't think about it :P Thanks for the feedback!
honestly this is Unary (the esoteric programming language) but overcomplicated
Oh that is nice, didn't know about it, thanks!
but we could also just look at any random c++ code and take the integer represented by the bytes no?