When given a number n, and you want to display them, how will you actually compute the numbers from 1 to n? If you precalculate a table - then how will you do n+1? What about n+2?

I don't agree that the language is turning complete due to the lack of any runtime dynamic aspect of the language.

It's akin to compiling a simple c program with loops unrolled and fully static variables and claiming the subset of generated assembly is turning complete.

> It's akin to compiling a simple c program with loops unrolled and fully static variables

Yes, and optimizing compilers take this to an extreme. It turns out that "computing" a factorial only requires my runtime language to have `mov` and `ret` instructions: https://godbolt.org/z/3Y6bGcsba

That isn't a thing.

This language can represent any function between finite sets of integers, I will believe that (proof left as exercise). But so can a lookup table.

P.S. precomputing even a single int64 space is going to be really really hard, 2^64 possible values is a lot

       Epigram 54.

       Beware of the Turing tar-pit
       in which everything is possible
       but nothing of interest is easy.

This should be able to interpret programs written in other Turing-complete languages, including in itself.

I was about to say, "choose an encoding for this language in terms of integers that it understands, and then show me an interpreter that can run any program so encoded". But we have another more basic problem first. As others have noted, there is no notion of program input.

Well, fine, we can solve that easily enough. Suppose that at the start of your program you have a variable called "in" containing a single integer of arbitrary size.

Now choose an encoding of these programs into integers, and show a program that will run one of them.

I assert that it isn't possible.

In fact, I think I even have a sketch at a proof. We know that Turing machines have an "undecideability" property, i.e., you can't know in general whether any given program will terminate. But we can prove that any program in this language terminates.

To do that we will need to be a little more formal about exactly what this language is. I think a reasonable choice would be to say that programs are written in static single assignment (SSA) form using only those functions. It is trivial to show that the program terminates in a number of steps equal to the number of lines of the program.

...

Is this post actually a clever troll to make the point that there's only so much that a feedforward neural net can do?

In the words of it's inventor, John Conway:

    Makes workday really easy!  FRACTRAN needs no complicate programming manual. It's entire syntax can be learned in 10 seconds, and programs for quite complex and interesting functions can be written almost at once.  The entire configuration of a FRACTRAN computer at any instant is a single integer - no messy "tapes" or other foreign concepts to be understood by the fledging programmer.