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Comment by hcs | Hacker News Reader

I don't think you have the right point, even with an infinite write-only output tape, you'd still need infinite states to output pi. How can your machine know how far along it is otherwise?

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The original thought was, that it wouldnt need to know, it would get the operands on a tail-call eliminated, recursive need to know basis- very slow, but very space efficient, basically lazy re-compute it every time its needed.

Still was wrong.

Even with tail calls, and purely functional code, it can be shown like this: imagine you see a purely functional, tail-called, all-state-as-argument call foo(A, B, REST-OF-STATE). Later on, you encounter the same call with exactly the same values - foo(A, B, REST-OF-STATE). At this point you know that the program looped, since you've proven (through observation) that the call to foo(A, B, REST-OF-STATE) will eventually yield another call to foo(A, B, REST-OF-STATE).

This is essentially equivalent to detecting cycles in a directed graph where each node has outdegree of 1. If you hit the same node twice, you've found a cycle.

It boils down to the question how dense you can encode, retrieve and write information- and this can be very dense.

Conways compression comes to mind, where a simple timing description and some start states can be used to describe a very complex outcome.

The problem is, that for every function computing a irrational number, you got to have the output as input again.

Imagine if it where otherwise, a completely lightweight PI-function, traveling for eternity, for which nothing in the universe could determinate if its going in a loop or going towards a circle constant.

> It boils down to the question how dense you can encode, retrieve and write information- and this can be very dense.

Your max density is 2^bits.

> Conways compression comes to mind, where a simple timing description and some start states can be used to describe a very complex outcome.

Eventually your 'time' number is so big it takes up the entirety of ram, and you can go no further.

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