Stephen Wolfram

Is it, in some sense, a definition of heat, perhaps?

Well, it's a combination of those things.

And it's the same thing with the principle of computational equivalence.

And in some sense, the principle of computational equivalence is at the heart of the definition of computation.

Because it's telling you there is a thing, there is a robust notion that is equivalent across all these systems and doesn't depend on the details of each individual system.

And that's why we can meaningfully talk about a thing called computation.

And we're not stuck talking about, oh, there's computation in Turing machine number 3785.

and et cetera, et cetera, et cetera.

That's why there is a robust notion like that.

Now, on the other hand, can we prove the principle of computational equivalence?

Can we prove it as a mathematical result?

Well, the answer is, actually, we've got some nice results along those lines that say, you know, throw me a random system with very simple rules.

Well, in a couple of cases, we now know that even the very simplest rules we can imagine of a

are universal and do sort of follow what you would expect from the principle of computational equivalence.

So that's a nice piece of sort of mathematical evidence for the principle of computational equivalence.

Right, so there are various indicators.

So for example, one thing would be, is it capable of universal computation?

That is, given the system, do there exist initial conditions for the system that can be set up to essentially represent programs to do anything you want, to compute primes, to compute pi, to do whatever you want, right?

So that's an indicator.

So we know in a couple of examples that yes,