3 ms·

All laws of physics aren't computable. None of them are, they are all defined over the reals. You've a bigger issue with talking about the computability of tho

All laws of physics aren't computable. None of them are, they are all defined over the reals. You've a bigger issue with talking about the computability of those laws. I am assuming that somehow the reals can be ditched (which, incidentally, seems unlikely). And in that reality, we still have a problem: the role the reals play is to make reality infinitely precisely measurable. Without this chaotic systems become non-deterministic over realtively short time horizons. Nearly all actual systems are chaotic in many of their properties. It isnt a matter of just adding an RNG to a CPU. A trivial object has 10^30 atoms, in organic stuff, much of that is behaving chaotically on environmental boundaries. The engery to simulate this, assuming 10^30 RNGs (etc.) is vastly greater than the target system and "cannot outrun" that system. Not only this but I think simulation of such things quickly surpasses physical bounds on physically possible computers, not least as the RNGs have an energy cost.

tsimionescu 5y ago
The real numbers used in physics are all computable (pi [0], e, sqrt(2) etc are all computable numbers). There are non-computable real numbers, but so far those have never been necessary for describing physical systems (e.g. Chaitin's constant). When talking about computability in the absolute, theoretical sense, the size of the universe is irrelevant - all we ask is if there exists some arbitrarily large, but finite, Turing machine that performs that computation and then halts after an arbitrarily large but finite amount of time. If we're taking about practical computability, than my base assertion that the physical world is itself a computer also implies that this computer is exactly fast enough to simulate physics. Since we know that the laws of physics are computable in the theoretical sense above, and since we know that the universe is accurately simulating itself, we have proven my assertion. Your other points about chaos and non-determinism I have addressed in another thread. [0] https://stackoverflow.com/questions/4134282/is-pi-a-turing-computable-number https://stackoverflow.com/questions/4134282/is-pi-a-turing-c...

tsimionescu 5y ago

The real numbers used in physics are all computable (pi [0], e, sqrt(2) etc are all computable numbers). There are non-computable real numbers, but so far those have never been necessary for describing physical systems (e.g. Chaitin's constant). When talking about computability in the absolute, theoretical sense, the size of the universe is irrelevant - all we ask is if there exists some arbitrarily large, but finite, Turing machine that performs that computation and then halts after an arbitrarily large but finite amount of time. If we're taking about practical computability, than my base assertion that the physical world is itself a computer also implies that this computer is exactly fast enough to simulate physics. Since we know that the laws of physics are computable in the theoretical sense above, and since we know that the universe is accurately simulating itself, we have proven my assertion. Your other points about chaos and non-determinism I have addressed in another thread. [0] https://stackoverflow.com/questions/4134282/is-pi-a-turing-computable-number https://stackoverflow.com/questions/4134282/is-pi-a-turing-c...