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I am thinking of a (greatly simplified) computer simulation of a universe that followed something like Newtonian rules. Inside the simulation are A.I.s that are made from those same rules, and can only use those rules interact the world around them. Would there be some fundamental limits on what those A.I.s could work out about their universe, like their own version of an uncertainty principle?

Sorry for phrasing this question in such a convoluted way. If anyone recognises what I am asking, and can point me in the right direction that would be appreciated.

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you want to be god, don't you? just kidding! :-) good question! – Pratik Deoghare Dec 15 '10 at 8:06
Is the premise; If the physical laws were deterministic - ie the future state of the universe could be completely determined by inserting a previous state + physical laws into a computer? And the question; If an AI within that universe could completely determine the future state of the universe? In that case, no, I will provide proof if assumption and question is correct. – TROLLKILLER Dec 15 '10 at 8:32
@kalle43 Yes, I think that is what I meant. I was going to use the word deterministic, but I hesitated. I don’t expect the AIs to be able to predict their own future (I would think that doing that would automatically change it, and result in some kind of infinite loop), but would they ultimately have to use a style of physics closer to quantum mechanics, just because they themselves are a part of their world. – user840 Dec 15 '10 at 12:22
There is a related question: A computer is running under the rules of our universe, so it should not possible for them to discover rules that do not exist in our world? We can consider: Given a finite time for a Turing machine, it should not possible simulate our world. Our world should be continue, but Turing machine can only has discrete state. – hwlau Dec 16 '10 at 15:39
Why the best answer get no upvotes? Always the fastest. This degrades the quality of this site, upvotes should be weighted by time etc. – TROLLKILLER Dec 17 '10 at 7:46

Newtonian mechanics is pretty good in producing deterministic chaos which is much more important factor in every-day uncertainty than the uncertainty principle.

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Classical chaos will always limit the ability to predict the future, no matter how good the AI, because there will always be some finite uncertainty in the measurement of the initial conditions. A remarkable number of seemingly simple systems are, in fact, sensitively dependent on the initial conditions, and will become unpredictable after a short time. There's a bit in Iain Banks's The State of the Art, one of his stories about the Culture, an interstellar civilization ruled by super-intelligent AI's where one of the Minds who run the Culture says "I’m the smartest thing for a hundred light year radius, and by a factor of about a million … but even I can’t predict where a snooker ball’s going to end up after more than six collisions." While I can't vouch for the exact accuracy of the snooker reference, the general idea is sound.

This uncertainty is not at all like quantum uncertainty a la the Heisenberg Uncertainty Principle, though. The Uncertainty Principle is, fundamentally, a consequence of the fact that quantum objects have both particle and wave properties, and any attempt to have both of those things at the same time will necessarily involve some uncertainty in both position and momentum (or any other pair of conjugate observables). A purely classical universe would not have that feature, but would have some uncertainty due to finite measurement capabilities.

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Well, Newtonian mechanics are abandoned in our world in some scales because it fails to explain certain phenomena. Which means that there are some stuff in work other than Newtonian mechanics, such as the properties of atoms, electrons, spectra, time, relative motion, limit of speed...

If the universe that the AIs that you proposed live in is a stable one(because electrons would spiral down to the core and annihilate in classical mechanics), and the energy was not quantized(i.e. h=0), uncertainty principle would no longer be valid. Wave-particle duality would not exist either as the de Broglie wavelengths of all particles would be 0. Similarly if c were equal to infinity, time dilation and length contraction would not exist in special relativity. To model gravity you would probably use springs between each and every object in the universe, so curvature of space time would disappear, hence the time dilation as well. Basically every single quantum and relativity based phenomena would not exist.

Let me note that I do not know if a stable universe capable of sustaining protons and electrons is possible solely by Newtonian mechanics without manipulating fundamental constants.

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Well, let's clear up some issues. contrary to the answers above I think you mean working out the uncertainty relationship (UR) itself as opposed to predicting the future.

If this is the case first of all you shouldn't expect your simulated intelligence to come up with a version of quantum uncertainty relation since you're limiting your system to the classical laws. Then even if you mean some sort of classical chaos then it would be tricky. There is no physical limitation to prevent your intelligence to come up with an UR. However, computationally it has to be capable of abstraction, categorisation and a rule-based reasoning. My intuition is some techniques such as adaptive resonance theory and a kind of evolutionary computation ( plus the introduction of some noise can work in principle. The reason for introducing noise is that this is something you gonna need to get to the state of complexity you need in your system which comes out of the evolutionary computation.

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"contrary to the answers above I think you mean working out the uncertainty relationship (UR) itself as opposed to predicting the future. " good observation – user346 Dec 16 '10 at 4:57

Some sources of uncertainties I could think of:

-They could never determine what physical laws governs their universe with 100% certainty. If an experiment has the same outcome a googleplex times, you still cannot know the same outcome will occur the next time.

-Computational problems, we dont even have a solution to the 3-body problem in CM.

-They would have to interact with their universe to measure it, how to account for what degree one at the same time impacts it ? Question: Sometimes the source of UP in QM is explained by the fact one cannot measure a system without interacting with it, but this same applies to calssical mechanics, so what is different?

-If the space they live in is continuous, it seems impossible to measure position etc exactly.

-Also as you mentioned, a complete model of the universe, within the universe, would include the model, which includes the model of the model etc. This leads to many problems..

-In the case one overcomes all these, Gödel's incompleteness theorem would ultimately rule over all.

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