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The Kolmogorov complexity of a deterministic universe is constant. The Kolmogorov complexity of a nondeterministic universe grows over time. It grows whenever something happens that is not predetermined by its laws of nature. E.g. randomness or free will.

Would it be possible to measure a difference? If the universe's information content grows, does that also increase its energy content?

Edit: By "complexity of a universe" I mean the amount of information required to simulate it up to some point in time.

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  • $\begingroup$ I think the question should be rephrased, because the statement "The Kolmogorov complexity of a nondeterministic universe grows over time. It grows whenever something happens that is not predetermined by its laws of nature." is not always correct as stated. For example, if we are studying a nondeterministic universe described by 100 bits, then its complexity cannot be greater then 100 + some overhead, and therefore cannot keep growing as nondeterministic events happen. $\endgroup$ Jan 11 at 3:11
  • $\begingroup$ Wouldn't that mean some information gets deleted? Isn't it impossible to delete information? en.wikipedia.org/wiki/…. $\endgroup$
    – LinusK
    Jan 11 at 12:46
  • $\begingroup$ That's only true for a universe governed by linear, unitary quantum mechanics $\endgroup$ Jan 11 at 17:15
  • $\begingroup$ @ReasonMeThis You are right. I can rephrase "complexity of a universe" more precisely: How much information is required to simulate a universe up to some point in time? Then your "100 bits" is the "frame size" and for every point in time there is such a "frame". So, how much can you compress a universe's history? Deterministic universes can be reduced to their constant-sized laws of nature and initial conditions. Nondeterministic universes require more data as time goes by and random events happen. $\endgroup$
    – LinusK
    Jan 15 at 3:04
  • $\begingroup$ @ReasonMeThis Nitpicking: Suppose we had a 100-bit universe containing only a single "1" bit which moves randomly left or right. Representing that bit's position requires about log2( 100 bit ) naively. However, in a deterministic universe this position isn't a random variable. So, for every point in time we can have different minimum space requirements. E.g. whenever our bit is at position 1 or 0 we can represent that state in a single bit because of determinism. $\endgroup$
    – LinusK
    Jan 15 at 4:35
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There are several reasons why this question doesn't make sense:

Algorithmic complexity is relative to some language, but you haven't specified one.

The universe is probably infinite, so its information content may be infinite.

The universe is not a classical system, so classical bits aren't a good way of measuring its information content. Qubits would make more sense. For a quantum system, we expect its information content to stay constant, because time evolution is unitary.

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  • $\begingroup$ The language is irrelevant. Any universe's algorithmic complexity is proportional to the number of nondeterministic events that occur within it. $\endgroup$
    – LinusK
    Jan 11 at 1:36
  • $\begingroup$ The answer to this question may be different for an observer in a quantum universe, whose world is determined, in effect, by a series of measurements performed by the observer. Each such measurement would reduce the amount of uncertainty in the observer's world. If "information" is the content of the universal wavefunction, then the amount of information must decrease for the observer. Of course, if the observer is included in the universal wave function (as in Many Worlds), then the amount of information in the universal wave function does not change. $\endgroup$
    – S. McGrew
    Jan 11 at 1:49
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Would it be possible to measure a difference?

No, if we are talking about the universe in which we live, and not some simple abstract universe. Why not? Because it's really hard to distinguish a good deterministic pseudo-random number generating algorithm from true randomness. And given that we can access only a minuscule portion of the information describing the universe, this task becomes impossible.

If the universe's information content grows, does that also increase its energy content?

No, there is no relation.

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  • $\begingroup$ Simulating a pseudo-random universe requires no more information than its finite laws of nature. A truly random universe requires more information the more randomness occurs. In our universe information cannot be destroyed, right? So, the information content would grow. And any information is represented in some form of energy, no? $\endgroup$
    – LinusK
    Jan 15 at 4:33
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    $\begingroup$ When the no hiding theorem says that information in our universe cannot be destroyed, it's using the term information to refer to the information content at a specific time, which is different from how you defined complexity in the revised version of your question (you defined it as the amount of information contained in the whole history of the universe up to a specific time). So if we continue to use the terms information and complexity in these distinct senses, then the correct thing to say would be: … $\endgroup$ Jan 15 at 22:28
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    $\begingroup$ … In our universe "information" stays the same (whether the correct interpretation of QM is deterministic or not), this is typically called conservation of information. "Complexity" (as you defined it here) stays the same if the ultimate laws of nature are deterministic, and keeps increasing otherwise, but we can never distinguish this case from the "pseudo- random" deterministic one (because of the argument i gave above). $\endgroup$ Jan 15 at 22:33
  • $\begingroup$ One more question: Say, our universe has some finite information content at a specific time. Information cannot be deleted. So, wouldn't a random event have to increase the universe's information content? And if it is impossible to increase the information content wouldn't that mean in reverse that the universe must be deterministic? $\endgroup$
    – LinusK
    Jan 16 at 19:59
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    $\begingroup$ If we are talking about information content at a specific time, then no, a random event does not have to increase it. That was what my very first comment under your question was about: "For example, if we are studying a nondeterministic universe described by 100 bits, then its [Information content] cannot be greater than 100 ... and therefore cannot keep growing as nondeterministic events happen." $\endgroup$ Jan 18 at 22:45

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