Estimating the kolmogorov complexity of the known laws of physics?

First, remember that Kolmogorov complexity is only well-defined up to a constant, which is determined by your model of computation.

And that is why it makes little sense to talk about the complexity of one object without comparing or using it in relation to other objects, like the OP asked. If you implement the law of physics in Brainfuck code, on a x86 instruction set, or on a machine specifically dedicated to have the law of physics as a command, you can get wildly different complexities, even down to a few bit (think about the UTM that has a specific instruction that emulates the law of physics).

\forall s\ |K_1(s) - K_2(s)| \leq c. "

First, remember that Kolmogorov complexity is only well-defined up to a constant, which is determined by your model of computation. [...]

Right, if you do something like (say) take away the addition instruction then the shortest program might get longer because it has to emulate addition using subtraction and negation (or use a totally different approach, or include a translation pass that expands the additions into negate+subtract or include a universal turing machine that runs the original machine or... yeah).

Second, the article addresses why you can have the paradoxical situation of the Universe being low-complexity while specific things in the Universe are of high complexity.

In this case I care about the complexity of the laws that govern the change in state (positions of atoms or values of the wave functions) with respect to time. I will not make you pay for the presumably absurd amount of data required for the initial state of >10^(82) atoms. That would make talking about the complexity of the laws laughably negligible. I realize that this is, in a sense, an arbitrary distinction but... that's the question I want to know the answer to.

According to string theory (which is a Universal theory in the sense that it is Turing-complete) the landscape of possible Universes is 2^500 or so, which leads to 500 bits of information. Perhaps this is where Eliezer got the figure from (though I admit that I don't exactly know where he got it from either).

Interesting guess at where the number could have come from. I just assumed he tallied up various aspects of the standard model somehow and got an answer between 450 and 510.

Well, several of the universal constants arguably define our units. For every base type of physical quantity (things like distance, time, temperature, and mass, but not, for example, speed, which can be constructed out of distance and time), you can set a physical constant to 1 if you're willing to change how you measure that property. For example, you can express distance in terms of time (measuring distance in light-seconds or light-years). By doing so, you can discard the speed of light: set it to 1. Speeds are now ratios of time to time: something moving at 30% the speed of light would move 0.3 (light) seconds per second: their speed would be the dimensionless quantity 0.3. You can drop many other physical constants in this fashion: Offhead, the speed of light, the gravitational constant, planks constant, the coulomb constant, and the Boltzmann constant can all be set to 1 without any trouble, and therefore don't count against your complexity budget.

I agree that the constants might be related in ways we don't know, which would allow compression. I'm more interested in an upper bound on the complexity than an exact value (which is likely incomputable for halting problem reasons), so I'm willing to be over by 100 bits because we fail to see a pattern.

As far variable constants: Sure, we can estimate the kolmogorov complexity where the "constants" are inputs. Or we can estimate the kolmogorov complexity of the current laws plus the state 13.7 billion years ago at the big bang. Or we can estimate the complexity of a program that runs all programs. All of these questions are interesting. But right now I want the answer to the one I asked, not the others.

edit clarified response

Assuming you're right about it only being a few more lines of code, and that you didn't use a lot of external libraries, that puts an upper bound at... a few dozen kilobits? I'm guessing that could be made a lot smaller, since you were likely not focused on minimizing the lines of code at all costs.

Attacking physics head on is going to be a bit hand-wavy, since we don't have the complete picture of a formal system that encompasses all physics yet.

How about for "the standard model" instead of "known physics"? That's a much better defined problem.

What do we know about estimating actual, stated-as-an-approximate-number-of-bits Kolmogorov complexities for formal systems which we can describe completely? Can we give an informed estimate about the number of bits needed for the rules of chess, or Newtonian-only toy physics?

Yes, we can (as long as you assume a particular instruction set). I don't know if anyone's done it, though. Also important to note that we can give upper bounds, but lower bounds are extremely difficult because of the inability to prove properties like "output matches physics" of programs in general.

I'm fine with a guess being included in the text, even one that isn't elaborated on. The post wasn't about making that particular estimate, and elaborating on it there would have been counter-productive.

However, that doesn't really answer my question. What factors need to be accounted for when estimating the kolmogorov complexity of physics? Which parts cost the most? Which parts cost almost nothing?

You're confusing chaos with incomputability. Chaos has to do with mixing, error amplification, and imperfect initial conditions. Incomputability has to do with abstract problems and whether or not they can solved by computers.

In any case, even if the standard model was an approximation of truly Turing-incomputable physics laws... I still want to know what its Kolmogorov complexity is.

My understanding is that solving Newton's equations for more than two bodies is not computable, to take just one example.

What do you mean?

They typically produce a program that, when run, plays video and audio in real time after a loading period.

Okay, I wouldn't normally do this, but.. what's with the downvote? I honestly have no idea what the problem is, which makes avoiding it hard. Please explain.

https://en.wikipedia.org/wiki/Speed_prior / http://www.idsia.ch/~juergen/speedprior.html

You're saying a Solomonoff Inductor would be outperformed by a variant that weighted quick programs more favorably, I think. (At the very least, it makes approximations computable.)

Whether or not penalizing for space/time cost increases the related complexity metric of the standard model is an interesting question, and there's a good chance it's a large penalty since simulating QM seems to require exponential time, but for starters I'm fine with just an estimate of the Kolmogorov Complexity.