2.5 years ago I made an attempt to calculate an upper bound for the complexity of the currently known laws of physics. Since the issue of physical laws and complexity keeps coming up, and my old post is hard to find with google searches, I'm reposting it here verbatim.

I would really like to see some solid estimates here, not just the usual hand-waving. Maybe someone better qualified can critique the following.

By "a computer program to simulate Maxwell's equations" EY presumably means a linear PDE solver for initial boundary value problems. The same general type of code should be able to handle the Schroedinger equation. There are a number of those available online, most written in Fortran or C, with the relevant code size about a megabyte. The Kolmogorov complexity of a solution produced by such a solver is probably of the same order as its code size (since the solver effectively describes the strings it generates), so, say, about 10^6 "complexity units". It might be much lower, but this is clearly the upper bound.

One wrinkle is that the initial and boundary conditions also have to be given, and the size of the relevant data heavily depends on the desired precision (you have to give the Dirichlet or Neumann boundary conditions at each point of a 3D grid, and the grid size can be 10^9 points or larger). On the other hand, the Kolmogorov complexity of this initial data set should be much lower than that, as the values for the points on the grid are generated by a piece of code usually much smaller than the main engine. So, in the first approximation, we can assume that it does not add significantly to the overall complexity.

Things get dicier if we try to estimate in a similar way the complexity of the models like General Relativity, the Navier-Stokes equations or the Quantum Field Theory, due to their non-linearity and a host of other issues. When no general-purpose solver is available, how does one estimate the complexity? Currently, a lot of heuristics are used, effectively hiding part of the algorithm in the human mind, thus making any estimate unreliable, as human mind (or "Thor's mind") is rather hard to simulate.

One can argue that the equations themselves for each of the theories are pretty compact, so the complexity cannot be that high, but then, as Feynman noted, all of physical laws can be written simply as A=0, where A hides all the gory details. We still have to specify the algorithm to generate the predictions, and that brings us back to numerical solvers.

I also cannot resist noting, yet again, that all interpretation of QM that rely on solving the Schrodinger equation have exactly the same complexity, as estimated above, and so cannot be distinguished by the Occam's razor. This applies, in particular, to MWI vs Copenhagen.

It is entirely possible that my understanding of how to calculate the Kolmogorov complexity of a physical theory is flawed, so I welcome any feedback on the matter. But no hand-waving, please.