information - Number of bits needed to express physical laws?

What is the minimum number of bits that would be needed to express a given physical law, like the law of universal gravitation? How many bits are needed to express each of the four fundamental forces? Is there a pattern here?

Answer

One productive way of thinking of the complexity of physical laws is in terms of the Kolmogorov complexity of the algorithm that simulates a given physical situation. This is defined as the length of the shortest code which does the simulation.

If you are given a law of nature, like Newton's law of universal gravitation, you can write a simulation of N-interacting particles. If you are only interested in an in-principle answer, you are looking for the best algorithm to simulate Newton's laws on a computer.

The problem of computing the Kolmogorov complexity precisely is in general the worst of all uncomputable problems. You can usually shrink a code written from scratch by a lot by cleverly rewriting the subroutines to make a specialized language for the description. You would never know if you have the optimal coding, since maybe you could compress things more by adding a special interpretation layer.

The rigorous version of this annoyance is the theorem that no axiomatic system can prove an algorithm is optimal (meaning of minimal length) if the length is significantly greater than the length of the program that does the deduction in this axiomatic system. The proof is a simple contradiction: suppose the axiomatic system proves program P is optimal. Write a program CONTRADICTION which goes through all the theorems of the axiomatic system until it finds a program which is proved optimal and whose length is greater than the length of CONTRADICTION. Then, run this program. By construction, CONTRADICTION is shorter than this program and yet has the same output.

If you think about what CONTRADICTION is doing, it is generating the code for a completely different program, and then running it. This gives a hint of the nature of the difficulty in finding a minimal description.

But if you are happy with a crude estimate of the complexity, you just write any old code to simulate the physics, and the length of this code is an estimate of the complexity. This is a useful heuristic that makes precise the desire for simple elegant theories.