Choosing a Physical Model: Why Symmetries?

One of the main applications of Kolmogorov complexity ideas to data processing is via the Minimum Description Length principle (see, e.g., 1, 11, 13] and references therein). According to this principle, if several diierent models (or theories) are consistent with the same observations, then we should choose a model with the shortest description, i.e., crudely speaking, a model with the smallest value of Kolmogorov complexity. This principle is in perfect agreement with the Occam principle (it actually formalizes Occam's principle), and it has been successfully applied to various problems. In particular, it has been successfully used in physics, where Occam's principle originated and where it has been successfully used. In modern fundamental physics, however, symmetry groups play such an important role that often, physicists choose not the simplest model, but the model which corresponds to the simplest symmetry group (see, e.g., 2{9], 12, 14]). At rst glance, this restriction to symmetry-deened models seem to prevent us from considering possible simple non-group models, and thus, make this symmetry-group version of Occam principle worse than the unrestricted one. However, the success of this direction in theoretical physics seems to indicate that this restriction does not bring any disadvantage at all. Our analysis shows that this restriction is indeed non-essential. To formalize the physicists' idea, we x a universal programming language (\computer") f, and deene a symmetry as a program from this language which transforms strings (e.g., binary strings) into strings and which is a bijection (1-1 and onto). By a complexity of a symmetry s, we mean the length len(s) of this program. We say that a symmetry s deenes a string x uniquely if s(x) = x and s(y) 6 = y for all y 6 = x. Now, for every string x, we can deene its group-symmetric complexity C sym (x) as the smallest complexity of a symmetry which deenes x uniquely. It turns out that this new complexity is asymptotically equivalent to the usual Kolmogorov complexity C(x) = C f (x): Proposition. jC(x) ? C sym (x)j = O(1).