The Incompressibility Method

Abstract

The incompressibility of random objects yields a simple but powerful proof technique. The incompressibility method is a general purpose tool and should be compared with the pidgeon hole principle or the probabilistic method. Whereas the older methods generally show the existence of an object with the required properties, the incompressibility argument shows that almost all objects have the required property. This follows immediately from the fact that the argument is typically used on a Kolmogorov random object. Since such objects are effectively undistinguishable, the proof holds for all such objects. Each class of objects has an abundance of relatively Kolmogorov random objects. In fact, they have probability about 1.