Kolmogorov Complexity and the Incompressibility Method

1. Introduction. What makes one object more complex than another? Kolmogorov complexity, or program-size complexity, provides one of many possible answers to this fundamental question. In this theory, whose foundations have been developed independently by R. the complexity of an object is defined as the length of its shortest effective description, which is the minimum number of symbols that must be specified such that the object can be reproduced from the specification by some abstract computing machine or formal system. The minimal length for an effective description of an object obviously depends on the exact method used for reproducing the object from the description. Since we want to measure the complexity of objects independently of any particular model of computation, we therefore define Kolmogorov complexity relative to a fixed (but unspecified) universal computable function. Then, using any model of computation, objects can be described at most a constant number of symbols shorter than their Kolmogorov complexity, where 'constant' refers to a value that depends on the method for reproducing the objects but not on the described objects themselves. We can ignore these additive constants because we are typically interested in asymptotic complexities rather than in numeric complexity values. The notion of Kolmogorov complexity yields a simple yet powerful proof technique, called the incompressibility method. The purpose of this paper is to explain the concepts on which this technique is based and, along the way, to provide a concise introduction to Kolmogorov complexity theory. The only prerequisite is a basic understanding of computable functions; the employed notation is summarized in a short paragraph after the main text. The definitions and theorems in this paper are more or less standard; if not stated otherwise, they have been taken (with some modifications) from the book by Li and Vitányi [9], which is a comprehensive reference on Kolmogorov complexity theory and it applications.