Kolmogorov Random Graphs and the Incompressibility Method

Kolmogorov Random Graphs and the Incompressibility Method

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of t...

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 mu...

Kolmogorov Incompressibility Method in Formal Proofs A Critical Survey

We compare the incompressibility method of Kolmogorov complexity that is used in formal proofs of mathematical and computational results with more traditional methods such as proofs by counting, proofs by probabilistic arguments and proofs by pumping lemmas for formal languages. We consider applications of Kolmogorov complexity in several diierent areas such as lower bounds, average case analys...

The Incompressibility Method

Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source. It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner. This is impossible for classical probability theory. Kol...

Kolmogorov random graphs only have trivial stable colorings