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 analysis of algorithms, formal language theory, and random graphs. We argue that the Kolmogorovcomplexity proofs are more intuitive, elegant and less lengthy than the other arguments. Furthermore, the former proofs are easier to construct and to understand because all of them have the same structure and they all use a similar argument based on the fact that \almost all" strings are not compressible at all.