Reconciling Data Compression and Kolmogorov Complexity

Abstract

While data compression and Kolmogorov complexity are both about effective coding of words, the two settings differ in the following respect. A compression algorithm or compressor, for short, has to map a word to a unique code for this word in one shot, whereas with the standard notions of Kolmogorov complexity a word has many different codes and the minimum code for a given word cannot be found effectively. This gap is bridged by introducing decidable Turing machines and a corresponding notion of Kolmogorov complexity, where compressors and suitably normalized decidable machines are essentially the same concept.

Kolmogorov complexity defined via decidable machines yields characterizations in terms of the intial segment complexity of sequences of the concepts of Martin-Löf randomness, Schnorr randomness, Kurtz randomness, and computable dimension. These results can also be reformulated in terms of time-bounded Kolmogorov complexity. Other applications of decidable machines are presented, such as a simplified proof of the Miller-Yu theorem (characterizing Martin-Löf randomness by the plain complexity of the initial segments) and a new characterization of computably traceable sequences via a natural lowness notion for decidable machines.