Distortion program-size complexity with respect to a fidelity criterion and rate-distortion function

A novel concept of distortion Kolmogorov-chaitin complexity is proposed. Bearing analogy to ordinary Kolmogorov-Chaitin complexity that represents the information content of a single message, distortion Kolmogorov-Chaitin complexity can be regarded as the amount of information about a single message that must be conveyed in order to reproduce it with a bounded distortion. This explanation is justified by the properties of distortion Kolmogorov-Chaitin complexity and the equivalences between distortion Kolmogorov-Chaitin complexity and the rate-distortion function, which are analogous to those between ordinary Kolmogorov-Chaitin complexity and the Shannon entropy. Some implications for universal almost sure data compression are also discussed