Kolmogorov Complexity & Compressibility

Structured patterns compress better — LZ complexity as a proxy for K-complexity

Pattern: Random LZ76 complexity: - Compression ratio: - Entropy (bits): -
About: Kolmogorov complexity K(x) is the length of the shortest program generating x — it is uncomputable in general. LZ76 (Lempel-Ziv 1976) complexity is a computable proxy: count the number of distinct substrings in a greedy parse. K(x) ≤ |LZ76 description| + O(log n). Random sequences have maximal K; periodic sequences have K ∝ log n; fractal sequences have intermediate complexity. Compression ratio = compressed/original is a practical upper bound on K/n.