RePair and All Irreducible Grammars are Upper Bounded by High-Order Empirical Entropy

Irreducible grammars are a class of context-free grammars with well-known representatives, such as Repair (with a few tweaks), Longest Match, Greedy, and Sequential. We show that a grammar-based compression method described by Kieffer and Yang (2000) is upper bounded by the high-order empirical entropy of the string when the underlying grammar is irreducible. Specifically, given a string $S$ over an alphabet of size $\sigma $ , we prove that if the underlying grammar is irreducible, then the length of the binary code output by this grammar-based compression method is bounded by $|S|H_{k}(S) + o(|S|\log \sigma)$ for any $k\in o(\log _\sigma |S|)$ , where $H_{k}(S)$ is the $k$ -order empirical entropy of $S$ . This is the first bound encompassing the whole class of irreducible grammars in terms of the high-order empirical entropy, with coefficient 1.