Local Grammar-Based Coding Revisited

In the setting of minimal local grammar-based coding, the input string is represented as a grammar with the minimal output length defined via simple symbol-by-symbol encoding. This paper discusses four contributions to this field. First, we invoke a simple harmonic bound on ranked probabilities, which reminds Zipf's law and simplifies universality proofs for minimal local grammar-based codes. Second, we refine known bounds on the vocabulary size, showing its partial power-law equivalence with mutual information and redundancy. These bounds are relevant for linking Zipf's law with the neural scaling law for large language models. Third, we develop a framework for universal codes with fixed infinite vocabularies, recasting universal coding as matching ranked patterns that are independent of empirical data. Finally, we analyze grammar-based codes with finite vocabularies being empirical rank lists, proving that that such codes are also universal. These results extend foundations of universal grammar-based coding and reaffirm previously stated connections to power laws for human language and language models.