Structured grammar-based codes for universal lossless data compression

Abstract

A grammar-based code losslessly compresses each finite-alphabet data string x by compressing a context-free grammar Gx which represents x in the sense that the language of Gx is fxg. In an earlier paper, we showed that if the grammar Gx is a type of grammar called ir- reducible grammar for every data string x, then the resulting grammar-based code has maximal redundancy/sample O(log logn=logn) for n data samples. To further reduce the maximal redun- dancy/sample, in the present paper, we first decompose a context-free grammar into its structure and its data content, then encode the data content conditional on the structure, and finally replace the irreducible grammar condition with a mild condition on the structures of all grammars used to repre- sent distinct data strings of a fixed length. The resulting grammar-based codes are called structured grammar-based codes. We prove a coding theorem which shows that a structured grammar-based code has maximal redundancy/sample O(1=logn) provided that a weak regular structure condition is satisfied.