On the university of grammar-based codes for sources with countably infinite alphabets

Grammar-based codes refer to a new class of universal lossless data compression algorithms. Originally, grammar-based codes were designed and analyzed for sources with finite alphabets. Grammar-based codes are very efficient in many applications like lossless audio compression and high-quality image compression, where data sequences to be compressed are often from large alphabets with cardinalities even comparable to the sequence lengths. In these cases, the assumption of a finite alphabet is hardly justified; instead, one may simply assume that data sequences come from a countably infinite alphabet. This paper investigates the universality and redundancy of grammar-based codes for sources with a countably infinite alphabet.