The universality of grammar-based codes for sources with countably infinite alphabets

In this paper, we investigate the performance of grammar-based codes for sources with countably infinite alphabets. Let Lambda denote an arbitrary class of stationary, ergodic sources with a countably infinite alphabet. It is shown that grammar-based codes can be modified so that they are universal with respect to any Lambda if and only if there exists a universal code for Lambda. Moreover, upper bounds on the worst case redundancies of grammar-based codes among large sets of length-n individual sequences from a countably infinite alphabet, are established. Depending upon the conditions satisfied by length-n individual sequences, these bounds range from O(log log n/ log n) to O(1/ log(1-alpha) n) for some 0 < alpha < 1. These results complement the previous universality and redundancy results in the literature on the performance of grammar-based codes for sources with finite alphabets.