The Structure Generating Function and Entropy of Tuple Languages

The structure generating function of a language enumerates the number of distinct words contained in the language with respect to their length. Given any unambiguous tuple grammar, a method is described which yields a system of equations, whose unique solution is the structure generating function. The entropy (channel capacity) is an important information theoretic quantity associated with a language. Since the entropy depends directly on the number of words contained in a language, the structure generating function of a tuple language allows to compute the entropy of the tuple language. The family of pseudolinear tuple grammars is considered. These pseudolinear tuple grammars allow rational structure generating functions easily computed by an algorithm. Strong statements on the entropy of tuple languages generated by pseudolinear tuple grammars can be made.