On the Vocabulary of Grammar-Based Codes and the Logical Consistency of Texts

This paper presents a new interpretation for Zipf–Mandelbrot's law in natural language which rests on two areas of information theory. Firstly, we construct a new class of grammar-based codes and, secondly, we investigate properties of strongly nonergodic stationary processes. The motivation for the joint discussion is to prove a proposition with a simple informal statement: If a text of length
$n$
describes
$n^{\beta} $
independent facts in a repetitive way then the text contains at least
$n^{\beta} /\log n$
different words, under suitable conditions on
$n$
. In the formal statement, two modeling postulates are adopted. Firstly, the words are understood as nonterminal symbols of the shortest grammar-based encoding of the text. Secondly, the text is assumed to be emitted by a finite-energy strongly nonergodic source whereas the facts are binary IID variables predictable in a shift-invariant way.