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arXiv · 0810.3125

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

Łukasz Dębowski
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Abstract

The article 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.

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Łukasz Dębowski. 2008-10-17. On the Vocabulary of Grammar-Based Codes and the Logical Consistency of Texts. https://doi.org/10.1109/tit.2011.2145170

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