Local generation of languages

TL;DR

This paper investigates how to generate fixed-length languages using local rules at each position, analyzing the communication needed between positions through a simplicial complex framework.

Contribution

It introduces a novel theoretical framework using simplicial complexes to characterize local generation of languages and identifies which communication structures can produce given languages.

Findings

Developed a theory linking communication structures to language generation

Applied the framework to various example languages

Provided criteria for the sufficiency of local rules in language production

Abstract

Given a language, which in this article is a set of strings of some fixed length, we study the problem of producing its elements by a procedure in which each position has its own local rule. We introduce a way of measuring how much communication is needed between positions. The communication structure is captured by a simplicial complex whose vertices are the positions and the simplices are the communication channels between positions. The main problem is then to identify the simplicial complexes that can be used to generate a given language. We develop the theory and apply it to a number of languages.