A short proof that $O_2$ is an MCFL

TL;DR

This paper offers a simplified proof that the language $O_2$ is a multiple context-free language, avoiding complex geometric concepts, and suggests potential for extending the result to higher dimensions, impacting linguistic syntax modeling.

Contribution

It provides a new, simpler proof that $O_2$ is an MCFL, avoiding geometric concepts, and opens avenues for higher-dimensional generalizations.

Findings

$O_2$ is an MCFL confirmed by a new proof

The proof avoids two-dimensional geometric concepts

Potential for extending results to higher dimensions

Abstract

We present a new proof that $O_2$ is a multiple context-free language. It contrasts with a recent proof by Salvati (2015) in its avoidance of concepts that seem specific to two-dimensional geometry, such as the complex exponential function. Our simple proof creates realistic prospects of widening the results to higher dimensions. This finding is of central importance to the relation between extreme free word order and classes of grammars used to describe the syntax of natural language.