Geometric representations for minimalist grammars

TL;DR

This paper reformulates minimalist grammars using geometric vector spaces, enabling the realization of language processing functions as piecewise linear operators and introducing a measure of processing complexity called harmony.

Contribution

It introduces a novel geometric framework for minimalist grammars, mapping structure-building functions into vector spaces and defining harmony as a complexity measure.

Findings

Language processing functions can be implemented as piecewise linear operators.

Harmony effectively measures processing complexity.

Illustrations include arithmetic and fractal representations.

Abstract

We reformulate minimalist grammars as partial functions on term algebras for strings and trees. Using filler/role bindings and tensor product representations, we construct homomorphisms for these data structures into geometric vector spaces. We prove that the structure-building functions as well as simple processors for minimalist languages can be realized by piecewise linear operators in representation space. We also propose harmony, i.e. the distance of an intermediate processing step from the final well-formed state in representation space, as a measure of processing complexity. Finally, we illustrate our findings by means of two particular arithmetic and fractal representations.