How to measure the optimality of word or gesture order with respect to the principle of swap distance minimization

TL;DR

This paper introduces a mathematical framework using permutohedra to measure how optimal word or gesture orderings are with respect to swap distance minimization, revealing significant crosslinguistic gesture optimality.

Contribution

It develops a novel method to quantify the optimality of language sequences based on swap distance, and integrates the quadratic assignment problem into linguistic analysis.

Findings

Crosslinguistic gestures are at least 77% optimal.

The framework shows that optimality is unlikely due to chance.

Establishes theoretical foundations for studying language order optimality.

Abstract

The structure of all the permutations of a sequence can be represented as a permutohedron, a graph where vertices are permutations and two vertices are linked if a swap of adjacent elements in the permutation of one of the vertices produces the permutation of the other vertex. It has been hypothesized that word orders in languages minimize the swap distance in the permutohedron: given a source order, word orders that are closer in the permutohedron should be less costly and thus more likely. Here we explain how to measure the degree of optimality of word order variation with respect to swap distance minimization. We illustrate the power of our novel mathematical framework by showing that crosslinguistic gestures are at least $77\%$ optimal. It is unlikely that the multiple times where crosslinguistic gestures hit optimality are due to chance. We establish the theoretical foundations for research on the optimality of word or gesture order with respect to swap distance minimization in communication systems. Finally, we introduce the quadratic assignment problem (QAP) into language research as an umbrella for multiple optimization problems and, accordingly, postulate a general principle of optimal assignment that unifies various linguistic principles including swap distance minimization.