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

Ramon Ferrer-i-Cancho

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.

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

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 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.

Paper Structure

This paper contains 37 sections, 9 theorems, 162 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Theoretical foundations
  3. How to measure the degree of optimality
  4. The problem of $\left< d \right>_{min}$
  5. Overview of theoretical foundations
  6. Methods
  7. Data
  8. Calculation of $\left< d \right>_{min}$
  9. The chance of the difference between $\left< d \right>$ and $\left< d \right>$
  10. The chance that $\left< d \right> = \left< d \right>_{min}$
  11. The chance of contiguity
  12. Results
  13. Evidence of swap distance minimization
  14. The degree of optimality of crosslinguistic gestures
  15. The epiphenomena of swap distance minimization
  16. ...and 22 more sections

Key Result

Theorem B.1

Given two vectors of real numbers $\mathbf{a}$ and $\mathbf{b}$Hardy1934a, $\blacktriangleleft$$\blacktriangleleft$

Figures (11)

  • Figure 1: The permutohedron of order 3. Vertices are labelled with all the possible permutations of SOV.
  • Figure 2: Hasse diagrams of four total orders (blue) on the permutohedron (black). Every Hasse diagram describes an arrangement of probabilities on the permutohedron. Vertices of the permutohedron are labelled with numbers from 1 to 6 in clockwise sense starting from the left-most vertex. Near each vertex, we show the probability of the vertex. $\pi_i$ stands for the $i$-th largest probability. A blue arrow from vertex $i$ to vertex $j$ represents the relation $p_i \geq p_j$. Top. A Hasse diagram (left) and a symmetric Hasse diagram (right) that minimize $\left< d \right>$ and $\left< d | 1 \right>$. Bottom. A Hasse diagram (left) and a symmetric Hasse diagram (right) that minimize $\left< d | 1 \right>$ but not $\left< d \right>$.
  • Figure 3: The Hasse diagram of a partial order that minimizes $\left< d | 1 \right>$ (blue) on the permutohedron. The permutohedron is labelled with numbers from $1$ to $6$ in a clockwise sense. An arrow from vertex $i$ to vertex $j$ means that $p_i \geq p_j$. The diagram illustrates the phenomenon of radiation from the most likely order. The total orders that minimize $\left< d \right>$ when vertex $1$ has maximum probability (Figure \ref{['fig:optimal_probability_arrangement']} top) are specializations of the diagram.
  • Figure 4: Arrangements with four non-zero probability orders (vertices with non-zero probability are marked in red). Top. The non-zero probability orders are contiguous, namely they form a path. Bottom. The non-zero probability orders are not contiguous.
  • Figure 5: The average swap distance ($\left< d \right>$) as a function of the random baseline ($\left< d \right>_r$) in crosslinguistic gestures. The dashed line is a control line to indicate identity, i.e. $\left< d \right> = \left< d \right>_r$. Points below the control line satisfy $\left< d \right> < \left< d \right>_r$.
  • ...and 6 more figures

Theorems & Definitions (27)

  • proof
  • Theorem B.1
  • proof
  • proof
  • proof
  • proof
  • proof
  • Corollary C.1
  • proof
  • proof
  • ...and 17 more