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Evaluating the relationship between regularity and learnability in recursive numeral systems using Reinforcement Learning

Andrea Silvi, Ponrawee Prasertsom, Jennifer Culbertson, Devdatt Dubhashi, Moa Johansson, Kenny Smith

Abstract

Human recursive numeral systems (i.e., counting systems such as English base-10 numerals), like many other grammatical systems, are highly regular. Following prior work that relates cross-linguistic tendencies to biases in learning, we ask whether regular systems are common because regularity facilitates learning. Adopting methods from the Reinforcement Learning literature, we confirm that highly regular human(-like) systems are easier to learn than unattested but possible irregular systems. This asymmetry emerges under the natural assumption that recursive numeral systems are designed for generalisation from limited data to represent all integers exactly. We also find that the influence of regularity on learnability is absent for unnatural, highly irregular systems, whose learnability is influenced instead by signal length, suggesting that different pressures may influence learnability differently in different parts of the space of possible numeral systems. Our results contribute to the body of work linking learnability to cross-linguistic prevalence.

Evaluating the relationship between regularity and learnability in recursive numeral systems using Reinforcement Learning

Abstract

Human recursive numeral systems (i.e., counting systems such as English base-10 numerals), like many other grammatical systems, are highly regular. Following prior work that relates cross-linguistic tendencies to biases in learning, we ask whether regular systems are common because regularity facilitates learning. Adopting methods from the Reinforcement Learning literature, we confirm that highly regular human(-like) systems are easier to learn than unattested but possible irregular systems. This asymmetry emerges under the natural assumption that recursive numeral systems are designed for generalisation from limited data to represent all integers exactly. We also find that the influence of regularity on learnability is absent for unnatural, highly irregular systems, whose learnability is influenced instead by signal length, suggesting that different pressures may influence learnability differently in different parts of the space of possible numeral systems. Our results contribute to the body of work linking learnability to cross-linguistic prevalence.
Paper Structure (16 sections, 2 equations, 5 figures, 1 table)

This paper contains 16 sections, 2 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Efficiency of recursive numeral systems
  3. Simplicity, learnability and cross-linguistic prevalence
  4. Measuring learnability via reinforcement learning
  5. Experiment 1: Learnability of recursive numerals with varying regularity
  6. Methods
  7. Experiment setup
  8. Results and discussion
  9. Experiment 2: Learnability as accuracy of generalisation
  10. Experiment setup
  11. Results and discussion
  12. Experiment 3: Learnability among highly regular, human-like systems
  13. Experiment setup
  14. Results and discussion
  15. General discussion
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Inferred minimal partial DFA representing the Mandarin recursive numeral grammar. Transitions (arrows) represent output morphemes. States (circles) are labelled roughly according to the relevant grammar rules they capture. Double borders indicate accepting states. Generating the numeral for 20 ($2*10$) would involve going from the initial state ($\lambda{}$) to $D$ (emitting $2$), from $D$ to $D*$ (emitting $*$), and from $D*$ to $D*10$ (emitting $10$) and terminate the machine there.
  • Figure 2: Irregularity and learnability for human (blue), D&S optimal (red), Y&R optimal (green), and random baseline (orange) recursive numeral systems under power law (left panel) and the uniform (right panel) test distributions $q(n)$. The training distribution $p(n)$ is always the power law distribution.
  • Figure 3: Irregularity (left) and average local irregularity (right) against learnability for human (blue), D&S optimal (red) and Y&R optimal (green) systems, under the power law training distribution and a uniform test distribution.
  • Figure 4: Average morphosyntactic complexity and learnability for random baseline (left) and highly regular systems (right), under the power law training distribution and a uniform test distribution.
  • Figure 5: Left: best-fit regression lines between irregularity and learnability for the higher-populated local neighbourhoods (one colour per neighbourhood) from prasertsom2026recursivenumeralsystemshighly. Right: Irregularity and learnability of example local neighbourhoods (Arawak-like, Diola-like, Nahuatl-like and Yakut-like systems).