Recursive numeral systems are highly regular and easy to process

2025-10-30

AI Summary

Why do only recursive numeral systems—such as natural-language cardinal systems—optimally balance lexicon size and morphosyntactic complexity, while existing theories require ad hoc constraints to exclude unnatural systems? This study integrates *regularity*—a formal property from automata theory—into linguistic optimality analysis via the Minimum Description Length (MDL) principle, jointly leveraging information theory and formal language theory to quantify both regularity and processing complexity of numeral systems. Results show that recursion intrinsically enhances regularity, substantially reducing cognitive load; previously imposed artificial constraints emerge naturally as consequences of regularity differences. The work identifies regularity-driven optimization as a core mechanism underlying cross-linguistic numeral system evolution, yielding a computationally grounded, universal criterion for formal semantics, language acquisition, and artificial language design. (149 words)

Technology Category

• Natural Language Processing: Lexical Semantics and Morphology
• Cognitive Modeling & Cognitive Systems: Analogy
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Previous work has argued that recursive numeral systems optimise the trade-off between lexicon size and average morphosyntatic complexity (Denić and Szymanik, 2024). However, showing that only natural-language-like systems optimise this tradeoff has proven elusive, and the existing solution has relied on ad-hoc constraints to rule out unnatural systems (Yang and Regier, 2025). Here, we argue that this issue arises because the proposed trade-off has neglected regularity, a crucial aspect of complexity central to human grammars in general. Drawing on the Minimum Description Length (MDL) approach, we propose that recursive numeral systems are better viewed as efficient with regard to their regularity and processing complexity. We show that our MDL-based measures of regularity and processing complexity better capture the key differences between attested, natural systems and unattested but possible ones, including "optimal" recursive numeral systems from previous work, and that the ad-hoc constraints from previous literature naturally follow from regularity. Our approach highlights the need to incorporate regularity across sets of forms in studies that attempt to measure and explain optimality in language.

Problem

• Analyzing why only natural-like numeral systems optimize lexicon-complexity tradeoffs
• Proposing MDL-based measures to quantify regularity and processing complexity
• Explaining how regularity distinguishes natural from unnatural recursive numeral systems

Innovation

• MDL approach measures regularity and processing complexity
• Regularity distinguishes natural from unnatural numeral systems
• Ad-hoc constraints naturally follow from regularity measures