💬 NLP

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

Using reinforcement learning, this study demonstrates that the high regularity of human recursive number systems facilitates their learnability from limited data, suggesting that this structural feature is a primary driver of their cross-linguistic spread.

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

Published 2026-04-30

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot how to count. You have two options for the "language" of numbers you give it:

The "regular" language: Like English or Mandarin. Once the robot learns that "ten" plus "one" is "eleven" and "ten" plus "two" is "twelve," it can easily guess the rest. It is like a cookbook where the instructions are consistent: "Add one cup of flour, then add one egg."
The "irregular" language: A made-up system where the rules change randomly. Maybe "ten plus one" is "eleven," but "ten plus two" is "banana" and "ten plus three" is "purple." There is no pattern to reuse; every number is a brand-new, confusing secret key.

This work poses a simple question: Is the reason humans use regular number systems (like English) that they are easier to learn?

To find out, the researchers did not ask humans; they built a digital "student" (a reinforcement-learning agent) and gave it thousands of different number systems to learn. They measured how quickly the robot could master the system.

Here is what they discovered, broken down into simple concepts:

1. The "Test" is Most Important

The researchers found that how they tested the robot changed the results.

Scenario A (The "Reality Test"): They trained the robot mainly with small numbers (1–10), since humans use these most frequently. Then they tested it on all numbers up to 99, including the rare, large ones.
- The Result: The robot learned the regular systems much faster. Because the rules were consistent, the robot could "generalize" (apply the logic) what it learned about small numbers to the large, rare numbers. The irregular systems were a nightmare; the robot had to memorize every single number from scratch.
- The Analogy: Think of regular systems like a Lego set. Once you know how to connect two bricks, you can build a tower, a car, or a castle. Irregular systems are like a box of random, unique puzzle pieces, where each piece fits only in one specific spot. If you want to build a large structure, the Lego set is much easier to learn.
Scenario B (The "Frequency Test"): They trained and tested the robot under the same skewed distribution (mainly small numbers, almost never large ones).
- The Result: In this specific case, it did not matter to the robot whether the system was regular or not. Since it was rarely tested on large numbers, it did not need to generalize. It simply memorized the small numbers it saw often.
- The Insight: Regularity helps you learn only if you later need to be able to handle new or rare situations.

2. The "Too Strange" Zone

The researchers also examined systems that were extremely chaotic and irregular.

The Result: With these "super strange" systems, regularity played almost no role. Instead, the main problem was length. If the words for numbers were too long and complicated to pronounce, the robot had difficulty learning them, regardless of whether there was a pattern or not.
The Analogy: If you try to memorize a list of random words, it does not matter whether they rhyme (regular) or not. If the words are simply incredibly long and hard to pronounce, you will forget them anyway. But if the words are short, a rhyming scheme (regularity) makes them much easier to remember.

3. The "Local" Pattern

The researchers found that a system sometimes looks chaotic overall but has small pockets of order.

The Result: The robot learned best when the rules were locally consistent. For example, if the rules for numbers 30–39 were consistent, even if the rules for 40–49 were different, the robot could still learn it well.
The Analogy: Imagine a city where every street in the "North District" follows a grid, but the "South District" is a labyrinth. You can still navigate the city easily if you simply stay in the North District. You do not need the entire city to be a grid to get around; you only need some local consistency.

The Big Conclusion

The work concludes that human number systems are likely not regular just because they are efficient to speak, but because regularity makes them easier to learn and pass on to the next generation.

If a language's number system is too chaotic, it is difficult for a child (or a robot) to figure out the rules for large numbers without being explicitly taught every single one. Regularity acts like a safety net, allowing learners to take a few examples and guess the rest correctly. This "learning pressure" is probably why human cultures worldwide naturally tend to use regular, pattern-based number systems.

1. Problem Statement

The paper investigates the evolutionary and cognitive pressures that shape recursive numeral systems (e.g., English base-10, Mandarin). While human numeral systems are highly regular, theoretically possible systems can be highly irregular. Previous research (e.g., Prasertsom et al., 2026) suggests that natural systems optimize a trade-off between lexicon size and morphosyntactic complexity but fail to account for regularity (the reuse of forms).

The central research question is: Does regularity facilitate learnability?
The authors hypothesize that regular systems are widespread because they are easier to learn, particularly when the learning task requires generalization from limited training data to correctly represent a broad range of integers. They intend to test this using Reinforcement Learning (RL) agents that simulate the learning process.

2. Methodology

The study uses Single-Agent Reinforcement Learning (RL) to quantify the learnability of various numeral systems.

Task Formulation: The task is formulated as a contextual bandit problem with a continuous action space.
- Input: A numeral string (sequence of morphemes).
- Output: A predicted integer value (1–99).
- Agent Architecture: A single-layer LSTM network. Morphemes are embedded (size 5), processed sequentially, and the final hidden state predicts a scalar $\mu \in [0, 1]$ . This is interpreted as the mean of a Gaussian policy $N(\mu, \sigma)$ .
- Reward Function: A bounded, exponentially decaying Gaussian function based on the distance between the predicted and target integer values. Predictions outside 1–99 receive a reward of zero.
- Training: Agents are trained over 30,000 epochs using the REINFORCE algorithm.
- Metric: Learnability is measured by the Area Under the Curve (AUC) of test accuracy versus training epochs. A higher AUC indicates faster learning.
System Datasets: The study evaluates four categories of systems:
1. Human Systems: 128 attested recursive numeral systems.
2. D&S Optimal: Artificially generated systems optimized for the trade-off between lexicon size and complexity (Denić & Szymanik, 2024).
3. Y&R Optimal: Artificial systems optimized for the same trade-off but constrained to human-like similarity (Yang & Regier, 2025).
4. Random Baseline: 300 unattested systems using human symbols.
Regularity Measurement: Irregularity is quantified by the size of the minimal Deterministic Finite Automaton (DFA) required to generate the system's grammar. The metric calculates the number of bits needed to encode the DFA's transition table, initial state, and accepting states.
Experimental Conditions:
- Experiment 1: Both the training ( $p(n)$ ) and test distributions ( $q(n)$ ) follow a power law ( $n^{-2}$ ), reflecting communicative frequency (strong skew toward small numbers).
- Experiment 2: Training follows the power law, but testing uses a uniform distribution ( $q(n) \propto 1/99$ ) to test generalization to all integers.
- Experiment 3: Focuses on "local neighborhoods" of human systems (only the arrangement of existing morphemes is varied) to isolate the effect of regularity while keeping lexicon size and average length constant.

3. Key Results

Experiment 1 (Power Law Training and Test):
- Result: No significant correlation between regularity and learnability ( $\beta = -0.020$ ).
- Interpretation: When the test distribution matches the skewed training distribution (focus on small numbers), agents can achieve high accuracy without needing to generalize to large numbers. Therefore, the structural advantages of regularity are not utilized.
Experiment 2 (Power Law Training, Uniform Test):
- Result: A strong negative correlation between irregularity and learnability ( $\beta = -0.828$ ). Highly regular, human-like systems are significantly easier to learn.
- Interpretation: When agents must generalize to large, rare numbers (the "long tail"), regular systems allow agents to efficiently reuse learned sub-parts (morphemes/rules). Irregular systems do not offer these reusable patterns.
- Local Regularity: Introducing a metric for "local regularity" (regularity within consecutive number ranges, e.g., 30s, 40s) further improved the correlation and explained outliers like French, which has inconsistent global bases but consistent local ranges.
Experiment 3 (Local Neighborhoods):
- Result: Within specific system families with fixed lexicon size and length, 21 out of 30 neighborhoods showed a negative correlation between irregularity and learnability.
- Nuance: In highly irregular systems (Random Baselines), regularity had little impact on learnability. Instead, morphosyntactic complexity (length) became the dominant factor; longer forms were harder to memorize regardless of regularity.

4. Key Contributions

RL as a Proxy for Learnability: It was demonstrated that Reinforcement Learning agents can effectively model the learnability of linguistic systems, particularly the trade-off between regularity and generalization.
The Role of Generalization: It was found that the relationship between regularity and learnability is conditional. Regularity offers an advantage for learnability only when the learning task requires generalization beyond the immediate training distribution (i.e., from frequent small numbers to rare large numbers).
Refinement of Complexity Metrics: It was shown that "irregularity" is not a monolithic predictor of learnability. In highly irregular systems, length/complexity dominates, whereas in regular systems, structural regularity dominates.
Local vs. Global Regularity: The concept of local regularity (consistency within consecutive ranges) was introduced as a critical factor for learnability, explaining why systems with mixed bases (like French) can still be learnable.

5. Significance and Implications

Language Evolution: The results support the hypothesis that the cross-linguistic prevalence of regular numeral systems is driven by learnability pressures, specifically the need to efficiently transmit knowledge across generations to represent all integers, not just the most frequently used ones.
Cognitive Modeling: The study suggests that human learning of number words likely relies on similar generalization mechanisms. The discrepancy with earlier human experiments (Holt & Barner, 2025) is attributed to a lack of pressure to generalize to large numbers in those specific experimental designs.
Future Directions: The authors suggest that counting routines (sequential exposure) artificially increase the frequency of larger numbers in human learning, thereby facilitating the extraction of general rules. They further propose that learnability is a multi-dimensional space where different complexity factors (length vs. regularity) dominate in different regions of the hypothesis space.

In summary, the paper provides computational evidence that regularity is an adaptive feature of numeral systems, selected because it enables robust generalization from limited data to a universal set of integers.