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.
Figures & tables
Number
Numeral (Mandarin)
Numeral (unnatural)
18
10+8
2∗9
19
10+9
10+9
20
2∗10
2∗10
21
2∗10+1
2∗9+3
22
2∗10+2
2∗10+2
23
2∗10+3
2∗9+5
Table 1: Numerals in Mandarin (first column) and in a hypothetical system (second column) that is matched for lexicon size and average morphosyntactic complexity for the number range 18−25 . Notice that Mandarin is consistently base-10, whereas the unnatural system has alternating bases (9, 10).
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 ( λ ) 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 Prasertsom et al. (2026) . Right: Irregularity and learnability of example local neighbourhoods (Arawak-like, Diola-like, Nahuatl-like and Yakut-like systems).
Many learning problems combine a dominant regularity with systematic exceptions. Motivated by U-shaped learning in language acquisition, we study this interaction in controlled quasiregular regression problems where regular and exceptional solutions are explicitly known. Neural networks can partially acquire exceptions, subsequently regress toward the dominant regularity, and finally recover. This overregularization becomes substantially stronger when exceptions are rare, despite their early acquisition, but does not emerge equally across all regularities considered. Our results isolate a simple form of competition between regularities and exceptions during neural learning.
Matthia Sabatelli
Department of Artificial Intelligence, University of Groningen Groningen, The Netherlands
Recent research argues that exact recursive numeral systems optimize communicative efficiency by balancing a tradeoff between the size of the numeral lexicon and the average morphosyntactic complexity (roughly length in morphemes) of numeral terms. We argue that previous studies have not characterized the data in a fashion that accounts for the degree of complexity languages display. Using data from 52 genetically diverse languages and an annotation scheme distinguishing between predictable and unpredictable allomorphy (formal variation), we show that many of the world's languages are decisively less efficient than one would expect. We discuss the implications of our findings for the study of numeral systems and linguistic evolution more generally.
Chundra Cathcart, Arne Rubehn, Katja Bocklage +7
Institute for the Interdisciplinary Study of Language Evolution University of Zurich, Zurich, Switzerland · Chair for Multilingual Computational Linguistics University of Passau, Passau, Germany
Language models, as multi-task learners, acquire a wide range of abilities during training. A fundamental question is how much task-specific data is needed to learn a given task. Answering this for natural language is difficult: tasks are hard to delineate and can confound one another. To rigorously investigate the relationship between data frequency and learnability, we turn to a controlled setting using formal languages induced from probabilistic finite automata. These serve as a methodological testbed to demonstrate that standard correlational evaluation practices are inherently flawed. To enable causal analysis, we introduce the binning semiring, an algebraic object that lets us control how often a targeted property occurs in a sampled corpus. We formulate the experimental pipeline as a causal graphical model and derive decomposed Kullback-Leibler divergence metrics to measure the learnability of specific sub-tasks. Our experiments show that evaluating learnability without causal intervention leads to incorrect conclusions due to confounders in correlational analysis, and serve as a warning about correlational pitfalls in natural-language settings.
Vésteinn Snæbjarnarson, Anej Svete, Josef Valvoda +3
1ETH Zürich · University of Copenhagen · *Now at Google