Across languages, numeral systems vary widely in how they construct and combine numbers. While humans consistently learn to navigate this diversity, large language models (LLMs) struggle with linguistic-mathematical puzzles involving cross-linguistic numeral systems, which humans can learn to solve successfully. We investigate why this task is difficult for LLMs through a series of experiments that untangle the linguistic and mathematical aspects of numbers in language. Our experiments establish that models cannot consistently solve such problems unless the mathematical operations in the problems are explicitly marked using known symbols ($+$, $\times$, etc., as in "twenty + three"). In further ablation studies, we probe how individual parameters of numeral construction and combination affect performance. While humans use their linguistic understanding of numbers to make inferences about the implicit compositional structure of numerals, LLMs seem to lack this notion of implicit numeral structure. We conclude that the ability to flexibly infer compositional rules from implicit patterns in human-scale data remains an open challenge for current reasoning models.

The numeral systems of Indo-Aryan languages such as Hindi, Gujarati, and Bengali are highly unusual in that unlike most numeral systems (e.g., those of English, Chinese, etc.), forms referring to 1--99 are highly non-transparent and are cannot be constructed using straightforward rules. As an example, Hindi/Urdu *ikyānve* `91' is not decomposable into the composite elements *ek* `one' and *nave* `ninety' in the way that its English counterpart is. This paper situates Indo-Aryan languages within the typology of cross-linguistic numeral systems, and explores the linguistic and non-linguistic factors that may be responsible for the persistence of complex systems in these languages. Using cross-linguistic data from multiple databases, we develop and employ a number of cross-linguistically applicable metrics to quantifies the complexity of languages' numeral systems, and demonstrate that Indo-Aryan languages have decisively more complex numeral systems than the world's languages as a whole, though individual Indo-Aryan languages differ from each other in terms of the complexity of the patterns they display. We investigate the factors (e.g., religion, geographic isolation, etc.) that underlie complexity in numeral systems, with a focus on South Asia, in an attempt to develop an account of why complex numeral systems developed and persisted in certain Indo-Aryan languages but not elsewhere. Finally, we demonstrate that Indo-Aryan numeral systems adhere to certain general pressures toward efficient communication found cross-linguistically, despite their high complexity. We call for this somewhat overlooked dimension of complexity to be taken seriously when discussing general variation in cross-linguistic numeral systems.

Numeral systems across the world's languages vary in fascinating ways, both regarding their synchronic structure and the diachronic processes that determined how they evolved in their current shape. For a proper comparison of numeral systems across different languages, however, it is important to code them in a standardized form that allows for the comparison of basic properties. Here, we present a simple but effective coding scheme for numeral annotation, along with a workflow that helps to code numeral systems in a computer-assisted manner, providing sample data for numerals from 1 to 40 in 25 typologically diverse languages. We perform a thorough analysis of the sample, focusing on the systematic comparison between the underlying and the surface morphological structure. We further experiment with automated models for morpheme segmentation, where we find allomorphy as the major reason for segmentation errors. Finally, we show that subword tokenization algorithms are not viable for discovering morphemes in low-resource scenarios.

Though Large Language Models (LLMs) have shown remarkable abilities in mathematics reasoning, they are still struggling with performing numeric operations accurately, such as addition and multiplication. Numbers can be tokenized into tokens in various ways by different LLMs and affect the numeric operations performance. Currently, there are two representatives: 1) Tokenize into $1$-digit, and 2) Tokenize into $1\sim 3$ digit. The difference is roughly equivalent to using different numeral systems (namely base $10$ or base $10^{3}$). In light of this, we study the scaling behavior of different numeral systems in the context of transformer-based large language models. We empirically show that a base $10$ system is consistently more data-efficient than a base $10^{2}$ or $10^{3}$ system across training data scale, model sizes under from-scratch training settings, while different number systems have very similar fine-tuning performances. We attribute this to higher token frequencies of a base $10$ system. Additionally, we reveal extrapolation behavior patterns on addition and multiplication. We identify that base $100$ and base $1000$ systems struggle on token-level discernment and token-level operations. We also sheds light on the mechanism learnt by the models.

The emergence of mathematical concepts, such as number systems, is an understudied area in AI for mathematics and reasoning. It has previously been shown Carlsson et al. (2021) that by using reinforcement learning (RL), agents can derive simple approximate and exact-restricted numeral systems. However, it is a major challenge to show how more complex recursive numeral systems, similar to the one utilised in English, could arise via a simple learning mechanism such as RL. Here, we introduce an approach towards deriving a mechanistic explanation of the emergence of recursive number systems where we consider an RL agent which directly optimizes a lexicon under a given meta-grammar. Utilising a slightly modified version of the seminal meta-grammar of Hurford (1975), we demonstrate that our RL agent can effectively modify the lexicon towards Pareto-optimal configurations which are comparable to those observed within human numeral systems.

Recent work (Xu et al., 2020) has suggested that numeral systems in different languages are shaped by a functional need for efficient communication in an information-theoretic sense. Here we take a learning-theoretic approach and show how efficient communication emerges via reinforcement learning. In our framework, two artificial agents play a Lewis signaling game where the goal is to convey a numeral concept. The agents gradually learn to communicate using reinforcement learning and the resulting numeral systems are shown to be efficient in the information-theoretic framework of Regier et al. (2015); Gibson et al. (2017). They are also shown to be similar to human numeral systems of same type. Our results thus provide a mechanistic explanation via reinforcement learning of the recent results in Xu et al. (2020) and can potentially be generalized to other semantic domains.

The sizes of deep neural networks (DNNs) are rapidly outgrowing the capacity of hardware to store and train them. Research over the past few decades has explored the prospect of sparsifying DNNs before, during, and after training by pruning edges from the underlying topology. The resulting neural network is known as a sparse neural network. More recent work has demonstrated the remarkable result that certain sparse DNNs can train to the same precision as dense DNNs at lower runtime and storage cost. An intriguing class of these sparse DNNs is the X-Nets, which are initialized and trained upon a sparse topology with neither reference to a parent dense DNN nor subsequent pruning. We present an algorithm that deterministically generates RadiX-Nets: sparse DNN topologies that, as a whole, are much more diverse than X-Net topologies, while preserving X-Nets' desired characteristics. We further present a functional-analytic conjecture based on the longstanding observation that sparse neural network topologies can attain the same expressive power as dense counterparts