What can large language models learn? By definition, language models (LM) are distributions over strings. Therefore, an intuitive way of addressing the above question is to formalize it as a matter of learnability of classes of distributions over strings. While prior work in this direction focused on assessing the theoretical limits, in contrast, we seek to understand the empirical learnability. Unlike prior empirical work, we evaluate neural LMs on their home turf-learning probabilistic languages-rather than as classifiers of formal languages. In particular, we investigate the learnability of regular LMs (RLMs) by RNN and Transformer LMs. We empirically test the learnability of RLMs as a function of various complexity parameters of the RLM and the hidden state size of the neural LM. We find that the RLM rank, which corresponds to the size of linear space spanned by the logits of its conditional distributions, and the expected length of sampled strings are strong and significant predictors of learnability for both RNNs and Transformers. Several other predictors also reach significance, but with differing patterns between RNNs and Transformers.

Studying language models (LMs) in terms of well-understood formalisms allows us to precisely characterize their abilities and limitations. Previous work has investigated the representational capacity of recurrent neural network (RNN) LMs in terms of their capacity to recognize unweighted formal languages. However, LMs do not describe unweighted formal languages -- rather, they define \emph{probability distributions} over strings. In this work, we study what classes of such probability distributions RNN LMs can represent, which allows us to make more direct statements about their capabilities. We show that simple RNNs are equivalent to a subclass of probabilistic finite-state automata, and can thus model a strict subset of probability distributions expressible by finite-state models. Furthermore, we study the space complexity of representing finite-state LMs with RNNs. We show that, to represent an arbitrary deterministic finite-state LM with $N$ states over an alphabet $\alphabet$, an RNN requires $\Omega\left(N |\Sigma|\right)$ neurons. These results present a first step towards characterizing the classes of distributions RNN LMs can represent and thus help us understand their capabilities and limitations.