Large language models (LLMs) operate in two fundamental learning modes – fine-tuning (FT) and in-context learning (ICL) – raising key questions about which mode yields greater language proficiency and whether they differ in their inductive biases. Prior studies comparing FT and ICL have yielded mixed and inconclusive results due to inconsistent experimental setups. To enable a rigorous comparison, we propose a formal language learning task – offering precise language boundaries, controlled string sampling, and no data contamination – and introduce a discriminative test for language proficiency, where an LLM succeeds if it assigns higher generation probability to in-language strings than to out-of-language strings.Empirically, we find that: (a) FT has greater language proficiency than ICL on in-distribution generalization, but both perform equally well on out-of-distribution generalization. (b) Their inductive biases, measured by the correlation in string generation probabilities, are similar when both modes partially learn the language but diverge at higher proficiency levels. (c) Unlike FT, ICL performance differs substantially across models of varying sizes and families and is sensitive to the token vocabulary of the language. Thus, our work demonstrates the promise of formal languages as a controlled testbed for evaluating LLMs, behaviors that are difficult to isolate in natural language datasets. Our source code is available at https://github.com/bishwamittra/formallm.

Do LLMs process text and mathematics as a unified skill, or do these components rely on distinct underlying mechanisms? We investigate this question by disentangling the textual interpretation and mathematical solving steps in word problems drawn from Brazil’s largest college entrance exam (ENEM) and GSM8K, a popular grade school-level benchmark. Using the symbolic solver SymPy, we transform word problems into equivalent purely mathematical representations, isolating equation formulation from textual comprehension. Our extended benchmarks enable a structured analysis of LLM performance across these two dimensions. Through empirical evaluations, we find that small-scale LLMs struggle significantly more with text interpretation than with equation solving, with accuracy dropping by a factor of 2 to 7 when solving full word problems compared to their math-only counterparts. Exploratory factor analysis confirms a bidimensional structure in LLM reasoning, where models exhibit distinct proficiencies in textual and mathematical components, underscoring the need for targeted improvements in language comprehension. By analyzing the latent factors associated with each model, our findings provide a framework for researchers and practitioners to make informed choices when selecting models based on computational costs and the nature of their tasks.