Density Measures for Language Generation

The recent successes of large language models (LLMs) have led to a surge of theoretical research into language generation. A recent line of work proposes an abstract view, called language generation in the limit, where generation is seen as a game between an adversary and an algorithm: the adversary generates strings from an unknown language $K$, chosen from a countable collection of candidate languages, and after seeing a finite set of these strings, the algorithm must generate new strings from $K$ that it has not seen before. This formalism highlights a key tension: the trade-off between validity (the algorithm should only produce strings from the language) and breadth (it should be able to produce many strings from the language). This trade-off is central in applied language generation as well, where it appears as a balance between hallucination (generating invalid utterances) and mode collapse (generating only a restricted set of outputs). Despite its importance, this trade-off has been challenging to study quantitatively. We develop ways to quantify this trade-off by formalizing breadth using measures of density. Existing algorithms for language generation in the limit produce output sets that can have zero density in the true language, and this important failure of breadth might seem unavoidable. We show, however, that such a failure is not necessary: we provide an algorithm for language generation in the limit whose outputs have strictly positive density in $K$. We also study the internal representations built by these algorithms, specifically the sequence of hypothesized candidate languages they consider, and show that achieving the strongest form of breadth may require oscillating indefinitely between high- and low-density representations. Our analysis introduces a novel topology on language families, with notions of convergence and limit points playing a key role.