Prefix parsing asks whether an input prefix can be extended to a complete string generated by a given grammar. In the weighted setting, it also provides prefix probabilities, which are central to context-free language modeling, psycholinguistic analysis, and syntactically constrained generation from large language models. We introduce the prefix grammar transformation, an efficient reduction of prefix parsing to ordinary parsing. Given a grammar, our method constructs another grammar that generates exactly the prefixes of its original strings. Prefix parsing is then solved by applying any ordinary parsing algorithm on the transformed grammar without modification. The reduction is both elegant and practical: the transformed grammar is only a small factor larger than the input, and any optimized implementation can be used directly, eliminating the need for bespoke prefix-parsing algorithms. We also present a strategy—based on algorithmic differentiation—for computing the next-token weight vector, i.e., the prefix weights of all one-token extensions, enabling efficient prediction of the next token. Together, these contributions yield a simple, general, and efficient framework for prefix parsing.

Extracting finite state automata (FSAs) fromblack-box models offers a powerful approachto gaining interpretable insights into complexmodel behaviors. To support this pursuit, wepresent a weighted variant of Angluin’s (1987)L* algorithm for learning FSAs. We stay faithful to the original formulation, devising a wayto exactly learn deterministic weighted FSAswhose weights support division. Furthermore,we formulate the learning process in a mannerthat highlights the connection with FSA minimization, showing how L* directly learns aminimal automaton for the target language.

Language models (LMs) are currently at the forefront of NLP research due to their remarkable versatility across diverse tasks. However, a large gap exists between their observed capabilities and the explanations proposed by established formal machinery. To motivate a better theoretical characterization of LMs’ abilities and limitations, this tutorial aims to provide a comprehensive introduction to a specific framework for formal analysis of modern LMs using tools from formal language theory (FLT). We present how tools from FLT can be useful in understanding the inner workings and predicting the capabilities of modern neural LM architectures. We cover recent results using FLT to make precise and practically relevant statements about LMs based on recurrent neural networks and transformers by relating them to formal devices such as finite-state automata, Turing machines, and analog circuits. Altogether, the results covered in this tutorial allow us to make precise statements and explanations about the observed as well as predicted behaviors of LMs, as well as provide theoretically motivated suggestions on the aspects of the architectures that could be improved.

The Bar-Hillel construction is a classic result in formal language theory. It shows, by a simple construction, that the intersection of a context-free language and a regular language is itself context-free. In the construction, the regular language is specified by a finite-state automaton. However, neither the original construction (Bar-Hillel et al., 1961) nor its weighted extension (Nederhof and Satta, 2003) can handle finite-state automata with ε-arcs. While it is possible to remove ε-arcs from a finite-state automaton efficiently without modifying the language, such an operation modifies the automaton’s set of paths. We give a construction that generalizes the Bar- Hillel in the case the desired automaton has ε-arcs, and further prove that our generalized construction leads to a grammar that encodes the structure of both the input automaton and grammar while retaining the asymptotic size of the original construction.