Autoformalization has emerged as a term referring to the automation of formalization in the context of the formalization of mathematics using interactive theorem provers (proof assistants). Its rapid development has been driven by progress in deep learning, especially large language models (LLMs). More recently, usage of the term has expanded beyond mathematics to describe tasks that involve translating natural language input into verifiable logical representations. At the same time, a growing body of research explores using LLMs to translate informal language into formal representations for reasoning, planning, and knowledge representation, but without explicitly referring to this process as autoformalization. As a result, despite addressing similar tasks, the largely independent development of these research areas has limited opportunities for shared methodologies, benchmarks, and theoretical frameworks that could accelerate progress. Our goal is to review - explicit or implicit - instances of what can be considered autoformalization and to propose a unified framework, encouraging cross-pollination between different fields to advance the development of next generation AI systems.

We describe our formalisation in the interactive theorem prover Isabelle/HOL of the Balog–Szemerédi–Gowers Theorem, a profound result in additive combinatorics which played a central role in Gowers’s proof deriving the first effective bounds for Szemerédi’s Theorem. The proof is of great mathematical interest given that it involves an interplay between different mathematical areas, namely applications of graph theory and probability theory to additive combinatorics involving algebraic objects. This interplay is what made the process of the formalisation, for which we had to develop formalisations of new background material in the aforementioned areas, more rich and technically challenging. We demonstrate how locales, Isabelle’s module system, can be employed to handle such interplays in mathematical formalisations. To treat the graph-theoretic aspects of the proof, we make use of a new, more general undirected graph theory library developed by Edmonds, which is both flexible and extensible. In addition to the main theorem, which, following our source, is formulated for difference sets, we also give an alternative version for sumsets which required a formalisation of an auxiliary triangle inequality. We moreover formalise a few additional results in additive combinatorics that are not used in the proof of the main theorem. This is the first formalisation of the Balog–Szemerédi–Gowers Theorem in any proof assistant to our knowledge.

Abstract We have formalised Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.