Improving Autoformalization using Type Checking

Large language models show promise for autoformalization, the task of automatically translating natural language into formal languages. However, current autoformalization methods remain limited. The last reported state-of-the-art performance on the ProofNet formalization benchmark for the Lean proof assistant, achieved using Codex for Lean 3, only showed successful formalization of 16.1% of informal statements. Similarly, our evaluation of GPT-4o for Lean 4 only produces successful translations 34.9% of the time. Our analysis shows that the performance of these models is largely limited by their inability to generate formal statements that successfully type-check (i.e., are syntactically correct and consistent with types) - with a whopping 86.6% of GPT-4o errors starting from a type-check failure. In this work, we propose a method to fix this issue through decoding with type-check filtering, where we initially sample a diverse set of candidate formalizations for an informal statement, then use the Lean proof assistant to filter out candidates that do not type-check. Using GPT-4o as a base model, and combining our method with self-consistency, we obtain a +18.3% absolute increase in formalization accuracy, and achieve a new state-of-the-art of 53.2% on ProofNet with Lean 4.

Neural-Network Guided Expression Transformation

Optimizing compilers, as well as other translator systems, often work by rewriting expressions according to equivalence preserving rules. Given an input expression and its optimized form, finding the sequence of rules that were applied is a non-trivial task. Most of the time, the tools provide no proof, of any kind, of the equivalence between the original expression and its optimized form. In this work, we propose to reconstruct proofs of equivalence of simple mathematical expressions, after the fact, by finding paths of equivalence preserving transformations between expressions. We propose to find those sequences of transformations using a search algorithm, guided by a neural network heuristic. Using a Tree-LSTM recursive neural network, we learn a distributed representation of expressions where the Manhattan distance between vectors approximately corresponds to the rewrite distance between expressions. We then show how the neural network can be efficiently used to search for transformation paths, leading to substantial gain in speed compared to an uninformed exhaustive search. In one of our experiments, our neural-network guided search algorithm is able to solve more instances with a 2 seconds timeout per instance than breadth-first search does with a 5 minutes timeout per instance.