Cobblestone: Iterative Automation for Formal Verification

Formal verification using proof assistants, such as Coq, is an effective way of improving software quality, but it is expensive. Writing proofs manually requires both significant effort and expertise. Recent research has used machine learning to automatically synthesize proofs, reducing verification effort, but these tools are able to prove only a fraction of the desired software properties. We introduce Cobblestone, a new proof-synthesis approach that improves on the state of the art by taking advantage of partial progress in proof synthesis attempts. Unlike prior tools, Cobblestone can produce multiple unsuccessful proofs using a large language model (LLM), identify the working portions of those proofs, and combine them into a single, successful proof, taking advantage of internal partial progress. We evaluate Cobblestone on two benchmarks of open-source Coq projects, controlling for training data leakage in LLM datasets. Fully automatically, Cobblestone can prove 48% of the theorems, while Proverbot9001, the previous state-of-the-art, learning-based, proof-synthesis tool, can prove 17%. Cobblestone establishes a new state of the art for fully automated proof synthesis tools for Coq. We also evaluate Cobblestone in a setting where it is given external partial proof progress from oracles, serving as proxies for a human proof engineer or another tool. When the theorem is broken down into a set of subgoals and Cobblestone is given a set of relevant lemmas already proven in the project, it can prove up to 58% of the theorems. We qualitatively study the theorems Cobblestone is and is not able to prove to outline potential future research directions to further improve proof synthesis, including developing interactive, semi-automated tools. Our research shows that tools can make better use of partial progress made during proof synthesis to more effectively automate formal verification.

Learn from Failure: Fine-Tuning LLMs with Trial-and-Error Data for Intuitionistic Propositional Logic Proving

Recent advances in Automated Theorem Proving have shown the effectiveness of leveraging a (large) language model that generates tactics (i.e. proof steps) to search through proof states. The current model, while trained solely on successful proof paths, faces a discrepancy at the inference stage, as it must sample and try various tactics at each proof state until finding success, unlike its training which does not incorporate learning from failed attempts. Intuitively, a tactic that leads to a failed search path would indicate that similar tactics should receive less attention during the following trials. In this paper, we demonstrate the benefit of training models that additionally learn from failed search paths. Facing the lack of such trial-and-error data in existing open-source theorem-proving datasets, we curate a dataset on intuitionistic propositional logic theorems and formalize it in Lean, such that we can reliably check the correctness of proofs. We compare our model trained on relatively short trial-and-error information (TrialMaster) with models trained only on the correct paths and discover that the former solves more unseen theorems with lower trial searches.