Faithful Autoformalization of Natural Language Assertions

When an LLM formalizes natural language, how do we know the output is faithful? We propose a roundtrip verification approach which does not require ground-truth annotations: formalize a statement, translate the result back to natural language, re-formalize, and use a formal tool to check logical equivalence. When the two formalizations agree, this provides evidence of a faithful formalization. When they disagree, a stage-level diagnosis localizes the error to a specific translation step, and a scoped repair operator attempts to correct that step. We evaluate the framework on two statutory domains (the Texas Transportation Code and the Texas Parks and Wildlife Code) using two LLMs (Claude Opus~4.6 and GPT-5.2) with three repair baselines. Diagnosis-guided scoped repair is the most effective method, with effectiveness contingent on the reliability of the diagnosis function. Across both domains and both models, under our full repair system, rules that fail the equivalence check show 1.4x-2.5x more NLI drift than rules that pass it.

AI coding agents are increasingly used to write real-world software, but ensuring that their outputs are correct remains a fundamental challenge. Formal verification offers a promising path: an agent generates code together with a machine-checked proof, guaranteeing that the code satisfies a formal specification. However, there is no guarantee that the formal spec itself matches the user's intent. In this work, we study specification autoformalization: whether LLM agents can translate informal programming problems into faithful formal specifications. We introduce Verus-SpecBench, a benchmark of 581 spec-writing tasks derived from Codeforces problems targeting Verus, a verifier for Rust, and Verus-SpecGym, an agentic environment in which models interact with Verus, bash, & the filesystem to develop these specs. The central challenge is evaluation: expert-written reference specs are expensive to write, & LLM judges can miss subtle mistakes. We address this by (a) extending Verus's exec_spec mechanism so that generated specs can be executed as Rust code, & (b) testing them against official Codeforces tests & adversarial cases extracted from Codeforces "hacks", which are edge cases written by competitors to break incorrect solutions. On Verus-SpecBench, the strongest model, Gemini 3.1 Pro, solves 77.8% of tasks, other frontier models solve 51.1--57.8% & OSS models reach only 21.5--25.5%. Our analysis of failure modes shows that model-generated specs can omit important input assumptions, accept incorrect outputs, & reject valid ones. We also find that LLM-as-a-judge evaluation misses 26% of the failures our evaluator catches. Overall, our results suggest that spec autoformalization is within reach for frontier agents but remains brittle even on problems where they can already generate correct code. The code, data, & logs can be found at https://github.com/formal-verif-is-cool/verus-spec-gym

Natural language arguments are compelling before they are formally explicit. A premise supports a claim through defeasible warrants, background commitments, and exception conditions that the text leaves implicit. However, formal verification requires the opposite. Making such arguments machine-checkable requires constructing the missing commitments, not only translating given sentences into logic. Construction, however, carries a risk that translation does not: a system free to add premises can make any claim provable, and a formally valid proof may assert the claim outright, prove it without the original premise, or establish more than the claim itself. We address this problem by formulating autoformalization for argumentative material inference as guard completion, in which non-monotonic material support is turned into monotonic formal inference relative to an explicitly constructed guard set. A completion is accepted only when its proof both passes the theorem prover and survives contrastive tests of premise dependence and claim selectivity. We implement this formulation in GUARD, a neuro-symbolic framework in which LLMs construct and formalize candidate guards, Isabelle/HOL verifies the resulting theories and returns step-level feedback for iterative refinement, and the system abstains when no faithful completion can be reached. Our empirical results on Debatepedia and ARCT using different LLMs demonstrate that GUARD yields significant improvements in verified-faithful (+35.3, +32.9 points) and substantial reductions in leakage (-25.9, -21.9 points) over the state-of-the-art LLM-driven theorem proving approach. Moreover, we show that the symbolic soft critique and the explicit assumption layer account for most of these gains, with the soft critique also improving the initial validity of the elicited context and reducing the number of iterations required for successful verification.