arxiv: 2604.25031 · v2 · submitted 2026-04-27 · 💻 cs.CL · cs.AI

Recognition: no theorem link

Faithful Autoformalization via Roundtrip Verification and Repair

Daneshvar Amrollahi , Jerry Lopez , Clark Barrett

Authors on Pith no claims yet

Pith reviewed 2026-05-12 00:59 UTC · model grok-4.3

classification 💻 cs.CL cs.AI

keywords autoformalizationroundtrip verificationLLM faithfulnesslogical equivalenceerror diagnosisscoped repairstatutory textnatural language inference

0 comments

The pith

Roundtrip verification detects and repairs unfaithful LLM formalizations of natural language by checking logical equivalence after back-translation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that formalizing a natural language statement, translating the formal result back to natural language, re-formalizing it, and checking whether the two formal versions are logically equivalent can verify faithfulness without any ground-truth annotations. This matters because LLMs frequently produce formal outputs that subtly change the original meaning, and creating reliable annotated examples for verification is costly especially in specialized domains such as statutory law. When the equivalence check fails, a diagnosis step identifies the faulty translation stage and a scoped repair operator corrects only that part of the process. Experiments on two Texas legal codes using two different LLMs demonstrate that this diagnosis-guided repair outperforms simpler baselines and that passing rules exhibit substantially lower natural-language-inference drift than failing ones.

Core claim

The central claim is that roundtrip verification supplies evidence of a faithful formalization whenever the original and roundtripped formalizations are logically equivalent, and that when they disagree a stage-level diagnosis can localize the error to a specific translation step so that a scoped repair operator can correct it, yielding higher overall faithfulness than repair methods that lack diagnosis.

What carries the argument

The roundtrip verification loop of formalization, back-translation, re-formalization, logical-equivalence checking, followed by diagnosis and scoped repair when the check fails.

If this is right

Diagnosis-guided scoped repair is the most effective repair strategy across both statutory domains and both tested LLMs.
Repair effectiveness depends directly on the accuracy of the diagnosis function that localizes the error.
Rules that fail the equivalence check exhibit 1.4x to 2.5x more natural-language-inference drift than rules that pass it.
The approach can be applied directly to formalizing rules from the Texas Transportation Code and the Texas Parks and Wildlife Code.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same back-translation-plus-equivalence pattern could be tested on other formalization tasks such as turning English specifications into code or turning mathematical text into proofs.
If the diagnosis step can be made reliable, self-verification loops of this kind may reduce the amount of human annotation needed to train or fine-tune formalization models.
The method suggests a general strategy for catching meaning drift whenever an LLM moves between informal and formal representations without external labels.

Load-bearing premise

Logical equivalence between the original and roundtripped formalizations reliably indicates that the formalization has preserved the original natural-language meaning rather than merely being equivalent for unrelated reasons.

What would settle it

A single statute where two formalizations are logically equivalent according to the tool yet produce different conclusions when fed into a downstream reasoner or when compared against human interpretation of the original text would refute the verification method.

Figures

Figures reproduced from arXiv: 2604.25031 by Clark Barrett, Daneshvar Amrollahi, Jerry Lopez.

**Figure 1.** Figure 1: Roundtrip loop. An input x is formalized (T1), reconstructed back to natural language (T2), and re-formalized (T3). The diagonal compares y orig and y rt for semantic equivalence under Σ. count while reducing cost-per-repair by 41 % relative to random repair. Our contributions are:1 (i) We propose a formalism-agnostic roundtrip verification framework that provides a signal on the truthfulness of autoforma… view at source ↗

**Figure 2.** Figure 2: Unified framework: roundtrip autoformalization, verification, and repair. view at source ↗

**Figure 3.** Figure 3: NLI diagnostic decision tree (first match wins). view at source ↗

**Figure 4.** Figure 4: Cumulative UNSAT rules per iteration for view at source ↗

**Figure 6.** Figure 6: Per-iteration stage selection (Full condition) view at source ↗

**Figure 7.** Figure 7: NLI category distribution for Claude (left) and view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Roundtrip verification gives a label-free signal on faithfulness for legal autoformalization, but equivalence can still reflect consistent LLM errors rather than accuracy to the source text.

read the letter

The main point is a practical loop for spotting unfaithful formalizations of statutes: formalize, back-translate to natural language, re-formalize, then use a theorem prover to test logical equivalence. When the two formal versions match, treat it as evidence the first one stayed faithful. When they diverge, diagnose which step went wrong and apply a targeted repair. They run this on the Texas Transportation Code and Parks and Wildlife Code with Claude Opus and GPT-5.2, pitting their diagnosis-guided repair against three baselines. The full system produces rules that show 1.4x–2.5x less NLI drift on the passing cases, which is a usable quantitative result on real statutory text. That combination of back-translation, stage diagnosis, and scoped repair is the clearest new framing here; most earlier autoformalization papers either assume labeled data or rely on direct human review. The evaluation uses external verifiers and a separate NLI model, so it avoids obvious circularity in the metrics themselves. The work is aimed at anyone trying to scale reliable formal reasoning over contracts or regulations, and the experiments give concrete numbers on two domains and two models that readers can try to replicate or extend. The soft spot is the central proxy. Logical equivalence between the original and roundtrip formalizations can pass even when both contain the same misreading of a quantifier or scope, because the back-translation step may preserve the error instead of surfacing it. The reported NLI drift difference is consistent with the method but does not rule out shared biases between the formalizer and the NLI checker. The paper itself notes that repair success hinges on diagnosis quality, yet the abstract gives no independent check of that diagnosis against human faithfulness judgments. Those gaps make the evidence for the strongest claim moderate rather than strong. The paper deserves peer review because the idea is executable, the domains are substantive, and the gaps are addressable with added validation runs. I would bring it to a reading group for the method details and the repair operators, but I would not cite it yet without seeing the full experimental controls and human correlation numbers.

Referee Report

3 major / 2 minor

Summary. The paper proposes a roundtrip verification and repair framework for faithful autoformalization of natural language (NL) statements by LLMs without ground-truth annotations. The method formalizes an NL statement to a formal representation F1, back-translates F1 to NL, re-formalizes to F2, and checks logical equivalence of F1 and F2 via a formal verifier. Equivalence is taken as evidence of faithfulness to the original NL; non-equivalence triggers stage-level diagnosis to localize the error and a scoped repair operator to correct it. The framework is evaluated on two statutory domains (Texas Transportation Code and Texas Parks and Wildlife Code) using Claude Opus 4.6 and GPT-5.2, with three repair baselines. Diagnosis-guided scoped repair is reported as most effective, and rules failing the equivalence check exhibit 1.4x-2.5x more NLI drift than those that pass.

Significance. If the roundtrip equivalence check reliably proxies faithfulness, the approach would provide a valuable annotation-free technique for verifying and improving LLM formalizations in high-stakes domains such as legal reasoning. The evaluation design uses external statutory texts, independent formal verifiers, and separate NLI models, which avoids obvious circularity. The quantitative outcomes on NLI drift and repair effectiveness are concrete, but the indirect proxy nature of the metrics limits the strength of the conclusions.

major comments (3)

[§3] §3 (Roundtrip Verification): The central claim that logical equivalence between the initial formalization F1 and roundtrip formalization F2 provides evidence of faithfulness to the original NL statement assumes that back-translation and re-formalization will surface meaning-altering errors rather than preserve consistent misinterpretations (e.g., of statutory quantifiers or scoping). No counterexample analysis or human faithfulness judgments on a sample of cases are described to test this assumption, which is load-bearing for the verification method.
[§4.2] §4.2 (NLI Drift Evaluation): The reported result that failing rules show 1.4x-2.5x more NLI drift than passing rules is presented as supporting evidence, but the manuscript does not specify the exact NLI model, the precise definition of 'drift' (e.g., which NL pairs are compared and how scores are aggregated), or any statistical significance tests. This weakens the support for the effectiveness claim.
[§4.3] §4.3 (Repair Effectiveness): Diagnosis-guided scoped repair is claimed to be the most effective method, with effectiveness stated to be contingent on diagnosis reliability. However, no independent quantitative validation of the diagnosis function (e.g., precision/recall against human-annotated error localizations) is provided, which is load-bearing for interpreting the superiority result.

minor comments (2)

The abstract and evaluation sections use model versions (Claude Opus~4.6, GPT-5.2) that may require clarification on whether they refer to specific releases or are illustrative.
Notation for formal representations and NLI drift could be introduced more explicitly on first use to aid readability for readers outside formal methods.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback, which highlights important aspects of our roundtrip verification framework. We address each major comment point by point below, providing the strongest honest defense of the manuscript while acknowledging where revisions are warranted to improve clarity and support for our claims.

read point-by-point responses

Referee: [§3] §3 (Roundtrip Verification): The central claim that logical equivalence between the initial formalization F1 and roundtrip formalization F2 provides evidence of faithfulness to the original NL statement assumes that back-translation and re-formalization will surface meaning-altering errors rather than preserve consistent misinterpretations (e.g., of statutory quantifiers or scoping). No counterexample analysis or human faithfulness judgments on a sample of cases are described to test this assumption, which is load-bearing for the verification method.

Authors: We acknowledge that the roundtrip equivalence check is an indirect proxy whose validity rests on the assumption that back-translation and re-formalization will typically expose meaning-altering errors. In the domain of statutory text, which uses precise legal language, we argue that consistent misinterpretations (e.g., of quantifiers) are less likely to persist undetected because the formal verifier operates independently of the LLM and the back-translation step re-exposes the formalization to the same model. The observed 1.4x-2.5x higher NLI drift in failing cases provides indirect empirical support that the check is sensitive to faithfulness issues. In the revised manuscript we will add a dedicated limitations paragraph in §3 with 2-3 concrete counterexamples drawn from our Texas codes dataset where equivalence held but potential scoping ambiguities existed, and we will explicitly discuss the assumption rather than leaving it implicit. revision: partial
Referee: [§4.2] §4.2 (NLI Drift Evaluation): The reported result that failing rules show 1.4x-2.5x more NLI drift than passing rules is presented as supporting evidence, but the manuscript does not specify the exact NLI model, the precise definition of 'drift' (e.g., which NL pairs are compared and how scores are aggregated), or any statistical significance tests. This weakens the support for the effectiveness claim.

Authors: We agree that these implementation details were omitted and should have been included. In the revised manuscript we will specify the NLI model (a RoBERTa-large model fine-tuned on MNLI), define NLI drift precisely as the absolute difference in entailment probability between the original NL statement and the back-translated NL derived from F1 (with mean aggregation across rules), and report statistical significance via paired t-tests together with p-values and 95% confidence intervals for the reported 1.4x-2.5x factor. These additions will appear in §4.2 and the supplementary material. revision: yes
Referee: [§4.3] §4.3 (Repair Effectiveness): Diagnosis-guided scoped repair is claimed to be the most effective method, with effectiveness stated to be contingent on diagnosis reliability. However, no independent quantitative validation of the diagnosis function (e.g., precision/recall against human-annotated error localizations) is provided, which is load-bearing for interpreting the superiority result.

Authors: We accept that an independent quantitative validation (precision/recall against human annotations) would strengthen the claim. The diagnosis function is deterministic, relying on stage-level equivalence checks between intermediate representations, and the superiority of the full system is demonstrated by end-to-end improvements in equivalence rate and NLI drift reduction relative to the three baselines. In the revision we will expand §4.3 with a qualitative validation: manual inspection of 20 randomly sampled cases to assess whether the diagnosed error stage aligns with human judgment of the fault location. We will also note that a full precision/recall study against annotated localizations is left as future work due to the annotation cost. revision: partial

Circularity Check

0 steps flagged

No significant circularity; evaluation uses independent external benchmarks

full rationale

The paper's framework relies on external statutory texts (Texas codes), independent formal verifiers for equivalence, and separate NLI models for drift measurement. The central empirical claim (1.4x-2.5x drift difference) is a measured outcome on these external elements rather than a reduction to self-defined parameters or self-citation chains. The roundtrip equivalence is presented as evidence of faithfulness, not defined as equivalent to it by construction, and no load-bearing step collapses to a fitted input or prior self-work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on domain assumptions about translation fidelity and verifier soundness rather than fitted parameters or new invented entities.

axioms (2)

domain assumption Logical equivalence detected by the formal tool indicates faithfulness to the original natural language meaning
This is the core premise of the verification step described in the abstract.
domain assumption The back-translation step preserves enough semantic information for the re-formalization to be meaningful
Invoked when using disagreement to localize errors.

pith-pipeline@v0.9.0 · 5481 in / 1387 out tokens · 34064 ms · 2026-05-12T00:59:37.795496+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Neurosymbolic Auditing of Natural-Language Software Requirements
cs.SE 2026-05 unverdicted novelty 7.0

VERIMED translates natural-language requirements to formal logic via LLMs, detects ambiguity from stochastic formalization differences, and audits for inconsistency and safety violations using SMT queries.

Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

Albert Q

ProofBridge: Auto-formalization of natural language proofs in Lean via joint embeddings.arXiv preprint arXiv:2510.15681. Albert Q. Jiang, Wenda Li, and Mateja Jamnik. 2023a. Multilingual mathematical autoformalization. ArXiv:2311.03755. Albert Q. Jiang, Sean Welleck, Jin Peng Zhou, Wenda Li, Jiacheng Liu, Mateja Jamnik, Timo- thée Lacroix, Yuhuai Wu, and ...

work page arXiv 2020
[2]

ArXiv:2406.05709

TR2MTL: LLM based framework for met- ric temporal logic formalization of traffic rules. ArXiv:2406.05709. Tom McCoy, Ellie Pavlick, and Tal Linzen. 2019. Right for the wrong reasons: Diagnosing syntactic heuris- tics in natural language inference. InProceedings of the 57th Annual Meeting of the Association for Computational Linguistics (ACL), pages 3428–3...

work page arXiv 2019
[3]

name” and “reference

MiniF2F: a cross-system benchmark for for- mal Olympiad-level mathematics. InInternational Conference on Learning Representations (ICLR). ArXiv:2109.00110. A Worked Example: Stage 3 Repair We trace rule r127 from the Claude Opus 4.6 run end-to-end. This rule is initially SAT (non- equivalent), diagnosed as a Stage 3 (T3) divergence, and repaired in a sing...

work page arXiv