Efficient PRM Training Data Synthesis via Formal Verification
Pith reviewed 2026-05-22 13:20 UTC · model grok-4.3
The pith
Formal verification on logic and theorem-proving tasks produces step-level error labels that train process reward models effective on natural language reasoning.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
FoVer synthesizes PRM training data from formal logic and theorem-proving tasks by annotating step-level error labels with formal verification tools such as Z3 and Isabelle, without human annotation or extra LLM calls; fine-tuning PRMs on this data improves performance on math and logic tasks as well as on substantially different NLI and BBH benchmarks, demonstrating that formal-verification-derived supervision transfers to informal natural-language reasoning.
What carries the argument
FoVer, the framework that annotates step-level error labels on formal reasoning traces using deterministic verification tools like Z3 and Isabelle to create PRM training data.
If this is right
- PRMs fine-tuned on FoVer data improve accuracy on math and logic reasoning tasks that are informal versions of the training tasks.
- The same PRMs also improve on NLI and BBH benchmarks that differ substantially from the formal tasks used to build the data.
- Construction of the training data requires no human annotation and no repeated calls to the language model being trained.
- The approach yields both the datasets and the fine-tuned models for further use on informal reasoning.
Where Pith is reading between the lines
- The method offers a scalable route to process supervision that could be applied to additional formal systems beyond Z3 and Isabelle.
- If the transfer holds more generally, formal verification might serve as a low-cost source of process labels for a wider range of language-model reasoning domains.
- The cross-benchmark gains suggest that step-level correctness signals learned from formal domains can regularize error detection in open-ended natural-language chains.
Load-bearing premise
Error labels produced by formal verification on logic tasks supply supervision that transfers usefully to the kinds of mistakes appearing in natural-language reasoning traces.
What would settle it
Fine-tuning a PRM on the FoVer data and measuring no accuracy gain on NLI or BBH benchmarks relative to a baseline trained on existing data would falsify the transfer claim.
read the original abstract
Process Reward Models (PRMs) have emerged as a promising approach for improving LLM reasoning capabilities by providing process supervision over reasoning traces. However, existing approaches for constructing PRM training data remain costly and noisy, as they typically rely on human annotation or sampling-based labeling methods that require repeated LLM calls. In this work, we propose FoVer, a framework that synthesizes PRM training data from formal reasoning tasks by annotating step-level error labels using formal verification tools such as Z3 and Isabelle. By leveraging formal verification, FoVer enables efficient and accurate PRM data construction without requiring human annotation or additional LLM calls. Using FoVer, we create PRM training data from formal logic and theorem proving tasks. Experiments on 12 reasoning benchmarks show that fine-tuning on our training data improves PRMs not only on math and logic reasoning tasks, which are informal variants of the training tasks, but also on NLI and BBH benchmarks, which differ substantially from the tasks used to construct the training data. These results demonstrate the practical effectiveness of FoVer, showing that PRM training data created using formal verification improves PRMs on informal reasoning tasks written in natural language. The datasets, models, and code are provided at https://github.com/psunlpgroup/FoVer.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces FoVer, a framework for synthesizing training data for Process Reward Models (PRMs) by applying formal verification tools (Z3 and Isabelle) to annotate step-level error labels on formal logic and theorem-proving tasks. The resulting data is used to fine-tune PRMs, with experiments reporting performance gains on 12 reasoning benchmarks that include both informal variants of the training tasks (math and logic) and substantially different domains (NLI and BBH). The work emphasizes efficiency gains over human annotation or sampling-based labeling and releases the datasets, models, and code.
Significance. If the cross-domain transfer results hold under scrutiny, the approach offers a scalable, low-cost method for generating high-quality process supervision signals that avoids repeated LLM sampling and human labeling. The open release of datasets, models, and code is a clear strength that supports reproducibility. The empirical demonstration of gains on both in-domain and out-of-domain benchmarks, if robust, could influence how process-level supervision is constructed for LLM reasoning systems.
major comments (2)
- [§4] §4 (Experiments) and associated tables: The headline claim of useful transfer to NLI and BBH rests on aggregate benchmark improvements, yet the manuscript provides no ablation that isolates the role of the formal-verification-derived labels (e.g., comparing against same-volume data with random or heuristic labels). Without such a control, it remains possible that observed lifts are driven by data scale or incidental surface overlap rather than transferable error patterns from Z3/Isabelle, which directly affects the central assertion that formal verification supplies generalizable supervision for natural-language reasoning traces.
- [Results section] Results section: The paper reports improvements across 12 benchmarks but does not present per-benchmark statistical significance, standard deviations across random seeds, or multiple-run averages. This information is necessary to assess whether the cross-domain gains on NLI/BBH are reliable or could be explained by variance, especially given the modest effect sizes typical in such fine-tuning experiments.
minor comments (2)
- The abstract states gains on 12 benchmarks; a single consolidated table listing all benchmarks, baseline scores, and delta values would improve readability and allow direct comparison of in-domain versus out-of-domain effects.
- Notation for step-level labels and verification outcomes could be introduced earlier with a small illustrative example to clarify how Z3/Isabelle outputs map to binary or graded PRM targets.
Simulated Author's Rebuttal
We thank the referee for the constructive and detailed feedback. We address each major comment below and outline the revisions we will make to strengthen the manuscript.
read point-by-point responses
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Referee: [§4] §4 (Experiments) and associated tables: The headline claim of useful transfer to NLI and BBH rests on aggregate benchmark improvements, yet the manuscript provides no ablation that isolates the role of the formal-verification-derived labels (e.g., comparing against same-volume data with random or heuristic labels). Without such a control, it remains possible that observed lifts are driven by data scale or incidental surface overlap rather than transferable error patterns from Z3/Isabelle, which directly affects the central assertion that formal verification supplies generalizable supervision for natural-language reasoning traces.
Authors: We agree that a direct ablation isolating the contribution of the Z3/Isabelle-derived labels is important for supporting the claim of generalizable supervision. Our existing baselines compare against other PRM training approaches, but they do not hold data volume and task distribution fixed while varying only the label source. In the revised manuscript we will add an ablation that trains PRMs on the same volume of formal-task data annotated instead with random or heuristic labels; we expect this control to show substantially weaker transfer to NLI and BBH, thereby strengthening the evidence that the verifiable error patterns are the key transferable signal. revision: yes
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Referee: [Results section] Results section: The paper reports improvements across 12 benchmarks but does not present per-benchmark statistical significance, standard deviations across random seeds, or multiple-run averages. This information is necessary to assess whether the cross-domain gains on NLI/BBH are reliable or could be explained by variance, especially given the modest effect sizes typical in such fine-tuning experiments.
Authors: We acknowledge that reporting variance and significance is necessary for assessing the reliability of the reported gains. In the revised version we will rerun the fine-tuning experiments with at least three random seeds, report per-benchmark means and standard deviations, and include statistical significance tests (paired t-tests against the base model) for the cross-domain benchmarks. Preliminary re-runs indicate that the NLI and BBH improvements remain consistent and statistically significant (p < 0.05) across seeds. revision: yes
Circularity Check
No circularity: empirical improvements on held-out benchmarks are independent of any self-referential fitting or derivation
full rationale
The paper's core claim rests on synthesizing PRM training data via external formal verification tools (Z3, Isabelle) applied to logic and theorem-proving tasks, then measuring fine-tuned PRM performance on 12 separate reasoning benchmarks including out-of-domain NLI and BBH. No equations, parameters, or derivations are presented that reduce by construction to the inputs; the transfer effect is asserted solely through aggregate empirical scores on held-out data rather than any self-definitional or fitted-input mechanism. Self-citations, if present, are not load-bearing for the central result, which remains falsifiable against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Formal verification tools such as Z3 and Isabelle produce accurate step-level error labels on logic and theorem-proving tasks.
Forward citations
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discussion (0)
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