Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs
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Computer Science > Computation and Language
arXiv:2610.02444 (cs)
[Submitted on 1 Oct 2026]
Title:Counterexample Generation via Per-Theorem Symbolic Verifiers: When Imitation Hurts and Reinforcement Repairs
Authors:Omar Farouk Zouak, Houssam Eddine Boukhalfa, Soumaya Lakehal, Shiv Katiyar, Samia Nefti-Meziani
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Abstract:Large language models often solve a theorem forward yet fail to disprove a closely related false one: a falsification gap that supervised fine-tuning does not close and can actively worsen. We frame counterexample generation as constrained witness emission against a deterministic per-theorem Python verifier, and release SymCE, a corpus of 4,707 false undergraduate-algebra and real-analysis conjectures, each paired with executable verifiers. The verifier also serves as the reward function, making SymCE a training environment. Training Qwen3-4B with SFT followed by GRPO under this oracle reveals an imitation trap: counterexample-only SFT collapses true-theorem recognition from 0.27 to 0.00, while RLVR with a sparse outcome-only reward repairs this and exceeds the base, to 0.66. The collapse replicates across four seeds and on Gemma-3-4B. Sparse and dense rewards yield statistically indistinguishable in-domain success yet diverge by 33 points on a held-out calibration probe, a dissociation we trace to the partial-credit term. Our 4B model outperforms every evaluated 7B open-weights math specialist, remains competitive with six frontier commercial APIs, and transfers under unchanged prompting to GSM8K, MATH-500 and MMLU-college-math. A human audit of 177 verifier decisions finds 97.7% accuracy. Code, data, verifier modules and annotations: this https URL.
| Comments: | Accepted at EMNLP 2026 Findings |
| Subjects: | Computation and Language (cs.CL); Artificial Intelligence (cs.AI) |
| Cite as: | arXiv:2610.02444 [cs.CL] |
| (or arXiv:2610.02444v1 [cs.CL] for this version) | |
| https://doi.org/10.48550/arXiv.2610.02444
arXiv-issued DOI via DataCite (pending registration)
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