Learning from Failed Hypotheses: Local Abduction for Neuro-Symbolic Self-Training

Combining language models with symbolic solvers enables explicit and verifiable reasoning, but learning to construct symbolic solutions is hindered by the scarcity of annotations. Weakly supervised self-training addresses this challenge by learning from the model's own predictions verified against observed answers, yet new supervision still depends on generating a complete correct solution. We observe that failed solutions can retain useful structure, suggesting that supervision may be recovered before the model can solve a problem on its own. We propose Local Abductive Self-Training (LAST), which formulates this recovery as local abductive inference. Given a failed hypothesis, LAST searches its neighborhood for symbolic solutions consistent with the observed answer and uses model likelihood to select promising corrections. These corrections complement successful neural predictions and accumulate in trajectory memory to support subsequent learning. Across five geometry and numerical reasoning benchmarks, LAST improves average accuracy after six rounds by 7.84–10.69 and 4.83–12.40 percentage points over self-training baselines with 2B and 8B models, respectively. Further analyses show that LAST broadens supervision coverage while preserving solution quality, highlighting the value of failed hypotheses as a source of supervision.