Abstract
First-order concept synthesis asks a system to infer one formula that classifies labeled objects consistently across several finite relational structures. Every candidate can be evaluated exactly, but quantified first-order formulas form a vast search space, and LLM outputs are often semantically promising without being fully correct. We introduce Hypothesis Frontier, a verifier-guided neurosymbolic framework that evaluates each LLM formula on every training object, retains the strongest verified hypothesis across rounds, and uses its remaining errors to guide subsequent generation. Symbolic processing repairs invalid formulas while remaining anchored to the LLM-generated hypothesis, and simplifies train-valid formulas without changing any training prediction. Under matched models, problem sets, and LLM-round budgets, Hypothesis Frontier solves substantially more problems than repeated original-prompt generation. After the final formulas are selected, exact simplification shortens many train-valid formulas while preserving every training prediction. Exact symbolic reasoning therefore helps both to solve more induction problems and to compress many of the resulting formulas.
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LLMs can solve program synthesis tasks but remain inefficient and unreliable on hard instances requiring large combinatorial search. Given a small set of reasoning traces, we use coding agents to compile them into reusable symbolic program synthesizers over constrained DSLs. The resulting solvers require no LLM calls at test time and are strong standalone systems: symbolic solver ensembles reach 91.3% accuracy on PBEBench-Lite and 84.7% on PBEBench-Hard, outperforming LLMs with test-time scaling for the latter by +16.3 percentage points at zero LLM inference cost. They also complement LLM search, improving PBEBench-Hard accuracy from 68.4% to 85.8% while reducing reported token usage by 78%, and raising SLR-Bench hard-tier accuracy from 34.4% to 58.0% in a neuro-symbolic hybrid setting. Compared to directly using coding agents as per-instance solvers, induced solvers are substantially more Pareto-efficient, amortizing a small one-time construction cost over many zero-token executions. Finally, most solvers transfer zero-shot to a real historical linguistics task - predicting sound changes in natural language data - reaching 80.1% accuracy under ensembling and recovering some plausible linguistic rules. Together, these results show that reasoning traces can be compiled into reusable symbolic solvers that solve many tasks directly, complement LLM inference on hard cases, and provide a scalable route to domain-general solver induction. We release code and data for reproducibility.
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