SAT-RL: Training Language Models for Logical Reasoning with Symbolic Solvers

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.

Natural language understanding requires interleaving textual and logical reasoning, yet large language models often fail to perform such reasoning reliably. Existing neurosymbolic systems combine LLMs with solvers but remain limited to fully formalizable tasks such as math or program synthesis, leaving natural documents with only partial logical structure unaddressed. We introduce Logitext, a neurosymbolic language that represents documents as natural language text constraints (NLTCs), making partial logical structure explicit. We develop an algorithm that integrates LLM-based constraint evaluation with satisfiability modulo theory (SMT) solving, enabling joint textual-logical reasoning. Experiments on a new content moderation benchmark, together with LegalBench and Super-Natural Instructions, show that Logitext improves both accuracy and coverage. This work is the first that treats LLM-based reasoning as an SMT theory, extending neurosymbolic methods beyond fully formalizable domains.

Large language models (LLMs) are increasingly used for tasks that implicitly reduce to Boolean satisfiability (SAT), yet their reasoning ability on SAT remains unclear. We present a systematic study of LLMs on 2-SAT and 3-SAT, together with two canonical reductions, Vertex Cover and discrete 3D packing, to probe representation-invariant reasoning. We first evaluate models using conventional metrics, including accuracy, precision, recall, and F1, as well as the SAT phase-transition setting. We find that these metrics can be misleading: many models obtain high scores by over-predicting satisfiable formulas, fail to reproduce the classical easy-hard-easy signature around the 3-SAT threshold, and degrade sharply as the number of variables grows. To address this problem, we introduce a paired-formula protocol based on minimally different satisfiable and unsatisfiable instances, together with Accurate Differentiation Rate (ADR), which requires both members of each pair to be classified correctly. ADR separates reasoning-oriented models from heuristic ones and correlates with witness validity. Beyond CNF, we test cross-representation consistency by converting CNF to Vertex Cover and 3-SAT to discrete 3D packing. Model decisions on CNF and on the corresponding graph or packing instances agree for most models on more than 80 percent of instances, suggesting stable decision rules across representations. Overall, our results show that SAT is a conservative probe for LLM reasoning, and that paired evaluation with ADR provides a more faithful and representation-robust assessment than conventional metrics.