Position: Certified Correctness in Neural Constraint Reasoning Requires Symbolic Integration

Abstract:Neural solvers for constraint satisfaction problems have achieved remarkable in-distribution accuracy, yet they suffer from a fundamental limitation persistent constraint violations occur under distribution shifts even when the model reports high confidence. This position paper argues that when hard constraints exist and the cost of verification is relatively low, neural constraint reasoning must prioritize symbolic integration over pure learning. We justify our focus on Sudoku as a representative NP-complete testbed because it exhibits a sharp asymmetry between easy verification and hard solving: checking a candidate solution requires only polynomial time $O(n^{2})$, while finding a solution may require exponential search. Through a comprehensive survey of solving methods spanning deterministic algorithms, metaheuristic optimization, learning-based approaches, and language-conditioned reasoning, we demonstrate that neural-only methods without instance-level certification fail to achieve the provable correctness that symbolic and neuro-symbolic approaches provide. We advocate for a bidirectional integration in which neural methods enhance symbolic solvers by learning heuristics and converting percepts into symbols, while symbolic methods verify neural outputs to ensure their reliability. To operationalize this position, we propose a multi-agent certified reasoning framework that demonstrates how this integration can achieve both computational efficiency and provable correctness.

Artificial Intelligence (cs.AI)

arXiv:2608.14569 [cs.AI]

(or arXiv:2608.14569v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2608.14569 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.