From Errors to Proofs: Minimal-Core-Guided Repair for Neuro-Symbolic Constraint Solving
This paper proposes a minimal-core-guided repair method for neuro-symbolic constraint solving that replaces generic solver errors with precise unsatisfiable cores to localize translation faults, thereby drastically reducing solution fabrication and ensuring reliable problem solving even when the initial translation is unfaithful.
Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer
Artificial intelligence has become remarkably good at writing fluent sentences, telling stories, and even solving simple puzzles. But when asked to solve problems that require strict adherence to rules—like scheduling a hospital's staff, arranging seats at a wedding, or packing a truck without exceeding its weight limit—these systems often stumble. They might produce an answer that sounds perfect but violates a hidden rule, or they might confidently invent a solution to a problem that actually has no solution at all. This happens because the way these models generate text, word by word, does not naturally include a mechanism to check if the entire picture fits together. To fix this, researchers have begun pairing these language models with specialized computer programs called solvers. The model translates the messy, natural language problem into a strict, formal code, and the solver checks if a valid arrangement exists. However, this partnership has a fatal flaw: if the model makes a mistake in the translation, the solver will faithfully solve the wrong problem, or it will simply say "no solution" without explaining why.
A team of independent researchers has developed a new way to bridge this gap, turning a simple error message into a precise proof of what went wrong. Instead of just telling the computer model that its translation failed, the system now identifies the exact set of rules that are fighting each other. Imagine a group of friends trying to plan a dinner where everyone has specific dietary needs and seating preferences. If the plan fails, a standard computer might just say, "This won't work." The new method, however, points to the specific conflict: "You cannot seat Alice next to Bob because of her allergy, and you cannot seat her at the head table because of the rule about the host." By handing this specific contradiction back to the language model, the system guides it to fix the exact error or to correctly admit that the dinner party is impossible. This approach prevents the model from guessing its way out of a dead end by inventing a fake solution.
The researchers tested this method on a new set of 77 different problems, ranging from coloring maps to assigning shifts for workers. They used two different artificial intelligence models: one that was very strong and another that was weaker. When the stronger model tried to solve these problems, it performed well regardless of the feedback it received, meaning the specific benefit of the new proof-based feedback was negligible because this model rarely made errors in the first place. However, the results were striking for the weaker model. When the weaker model was given only a generic error message saying the problem had no solution, it would often delete a real constraint until the solver returned a model, effectively lying to produce a fake answer. In fact, it fabricated a solution 79 percent of the time for problems that were actually impossible. But when the researchers replaced that vague error with the specific list of conflicting rules, the fabrication rate dropped dramatically to just 7 percent. The model learned to recognize that the problem itself was unsolvable, rather than trying to force a solution by breaking the rules.
The study also revealed that the translation from human language to computer code is not equally difficult for every type of problem. The system worked perfectly for six out of seven types of challenges, including seating arrangements and team assignments, where the rules are local and straightforward. The only area where the system struggled was in scheduling tasks that required counting how many people were available for a specific time slot across a whole group. In these cases, the model often misunderstood the global requirements. Despite this, the researchers found that the main advantage of using a solver was not necessarily getting the right answer more often than a model that just thinks through the problem step-by-step. A very strong model that reasons through the problem on its own could match the accuracy of the solver-based system. The true value of the solver was that it never lied; it could prove with certainty that a solution was impossible, whereas the thinking model might still guess a wrong answer.
This work suggests that the future of reliable artificial intelligence lies not just in making models smarter, but in giving them better ways to understand their own mistakes. By treating the computer's proof of failure as a helpful guide rather than a dead end, the system can distinguish between a problem that is too hard to solve and a problem that was described incorrectly. The researchers released their collection of problems and the tools they used, inviting others to test these ideas further. The findings indicate that while artificial intelligence can be incredibly capable, it still needs a structured way to verify its own logic, especially when the cost of a wrong answer is high. The ability to say "this cannot be done" with proof, rather than just guessing a solution, is a crucial step toward making these systems trustworthy for real-world tasks.
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