Neuro-symbolic Learning Yielding Logical Constraints

Neuro-symbolic systems combine neural perception and logical reasoning, representing one of the priorities of AI research. End-to-end learning of neuro-symbolic systems is highly desirable, but remains to be challenging. Resembling the distinction and cooperation between System 1 and System 2 of human thought (à la Kahneman), this paper proposes a framework that fuses neural network training, symbol grounding, and logical constraint synthesis to support learning in a weakly supervised setting. Technically, it is cast as a game with two optimization problems which correspond to neural network learning and symbolic constraint learning respectively. Such a formulation naturally embeds symbol grounding and enables the interaction between the neural and the symbolic part in both training and inference. The logical constraints are represented as cardinality constraints, and we use the trust region method to avoid degeneracy in learning. A distinguished feature of the optimization lies in the Boolean constraints for which we introduce a difference-ofconvex programming approach. Both theoretical analysis and empirical evaluations substantiate the effectiveness of the proposed framework.