LogicKAN: Alternating Differentiable Rule Learning and Structural Repair for Compact Boolean Program Induction

Boolean rule induction seeks to recover an executable logical program from labeled binary examples. Yet high held-out accuracy can conceal disagreements with the target rule elsewhere in the Boolean domain, and learning with a fixed clause budget can produce unnecessarily large programs. We propose Logic-KAN, a differentiable model that assembles positive and negated literals into shared conjunctive clauses and combines them through OR and XOR heads. Continuous parameters learn which literals and clauses to select and how each head combines its clauses, while a thresholded version exposes the current rule during training. Using only labeled training examples, Logic-KAN scores local edits that add, refine, replace, or remove clauses, adapting its structure as gradient learning proceeds. After training, the discrete model is compiled into an executable symbolic program whose clauses and operators can be inspected directly. Separate checks compare the exported program with the discrete model and, when a target formula is available, use SAT to test equivalence over the full Boolean domain. On synthetic DNF and mixed OR/XOR tasks, SAT finds counterexamples in some runs with perfect held-out accuracy. Compared with existing models, Logic-KAN produces smaller programs on both task families and achieves better exact recovery on DNF.