🤖 AI Summary
Standard neural networks completely fail on compositional generalization tasks (0% accuracy) due to their inability to intrinsically model structured regularities. This work establishes, for the first time, a rigorous correspondence between neural network generalization and ternary Gamma semirings from abstract algebra, leveraging this algebraic framework to impose logical constraints that guide the network to learn representations satisfying symmetry, idempotence, and majority rules. Under identical architectures, these constraints elevate compositional generalization accuracy from 0% to 100%. Furthermore, the learned representations are proven to be isomorphic to the unique Boolean quaternary ternary Gamma semiring (|T| = 4, |Γ| = 1). This study pioneers a novel neuro-symbolic direction—computational Gamma algebra—bridging deep learning with formal algebraic structures.
📝 Abstract
This paper establishes a theoretical framework connecting neural network learning with abstract algebraic structures. We first present a minimal counterexample demonstrating that standard neural networks completely fail on compositional generalization tasks (0% accuracy). By introducing a logical constraint -- the Ternary Gamma Semiring -- the same architecture learns a perfectly structured feature space, achieving 100% accuracy on novel combinations. We prove that this learned feature space constitutes a finite commutative ternary $Γ$-semiring, whose ternary operation implements the majority vote rule. Comparing with the recently established classification of Gokavarapu et al., we show that this structure corresponds precisely to the Boolean-type ternary $Γ$-semiring with $|T|=4$, $|Γ|=1$}, which is unique up to isomorphism in their enumeration. Our findings reveal three profound conclusions: (i) the success of neural networks can be understood as an approximation of mathematically ``natural'' structures; (ii) learned representations generalize because they internalize algebraic axioms (symmetry, idempotence, majority property); (iii) logical constraints guide networks to converge to these canonical forms. This work provides a rigorous mathematical framework for understanding neural network generalization and inaugurates the new interdisciplinary direction of Computational $Γ$-Algebra.