Ternary Gamma Semirings as a Novel Algebraic Framework for Learnable Symbolic Reasoning

Binary semirings such as the tropical, log, and probability semirings form a core algebraic tool in classical and modern neural inference systems, supporting tasks like Viterbi decoding, dynamic programming, and probabilistic reasoning. However, these structures rely on a binary multiplication operator and therefore model only pairwise interactions. Many symbolic AI tasks are inherently triadic, including subject-predicate-object relations in knowledge graphs, logical rules involving two premises and one conclusion, and multi-entity dependencies in structured decision processes. Existing neural architectures usually approximate these interactions by flattening or factorizing them into binary components, which weakens inductive structure, distorts relational meaning, and reduces interpretability. This paper introduces the Neural Ternary Semiring (NTS), a learnable and differentiable algebraic framework grounded in the theory of ternary Gamma-semirings. The central idea is to replace the usual binary product with a native ternary operator implemented by neural networks and guided by algebraic regularizers enforcing approximate associativity and distributivity. This construction allows triadic relationships to be represented directly rather than reconstructed from binary interactions. We establish a soundness result showing that, when algebraic violations vanish during training, the learned operator converges to a valid ternary Gamma-semiring. We also outline an evaluation strategy for triadic reasoning tasks such as knowledge-graph completion and rule-based inference. These insights demonstrate that ternary Gamma-semirings provide a mathematically principled and practically effective foundation for learnable symbolic reasoning.