Ternary Gamma Semirings as a Novel Algebraic Framework for Learnable Symbolic Reasoning
Chandrasekhar Gokavarapu, D. Madhusudhana Rao
TL;DR
This work addresses the limitation of binary semiring frameworks in AI by introducing Neural Ternary Semiring (NTS), a learnable framework grounded in ternary Gamma-semirings that can represent native triadic interactions via the operator $[x,y,z]$ with context $\gamma$. It presents two parameterizations (tensor-based and attention-based) for differentiable triadic fusion, combined with algebraic regularizers enforcing approximate associativity and distributivity, and proves a soundness theorem: if these regularizers vanish, the learned operator satisfies the ternary Gamma-semiring axioms. A practical evaluation strategy is proposed for triadic reasoning tasks, including knowledge-graph completion and rule-based inference, with comparisons to binary and triadic baselines using standard metrics such as $\mathrm{MRR}$ and $\mathrm{Hits}@k$. The framework promises more faithful modeling of higher-order relational structure, improved generalization, and stability in sparse data regimes, and opens avenues for extending to $n$-ary semirings, categorical semantics, and neural interpretations of semantic layers. Overall, NTS provides a mathematically principled, scalable, and expressive foundation for learnable symbolic reasoning that directly captures triadic relationships in structured data.
Abstract
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.