A tensor network formalism for neuro-symbolic AI

2026-01-21

AI Summary

This work addresses the long-standing challenge in neuro-symbolic artificial intelligence of integrating logical reasoning and probabilistic learning due to the absence of a unified mathematical framework. The authors propose a novel formalism based on tensor networks, wherein logical formulas and probability distributions are jointly represented as structured tensor decompositions. For the first time, tensor contraction is employed as a universal inference mechanism, enabling the construction of Hybrid Logic Network—a trainable hybrid logic-probabilistic model. Through tensor decomposition, basis encoding, contraction-based message passing algorithms, and a custom Python library (tnreason), the framework achieves a unified and efficient computational treatment of both logical and probabilistic reasoning. This approach establishes a scalable, differentiable paradigm for neuro-symbolic systems, bridging symbolic expressiveness with data-driven learning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The unification of neural and symbolic approaches to artificial intelligence remains a central open challenge. In this work, we introduce a tensor network formalism, which captures sparsity principles originating in the different approaches in tensor decompositions. In particular, we describe a basis encoding scheme for functions and model neural decompositions as tensor decompositions. The proposed formalism can be applied to represent logical formulas and probability distributions as structured tensor decompositions. This unified treatment identifies tensor network contractions as a fundamental inference class and formulates efficiently scaling reasoning algorithms, originating from probability theory and propositional logic, as contraction message passing schemes. The framework enables the definition and training of hybrid logical and probabilistic models, which we call Hybrid Logic Network. The theoretical concepts are accompanied by the python library tnreason, which enables the implementation and practical use of the proposed architectures.

Problem

• neuro-symbolic AI
• tensor networks
• neural-symbolic integration
• logical reasoning
• probabilistic inference

Innovation

• tensor network
• neuro-symbolic AI
• tensor decomposition
• logical reasoning
• probabilistic inference