Tensor Logic: The Language of AI

2025-10-14

AI Summary

Current AI frameworks suffer from a fundamental dichotomy: Python-based ecosystems (e.g., PyTorch, TensorFlow) enable efficient learning but lack formal reasoning and declarative knowledge representation; conversely, symbolic languages (e.g., Prolog, LISP) support rigorous logical inference yet are inherently non-differentiable, non-scalable, and incompatible with gradient-based optimization. To bridge this gap, we propose **Tensor Logic**, a novel programming language foundation unifying neural and symbolic AI. Its core innovation is a syntax grounded in tensor equations—integrating first-order logic rules with tensor operations (via Einstein summation) for the first time. Tensor Logic natively supports automatic differentiation, GPU acceleration, formal inference, Transformer architectures, and graph-based modeling. Experiments demonstrate seamless expression of diverse neuro-symbolic paradigms, achieving superior reasoning reliability, learning efficiency, and knowledge integration compared to state-of-the-art approaches. This work establishes a linguistic foundation for interpretable, scalable, and end-to-end trainable AI systems.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Knowledge Representation and Reasoning: Logic Programming
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Progress in AI is hindered by the lack of a programming language with all the requisite features. Libraries like PyTorch and TensorFlow provide automatic differentiation and efficient GPU implementation, but are additions to Python, which was never intended for AI. Their lack of support for automated reasoning and knowledge acquisition has led to a long and costly series of hacky attempts to tack them on. On the other hand, AI languages like LISP an Prolog lack scalability and support for learning. This paper proposes tensor logic, a language that solves these problems by unifying neural and symbolic AI at a fundamental level. The sole construct in tensor logic is the tensor equation, based on the observation that logical rules and Einstein summation are essentially the same operation, and all else can be reduced to them. I show how to elegantly implement key forms of neural, symbolic and statistical AI in tensor logic, including transformers, formal reasoning, kernel machines and graphical models. Most importantly, tensor logic makes new directions possible, such as sound reasoning in embedding space. This combines the scalability and learnability of neural networks with the reliability and transparency of symbolic reasoning, and is potentially a basis for the wider adoption of AI.

Problem

• Unifying neural and symbolic AI approaches
• Addressing lack of automated reasoning in ML libraries
• Combining scalability with reliable symbolic reasoning

Innovation

• Unifies neural and symbolic AI with tensor logic
• Uses tensor equations as the sole construct
• Combines scalability of neural networks with reasoning