# Tensor Logic: The Language of AI

arXiv: https://arxiv.org/abs/2510.12269 · HTML: https://mlsanity.com/paper/2510.12269/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 18 references that are in the map.
- First posted on arXiv 14 Oct 2025.

## Builds on (most cited first, 3 of 3)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Introduction to Tensor Decompositions and their Applications in Machine Learning](https://mlsanity.com/paper/1711.10781/) (2017)
- [A Survey on Retrieval And Structuring Augmented Generation with Large Language Models](https://mlsanity.com/paper/2509.10697/) (2025)

## Built on by (most cited first, 8 of 8)

- [State over Tokens: Characterizing the Role of Reasoning Tokens](https://mlsanity.com/paper/2512.12777/) (2025)
- [Explicit Fuzzy Logic in the Feed-Forward Layer: Self-Forgetting Quantifiers Discover Legible Grammatical-Licensing Detectors](https://mlsanity.com/paper/2606.31845/) (2026)
- [Incorporating Deep Learning Design in Database Queries](https://mlsanity.com/paper/2605.24207/) (2026)
- [Training, Reading, and Editing Legible Transformers](https://mlsanity.com/paper/2607.08946/) (2026)
- [Computational Concept of the Psyche](https://mlsanity.com/paper/2603.15586/) (2026)
- [ClosureBench: A Constructive Benchmark for Compositional Graph Reasoning](https://mlsanity.com/paper/2608.18242/) (2026)
- [Tensor network representations of discrete maximum entropy distributions via mean polytopes](https://mlsanity.com/paper/2609.07184/) (2026)
- [Exactness at Inference: A Representational Criterion for Out-of-Distribution Generalization](https://mlsanity.com/paper/2609.24942/) (2026)

## Abstract (arXiv)

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 and 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.