Tensor Logic: The Language of AI (2510.12269v2)

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.

• The paper proposes tensor logic as a minimal, expressive language that unifies neural, symbolic, and statistical AI by mapping logical rules to tensor equations.
• The methodology employs generalized tensor joins, projections, and elementwise nonlinearities to concisely implement neural architectures, symbolic reasoning, and graphical models.
• The framework enables efficient learning and transparent reasoning in embedding space through backpropagation via tensor equations and scalable GPU integration.

Tensor Logic: A Unified Language for Neural and Symbolic AI

Motivation and Foundations

The paper "Tensor Logic: The Language of AI" (2510.12269) introduces tensor logic as a programming language designed to unify neural, symbolic, and statistical AI paradigms. The motivation stems from the limitations of existing languages: Python-based frameworks (e.g., PyTorch, TensorFlow) offer efficient tensor computation and autodiff but lack native support for automated reasoning and knowledge representation, while traditional AI languages (LISP, Prolog) are not scalable and do not support learning. Tensor logic is proposed as a minimal, expressive language whose sole construct is the tensor equation, leveraging the equivalence between logical rules (as in Datalog) and Einstein summation (einsum) over tensors.

The core insight is that relations in logic programming correspond to sparse Boolean tensors, and Datalog rules are einsums over these tensors, followed by a step function. This equivalence enables a direct mapping between symbolic reasoning and tensor algebra, allowing both to be implemented and composed within a single language.

Language Design and Semantics

Tensor logic programs consist of tensor equations, where the left-hand side (LHS) is the tensor being computed, and the right-hand side (RHS) is a sequence of tensor joins (generalized einsums), projections (summations over indices), and elementwise nonlinearities. The language supports both dense and sparse tensors, with syntactic sugar for common operations (e.g., slices, normalization, index functions).

Key constructs:

• Tensor Join: Generalizes database join; for tensors $U_{\alpha\beta}$ and $V_{\beta\gamma}$, the join is $U_{\alpha\beta} V_{\beta\gamma}$.
• Tensor Projection: Generalizes marginalization; for tensor $T_{\alpha\beta}$, projection onto $\alpha$ is $\sum_\beta T_{\alpha\beta}$.
• Nonlinearity: Elementwise application (e.g., step, sigmoid, softmax).

Tensor logic subsumes Datalog syntax for Boolean tensors, enabling compact representation of sparse relations as sets of facts. The language is expressive enough to encode neural architectures, symbolic reasoning, kernel machines, and probabilistic graphical models.

Inference and Learning

Inference in tensor logic generalizes forward and backward chaining from logic programming:

• Forward Chaining: Tensor equations are executed iteratively, propagating values until fixpoint.
• Backward Chaining: Queries are recursively decomposed into subqueries, with default values for missing data.

Learning is facilitated by the uniformity of tensor equations: autodiff is straightforward, as the gradient of a tensor equation with respect to its inputs is the product of the other tensors on the RHS. The gradient program is itself a tensor logic program. The framework supports backpropagation through structure, enabling learning in architectures where the computation graph varies per example (e.g., recursive neural networks).

Tensor decomposition (e.g., Tucker) is naturally expressed, generalizing predicate invention in inductive logic programming. This enables learning compact, interpretable representations from data.

Implementation of AI Paradigms

Neural Networks

Tensor logic provides concise implementations for MLPs, CNNs, RNNs, and GNNs. For example, a convolutional layer is expressed as:

Features[x,y] = relu(Filter[dx,dy,ch] * Image[x+dx, y+dy, ch])

Pooling, attention, and transformer architectures are similarly encoded as tensor equations, with explicit index manipulation and normalization.

Symbolic AI

Datalog programs are directly expressible, supporting sound reasoning and planning in function-free domains. Unification for function symbols (as in Prolog) can be implemented within tensor logic.

Kernel Machines

Support vector machines and kernel methods are encoded via tensor equations for the kernel function and output computation. Structured prediction is supported by output tensors and interaction equations.

Probabilistic Graphical Models

Graphical models are represented as products of factor tensors, with marginalization and pointwise product corresponding to projection and join. Bayesian networks are encoded as tensor equations for conditional probability tables. Inference algorithms (belief propagation, sampling) are mapped to forward/backward chaining and selective projection.

Reasoning in Embedding Space

A novel contribution is the formalization of reasoning in embedding space. Objects and relations are embedded as tensors, enabling superposition and retrieval via tensor products. Reasoning over embedded rules is performed by chaining tensor equations, with the error probability controlled by embedding dimension. This approach supports analogical reasoning, compositionality, and transparency.

The temperature parameter in sigmoid nonlinearities modulates the deductive vs. analogical nature of inference. At $T=0$, reasoning is purely deductive and immune to hallucinations, contrasting with LLMs. At higher $T$, inference becomes analogical, supporting generalization.

Scaling and Implementation Considerations

Tensor logic is designed for scalability:

• Dense Tensors: Direct GPU implementation.
• Sparse Tensors: Two approaches: (1) separation of concerns, using database engines for sparse operations and GPUs for dense; (2) conversion of sparse tensors to dense via Tucker decomposition, enabling efficient GPU computation with controlled error.

The language is amenable to integration with existing Python frameworks, facilitating adoption. Efficient implementations require optimized einsum and tensor decomposition routines, as well as support for sparse data structures.

Implications and Future Directions

Tensor logic provides a unified formalism for AI, bridging neural, symbolic, and statistical paradigms. Its minimal design enables direct translation of mathematical models to code, facilitating transparency, reliability, and learnability. The framework supports new directions in reasoning, particularly in embedding space, with potential applications in knowledge representation, scientific computing, and interpretable AI.

Future work includes CUDA-based implementations, development of libraries and IDEs, and exploration of tensor logic in diverse AI applications. The language's compatibility with Python and its expressive power position it as a candidate for the standard language of AI.

Conclusion

Tensor logic offers a principled, expressive language for AI, unifying neural and symbolic computation via tensor equations. Its design enables concise implementation of diverse AI models, efficient learning and inference, and transparent reasoning in embedding space. The framework addresses key limitations of existing languages and opens new avenues for research and application in AI.