Tensor Logic: Unified AI Framework

• Tensor Logic is a unified computational language that encodes neural, symbolic, and probabilistic reasoning as high-dimensional tensor equations.
• It employs generalized Einstein summation to perform logical joins, projections, and inference, ensuring efficient and scalable operations on modern hardware.
• The framework offers end-to-end differentiability and transparency, facilitating rigorous, traceable reasoning critical for safety-sensitive AI applications.

Tensor logic provides a unified mathematical and computational language that expresses neural, symbolic, and statistical reasoning as tensor equations, primarily via generalized Einstein summation. In tensor logic, entities, relations, and logical rules are encoded as high-dimensional tensors (which may be Boolean or real-valued), and operations such as logical joins, inference, and learning are cast as tensor contractions, projections, and nonlinearities. This approach reveals that the same algebraic structures underpin the forward passes of neural networks, symbolic deduction, database joins, and probabilistic inference. It enables scalable, differentiable reasoning in both the discrete symbolic space and continuous embedding spaces, offering a pathway to integrate reliability and transparency of symbolic logic with the flexibility and scalability of machine learning (Domingos, 14 Oct 2025).

1. Tensor Equations as a Unifying Language

Tensor logic centers on the principle that all forms of AI—neural, symbolic, probabilistic—can be described by tensor equations of the general form: $$O_{i_1 \dots i_k} = f\left( \text{einsum}(A^{(1)}, A^{(2)}, \dots, A^{(n)}) \right)$$ where “einsum” denotes generalized Einstein summation over one or more shared indices, and $f$ is an optional nonlinearity (e.g., step, sigmoid, relu).

• Symbolic Logic: Database relations are sparse Boolean tensors (e.g., matrices for binary relations), and Datalog or logic programming rules become tensor join equations. For instance, the rule

$$\text{Aunt}(x,z) \leftarrow \text{Sister}(x,y), \text{Parent}(y,z)$$

is re-expressed as:

$A_{xz} = H\left( \sum_{y} S_{xy} P_{yz} \right )$

where $H$ is the Heaviside step function.

• Neural Networks: Layers are expressed as tensor contractions; for example, a multi-layer perceptron layer:

$y = \text{act}(W_{ij} x_j)$

and convolutional or attention mechanisms in transformers follow similar principles, with multiple indices summed and projected to obtain outputs.

• Graphical Models and Probabilistic Inference: Factors in probabilistic graphical models are tensors, and reasoning (e.g., marginalization) is performed by summing over appropriate indices.

All of these paradigms, regardless of their original conceptual differences, are mapped to the same underlying framework based on tensor equations.

2. Symbolic Reasoning as Tensor Operations

The main insight is that logical conjunctions (joins) are tensor products/summations over shared indices, and logical rules are tensor equations augmented with nonlinearities for thresholding or probabilistic weighting (Domingos, 14 Oct 2025). The steps of symbolic inference align as follows:

Symbolic Operation	Tensor Logic Equivalent
Relation (e.g., R(x,y))	Boolean/Sparse tensor $R_{xy}$
Logical join (conjunction)	Einstein summation over shared indices
Projection	Summing out (contracting) indices
Deductive closure	Iterating tensor equations

This allows arbitrary symbolic deduction, including forward- and backward-chaining, to be implemented as recursive or iterative tensor contractions and thresholdings.

3. Neural, Kernel, and Statistical Computation

Neural architectures in tensor logic are sequences of tensor equations, with every layer being a contraction plus nonlinearity (e.g., $Y = \text{relu}(W_{ij} X_j + b_i)$). More advanced models—transformer attention or kernel machines—also reduce to tensor equations:

• Transformers: Attention layers are written as

$$Q_{pd_k} = W^Q_{d_k d} X_{p d}, \quad K_{p' d_k} = W^K_{d_k d} X_{p' d}$$

$$C_{p p'} = \text{softmax}(Q_{p d_k} K_{p' d_k}/\sqrt{d_k})$$

$\text{Attn}_{p d_v} = C_{p p'} V_{p' d_v}$

• Kernel Machines: Kernel matrices are tensor equations, e.g.,

$K_{ii'} = ( X_{ij} X_{i'j} )^n$

• Graphical Models: Marginals and conditionals are projections and tensor joins, e.g.,

$\phi'(X) = \sum_Y \phi(X, Y)$

In each case, the framework ensures efficient GPU/TPU computation and end-to-end differentiability.

4. Reasoning in Embedding Space

Tensor logic extends logical reasoning to continuous embedding spaces, facilitating analogical and fuzzy inference. Objects are assigned learned embedding vectors ($\text{Emb}[x, d]$). Logical queries and set memberships are realized via dot products and tensor contractions: $D[A] = S[d] \text{Emb}[A, d]$ where $S[d]$ may be a sum over a selector for the set. Relations $R(x,y)$ can be represented by: $$\text{EmbR}[i,j] = \sum_{x,y} R(x,y) \text{Emb}[x,i] \text{Emb}[y,j]$$ Membership queries are again dot products. By controlling the embedding temperature and periodic re-thresholding, one can interpolate between analogical (fuzzy) and purely symbolic (deductive) reasoning in the embedding space. This allows scalable, learnable, and partially interpretable reasoning directly in high-dimensional vector spaces, while still supporting sound deductive steps when required (Domingos, 14 Oct 2025).

5. Scalability, Transparency, and Differentiability

Tensor logic exploits the inherent parallelism and efficiency of tensor computation (e.g., einsum, matrix multiply, and sparse operations) on modern hardware. Key properties include:

• Scalability: Tensor operations are highly parallel and map efficiently to GPU/TPU architectures. Sparse tensors representing logic programs correspond to query structures in databases, while dense tensors are streamed through neural hardware.
• Transparency: Every step is an explicit computation via a sequence of tensor equations, unlike black-box neural networks. Intermediate tensors (joins, projections, embeddings) are accessible for inspection.
• Differentiability: Since each tensor operation is differentiable, systems (including rules, predicates, and embeddings) can be trained end-to-end via gradients, providing natural integration of learning and structured reasoning.
• Soundness: Symbolic reasoning is explicitly implemented and can be made exact (Boolean limit), eliminating the risk of “hallucinations” that affect purely neural approaches.

A plausible implication is that tensor logic, being fundamentally transparent and sound, supports rigorous reasoning and learning in safety-critical systems, where traceability is essential.

6. Comparison with Other Neurosymbolic Paradigms

Tensor logic stands out by using a single language to unify neural, probabilistic, and symbolic methods, which are traditionally treated with heterogeneous tools. In comparison:

• Logic Tensor Networks (LTNs): LTNs ground logical terms and predicates into real-valued tensors in a continuous space for differentiable reasoning (Serafini et al., 2016, Badreddine et al., 2020, Carraro et al., 24 Sep 2024).
• T-PRISM: Uses tensor embeddings for atoms and an algebraic framework based on Einstein’s notation, but retains separation between symbolic and numerical computation (Kojima et al., 2019).
• Differentiable Probabilistic Logic Networks: Focus on chain-of-reasoning as differentiable computation graphs over tensor-valued truth values (Potapov et al., 2019).
• TensorLog: Expresses logic programs as differentiable tensor operations by unrolling belief propagation on factor graphs (Cohen, 2016, Cohen et al., 2017).

Tensor logic is distinct in its claim that all these operations—not only symbolic or neural, but also kernel, graphical, and embedding-based models—can be subsumed within a common language of tensor equations with Einstein summation as its sole primitive.

7. Implications for the Future of AI

The tensor logic framework potentially provides the foundation for a new generation of AI languages and systems:

• Unified Language: Removes the fragmentation between logic programming, neural computation, and statistical inference by expressing all in tensor equations.
• Learnable, Reasonable, and Scalable: Enables seamless, end-to-end differentiable learning and sound reasoning at the scale of the largest neural architectures.
• Reliability and Transparency: Supports explicit tracing, debugging, and validation of both neural and symbolic reasoning chains, addressing critical limitations in current deep learning systems.
• Adoption in Safety-Critical AI: The explicit, inspectable, and verifiable nature of tensor logic suggests strong applicability in domains where correctness guarantees are required.

Tensor logic, by combining the mathematical rigor of symbolic AI with the scalability and data-adaptivity of modern deep learning, delineates a path toward robust, transparent, and integrated AI systems (Domingos, 14 Oct 2025).