Library / AI And Mathematics

Tensor Expression Graphs

Modern AI systems depend on matrix and tensor expressions. Those expressions can also be treated symbolically, which makes it possible to rewrite, compare, and optimize them before execution.

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Tensor Perspective

Expressions Before Kernels

A tensor program can be viewed as an expression graph built from operations such as matrix multiply, elementwise multiplication, addition, transpose, normalization, and activation functions. Before those operations are lowered to kernels or compiled code, they still have algebraic structure.

That structure is useful. It allows a symbolic system to detect equivalent forms, identify fusion opportunities, and compare alternatives using a cost model that reflects shape, operation count, or execution goals.

Sym Relevance

What Sym Already Handles

Sym includes tensor-oriented operations and a tensor cost model. The codebase works with operators such as MatMul, TensorAdd, TensorMul, and Transpose, and the tests include rewrite-based fusion of expressions like:

Relu(TensorAdd(MatMul(A, B), C)) -> FusedMatMulAddRelu(A, B, C)

That is important in an AI context because it shows symbolic reasoning applied to the kinds of tensor equations that appear in real model pipelines.

Optimization

Why Rewrite Tensor Graphs

The same tensor computation can often be expressed in multiple equivalent ways. A rewrite system can search those alternatives and prefer one with better fusion, lower cost, or clearer structure.

That makes tensor graphs a natural target for the same kinds of techniques used in symbolic optimization more generally: rewriting, equivalence classes, and cost-guided extraction.

AI Link

Symbolic Reasoning Meets ML Infrastructure

This is one place where symbolic computation and AI overlap directly. AI uses tensor programs, and symbolic systems can reason about those programs before they become low-level execution plans.

That overlap matters because it gives symbolic methods a role in modern workloads that is concrete rather than merely historical.

Cost Models

Equivalent Forms Are Not Equally Useful

Two tensor expressions may be mathematically equivalent while differing significantly in runtime cost, memory traffic, or fusion opportunities. A symbolic optimizer therefore needs more than rewrite rules. It needs a way to compare results. That is where cost models come in. A cost model can reflect shape, operation count, extraction preference, or another domain-specific objective.

This is one reason tensor optimization fits naturally with symbolic methods. Once the expressions are represented structurally, equivalence and preference can be handled separately.

E-Graphs Tensor Contraction Open Tensor Demo

GPU Context

Why This Matters For AI Infrastructure

GPU workloads are often described operationally in terms of kernels, launches, and memory layouts. But before those implementation details take over, there is usually a higher-level tensor expression that can be reasoned about. That is the layer where symbolic optimization can detect fusion patterns, expose shared structure, and choose more efficient forms before execution is fixed.

In Sym, this shows up in a concrete way through tensor-aware operators, rule packs, and cost models. The system is already able to recognize and improve some fused tensor-style expressions, which makes it a useful example of symbolic computation meeting AI-oriented workloads.

Representation

Why Graph Form Matters Before Execution

Once a tensor computation is represented as a graph, subexpressions, dependencies, and possible fusion boundaries become visible in a way that flat code often obscures. That visibility matters for both optimization and explanation.

It also makes the connection to symbolic computation more direct: the graph is a structured object that can be rewritten and compared rather than a fixed sequence of opaque steps.

Bigger Picture

Why This Topic Belongs In The Library

Tensor expression graphs are a good example of symbolic ideas leaving the textbook and entering modern infrastructure. They show that representation, equivalence, and extraction are not only concerns for algebra systems. They matter in the optimization of real AI workloads too.

That makes this topic a bridge between the symbolic-computation section and the broader AI section.