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The Program Hypergraph: Multi-Way Relational Structure for Geometric Algebra, Spatial Compute, and Physics-Aware Compilation

Houston Haynes

Abstract

The Program Semantic Graph (PSG) introduced in prior work on Dimensional Type Systems and Deterministic Memory Management encodes compilation-relevant properties as binary edge relations between computation nodes. This representation is adequate for scalar and tensor computations, but becomes structurally insufficient for two classes of problems central to heterogeneous compute: tile co-location and routing constraints in spatial dataflow architectures, which are inherently multi-way; and geometric algebra computation, where graded multi-way products cannot be faithfully represented as sequences of binary operations without loss of algebraic identity. This paper introduces the Program Hypergraph (PHG) as a principled generalization of the PSG that promotes binary edges to hyperedges of arbitrary arity. We demonstrate that grade in Clifford algebra is a natural dimension axis within the existing DTS abelian group framework, that grade inference derives geometric product sparsity eliminating the primary performance objection to Clifford algebra neural networks without manual specialization, and that the k-simplex structure of mesh topology is a direct instance of the hyperedge formalism. We assess the existing geometric algebra library ecosystem, identify the consistent type-theoretic gap that no current system addresses, and show that the PHG closes it within the Fidelity compilation framework. The result is a compilation framework where geometric correctness, memory placement, numerical precision selection, and hardware partitioning are jointly derivable from a single graph structure exposed as interactive design-time feedback.

The Program Hypergraph: Multi-Way Relational Structure for Geometric Algebra, Spatial Compute, and Physics-Aware Compilation

Abstract

The Program Semantic Graph (PSG) introduced in prior work on Dimensional Type Systems and Deterministic Memory Management encodes compilation-relevant properties as binary edge relations between computation nodes. This representation is adequate for scalar and tensor computations, but becomes structurally insufficient for two classes of problems central to heterogeneous compute: tile co-location and routing constraints in spatial dataflow architectures, which are inherently multi-way; and geometric algebra computation, where graded multi-way products cannot be faithfully represented as sequences of binary operations without loss of algebraic identity. This paper introduces the Program Hypergraph (PHG) as a principled generalization of the PSG that promotes binary edges to hyperedges of arbitrary arity. We demonstrate that grade in Clifford algebra is a natural dimension axis within the existing DTS abelian group framework, that grade inference derives geometric product sparsity eliminating the primary performance objection to Clifford algebra neural networks without manual specialization, and that the k-simplex structure of mesh topology is a direct instance of the hyperedge formalism. We assess the existing geometric algebra library ecosystem, identify the consistent type-theoretic gap that no current system addresses, and show that the PHG closes it within the Fidelity compilation framework. The result is a compilation framework where geometric correctness, memory placement, numerical precision selection, and hardware partitioning are jointly derivable from a single graph structure exposed as interactive design-time feedback.
Paper Structure (48 sections, 2 theorems, 10 equations, 2 figures)

This paper contains 48 sections, 2 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Binary Edge Boundary
  3. Contributions
  4. Scope and Context
  5. Intellectual Continuity: The DCont/Inet Headwaters
  6. The Program Hypergraph: Formal Structure
  7. From Binary Edges to Hyperedges
  8. Hyperedge Annotation Categories
  9. Saturation Semantics for Hyperedges
  10. The Binary Edge as Degenerate Hyperedge
  11. Geometric Algebra as a PHG Domain
  12. A Primer on Clifford Algebras
  13. The Graded Structure of Clifford Algebras
  14. Geometric Product Sparsity from Grade Inference
  15. The $k$-Simplex as a Native PHG Hyperedge
  16. ...and 33 more sections

Key Result

Proposition 3.1

The grade of a Clifford algebra element satisfies the axioms of a DTS dimension: it is an element of a finitely generated abelian group $(\mathbb{Z}, +)$, it is preserved under the outer product (addition), and its consistency is decidable in $O(1)$ per operation. The DTS inference algorithm of Sect

Figures (2)

  • Figure 1: The PHG within the Fidelity/Composer compilation pipeline. Three primary application domains feed into the PHG through grade, co-location, and topology hyperedges. The pipeline fans out to LLVM, CIRCT, and MLIR-AIE backends from a shared MLIR middle-end.
  • Figure 2: Route topology for the $2\times2$ tile co-location hyperedge. The DMA channel (dashed) connects load$_A$ and reduce$_D$ directly, in addition to the data flow through compute stages $B$ and $C$.

Theorems & Definitions (3)

  • Definition 2.1: Program Hypergraph
  • Proposition 3.1: Grade is a DTS Dimension Axis
  • Proposition 3.2: BSP Predicate Exactness from PHG Grade Inference