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

TL;DR

This paper introduces the Program Hypergraph, a multi-way relational structure that extends previous graph representations to better model geometric algebra, spatial compute, and hardware compilation constraints.

Contribution

It generalizes the Program Semantic Graph to hyperedges, enabling accurate representation of multi-way relations in geometric algebra and spatial dataflow architectures.

Findings

Grade in Clifford algebra aligns with existing type-theoretic frameworks.

Hypergraph formalism captures mesh topology as k-simplex structures.

The framework enables joint derivation of geometric correctness and hardware partitioning.

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.