Rewriting Graphically with Symmetric Traced Monoidal Categories

TL;DR

This paper introduces interfaced linear hypergraphs as a new graphical language for symmetric traced monoidal categories, enabling sound and complete graph rewriting suitable for Cartesian structures like digital circuits.

Contribution

It develops a novel hypergraph-based language for STMCs that is sound, complete, and compatible with Cartesian structures, and applies adhesive category theory for graph rewriting.

Findings

Interfaced linear hypergraphs accurately model STMC morphisms.

The language supports sound and complete graph rewriting.

Applied to digital circuits as a case study.

Abstract

We examine a variant of hypergraphs that we call interfaced linear hypergraphs, with the aim of creating a sound and complete graphical language for symmetric traced monoidal categories (STMCs) suitable for graph rewriting. In particular, we are interested in rewriting for categorical settings with a Cartesian structure, such as digital circuits. These are incompatible with previous languages where the trace is constructed using a compact closed or Frobenius structure, as combining these with Cartesian product can lead to degenerate diagrams. Instead we must consider an approach where the trace is constructed as an atomic operation. Interfaced linear hypergraphs are defined as regular hypergraphs in which each vertex is the source and target of exactly one edge each, equipped with an additional interface edge. The morphisms of a freely generated STMC are interpreted as interfaced linear hypergraphs, up to isomorphism (soundness). Moreover, any linear hypergraph is the representation of a unique STMC morphism, up to the equational theory of the category (completeness). This establishes interfaced linear hypergraphs as a suitable combinatorial language for STMCs. We then show how we can apply the theory of adhesive categories to our graphical language, meaning that a broad range of equational properties of STMCs can be specified as a graph rewriting system. The graphical language of digital circuits is presented as a case study.