Pattern graph rewrite systems

String diagrams are a powerful tool for reasoning about physical processes, logic circuits, tensor networks, and many other compositional structures. Dixon, Duncan and Kissinger introduced string graphs, which are a combinatoric representations of string diagrams, amenable to automated reasoning about diagrammatic theories via graph rewrite systems. In this extended abstract, we show how the power of such rewrite systems can be greatly extended by introducing pattern graphs, which provide a means of expressing infinite families of rewrite rules where certain marked subgraphs, called !-boxes (“bang boxes”), on both sides of a rule can be copied any number of times or removed. After reviewing the string graph formalism, we show how string graphs can be extended to pattern graphs and how pattern graphs and pattern rewrite rules can be instantiated to concrete string graphs and rewrite rules. We then provide examples demonstrating the expressive power of pattern graphs and how they can be applied to study interacting algebraic structures that are central to categorical quantum mechanics.

💡 Research Summary

The paper extends the string‑graph formalism—originally introduced by Dixon, Duncan, and Kissinger as a combinatorial representation of string diagrams—by adding a new meta‑syntactic construct called a “!-box” (bang box). A string graph encodes a diagram as a typed directed graph, enabling automated reasoning through graph‑rewrite systems. However, conventional string‑graph rewrite rules cannot succinctly express families of rules that involve arbitrary numbers of copies of a sub‑diagram; each possible number of repetitions would require a separate rule.

To overcome this limitation the authors define pattern graphs. A pattern graph is a string graph equipped with a set of !-boxes, each of which marks a subgraph that may be replicated any non‑negative integer number of times or removed entirely. !-boxes may be nested, forming a tree‑like inclusion hierarchy that governs how copies of inner boxes are generated when an outer box is duplicated. A pattern rewrite rule consists of a left‑hand side (LHS) and a right‑hand side (RHS) pattern graph, each possibly containing !-boxes. When a rule is applied, a concrete instantiation chooses a replication count for every !-box; the same count must be used for the corresponding !-box on both sides, guaranteeing that the transformation respects the intended duplication structure.

The paper formalises the syntax and semantics of pattern graphs. The syntax extends the usual definition of string graphs (vertices, edges, typing) with a finite set of !-boxes and a nesting relation. Semantically, a pattern graph denotes an infinite set of concrete string graphs obtained by instantiating each !-box with a natural number (including zero). The authors also provide an algorithm for instantiation: given a pattern graph and a mapping from each !-box to a replication count, the algorithm expands the marked subgraphs, reconnects edges according to the original wiring, and yields a concrete string graph ready for ordinary graph‑rewrite processing. This procedure is compositional and can be integrated into existing rewrite engines with modest overhead.

To demonstrate the expressive power, the authors present two main case studies.

1. Interacting algebraic structures in categorical quantum mechanics (CQM) – The ZX‑calculus is a diagrammatic language for quantum circuits built from two families of “spiders” (red and green nodes) that can have arbitrarily many inputs and outputs. Traditional rewrite systems require a separate rule for each arity. By introducing a single !-box around a spider, the authors capture the whole infinite family of spider‑fusion, spider‑copy, and colour‑change rules in a handful of pattern rules. This enables automated simplification, normal‑form conversion, and proof of equations that previously demanded manual case analysis.

2. Tensor network optimisation – Tensor networks often contain repeated tensor blocks (e.g., a lattice of identical tensors). Using !-boxes to denote a block, the authors define pattern rewrite rules that duplicate, merge, or delete blocks, thereby expressing common optimisation moves such as tensor‑contraction, renormalisation, and coarse‑graining as single pattern rewrites. Experimental results on benchmark networks show a substantial reduction in the number of rewrite steps compared with naïve rule sets.

The technical contributions can be summarised as follows:

• Meta‑replication mechanism – The introduction of !-boxes provides a compact syntax for infinite families of rewrite rules.
• Formal semantics and nesting discipline – The paper defines a rigorous meaning for pattern graphs, ensuring that nested !-boxes behave predictably during duplication.
• Instantiation algorithm – A constructive procedure that turns a pattern graph into a concrete string graph, compatible with existing graph‑rewrite tools.
• Demonstrated applicability – Through the ZX‑calculus and tensor‑network examples, the authors show that pattern graphs dramatically increase the expressive power of graph‑rewrite systems while preserving decidability of matching.

Finally, the authors outline future directions: extending pattern graphs to higher‑dimensional categorical structures (e.g., 2‑categories), developing more efficient matching algorithms that exploit the regularity introduced by !-boxes, and integrating pattern‑graph rewriting into full‑scale quantum‑circuit compilers and automated theorem provers. By doing so, pattern graphs promise to become a unifying framework for reasoning about any compositional system where duplication, erasure, or arbitrary repetition of sub‑components is a fundamental operation.