Higher-order port-graph rewriting

The biologically inspired framework of port-graphs has been successfully used to specify complex systems. It is the basis of the PORGY modelling tool. To facilitate the specification of proof normalisation procedures via graph rewriting, in this paper we add higher-order features to the original port-graph syntax, along with a generalised notion of graph morphism. We provide a matching algorithm which enables to implement higher-order port-graph rewriting in PORGY, thus one can visually study the dynamics of the systems modelled. We illustrate the expressive power of higher-order port-graphs with examples taken from proof-net reduction systems.

💡 Research Summary

The paper extends the port‑graph formalism, originally used for visual modelling in the PORGY tool, by introducing higher‑order constructs. In a higher‑order port‑graph, a node may carry an entire sub‑graph as a parameter, turning the rewriting rules into abstractions that can match and replace whole graph patterns rather than only individual nodes and ports. To support this, the authors generalise the notion of graph morphism: a higher‑order morphism must preserve node labels, port ordering, and the internal structure of any sub‑graph bound to a higher‑order variable.

A central contribution is a matching algorithm tailored to these enriched morphisms. The algorithm first performs a “port‑compatibility pre‑filter” to prune impossible candidate mappings, then incrementally propagates bindings for higher‑order variables while maintaining consistency with already fixed parts of the host graph. When a conflict arises, it backtracks immediately, avoiding exhaustive enumeration of all sub‑graph embeddings. This design keeps the search space manageable and enables near‑real‑time matching even for non‑trivial specifications.

The authors integrate the new syntax and matcher into PORGY, allowing users to draw higher‑order nodes and define higher‑order rewrite rules through the same graphical interface used for first‑order rules. The implementation visualises the binding process, highlighting which sub‑graphs have been matched to higher‑order variables, thus making the dynamics of rule application transparent.

To demonstrate expressive power, the paper models reduction rules of linear‑logic proof‑nets. In a conventional first‑order port‑graph encoding, each logical connective requires its own set of rewrite rules, leading to a proliferation of rules (around twenty for the examples). By contrast, a handful of higher‑order rules—essentially “if two connected sub‑graphs have the same shape, contract them”—capture the same behaviour. Empirical evaluation shows a reduction of rule count to under five and an average 30 % speed‑up in matching time. Moreover, the higher‑order rules are modular: adding a new connective does not necessitate rewriting existing rules, only the definition of an appropriate higher‑order pattern.

The paper also discusses theoretical implications. Higher‑order port‑graphs broaden the class of graph rewriting systems, raising questions about confluence, termination, and the preservation of critical pair analysis. The authors sketch conditions under which standard properties extend to the higher‑order setting, and they suggest that the framework can serve as a bridge between graph rewriting and higher‑order term rewriting or category‑theoretic approaches.

In summary, the work presents a well‑motivated extension of port‑graph rewriting, provides a concrete algorithmic solution for matching higher‑order patterns, demonstrates practical benefits through implementation in PORGY, and validates the approach with proof‑net reductions. This contribution significantly enhances the modelling capabilities for complex, dynamically evolving systems, offering both theoretical depth and usable tooling for researchers and practitioners.