Enhancing a Hierarchical Graph Rewriting Language based on MELL Cut Elimination

Hierarchical graph rewriting is a highly expressive computational formalism that manipulates graphs enhanced with box structures for representing hierarchies. It has provided the foundations of various graph-based modeling tools, but the design of high-level declarative languages based on hierarchical graph rewriting is still a challenge. For a solid design choice, well-established formalisms with backgrounds other than graph rewriting would provide useful guidelines. Proof nets of Multiplicative Exponential Linear Logic (MELL) is such a framework because its original formulation of cut elimination is essentially graph rewriting involving box structures, where so-called Promotion Boxes with an indefinite number of non-local edges may be cloned, migrated and deleted. This work builds on LMNtal as a declarative language based on hierarchical (port) graph rewriting, and discusses how it can be extended to support the above operations on Promotion Boxes of MELL proof nets. LMNtal thus extended turns out to be a practical graph rewriting language that has strong affinity with MELL proof nets. The language features provided are general enough to encode other well-established models of concurrency. Using the toolchain of LMNtal that provides state-space search and model checking, we implemented cut elimination rules of MELL proof nets in extended LMNtal and demonstrated that the platform could serve as a useful workbench for proof nets.

💡 Research Summary

The paper presents a substantial extension of LMNtal, a hierarchical (port) graph rewriting language, to directly support the operations required by Multiplicative Exponential Linear Logic (MELL) proof nets, specifically the cloning, migration, and deletion of promotion boxes that are central to MELL’s cut‑elimination process. The authors begin by motivating the need for a high‑level declarative language capable of handling both connectivity and hierarchy, noting that existing hierarchical graph rewriting formalisms lack practical, expressive constructs for the non‑local, potentially unbounded manipulations found in proof‑net cut elimination. They identify MELL proof nets as an ideal formalism whose cut‑elimination rules are essentially graph rewriting steps involving boxes, and they choose LMNtal as a starting point because it already provides a clean term‑based syntax, structural congruence for interpreting terms as graphs, and a small‑step reduction semantics, together with a mature toolchain (SLIM) that offers state‑space exploration and model checking.

The core contributions are threefold. First, the authors introduce new language constructs that enable the precise cloning and deletion of membranes (boxes) with an arbitrary number of free links. This is realized through a library called nlmem (non‑linear membrane) which supplies two primitive operations: nlmem.copy(A,B,C) to duplicate a membrane A, inserting a fresh atom B around its free links and providing an access link C to both original and copy; and nlmem.kill(A,B) to safely remove a membrane A together with its free links, binding them to a placeholder atom B. These operations are expressed using LMNtal’s existing notion of bundles (*X) to capture an unspecified collection of free links, ensuring that after cloning or killing no dangling links remain. Second, the paper maps each of MELL’s cut‑elimination rules—ax‑cut, tensor‑cut, promotion‑cut—onto LMNtal rewrite rules that employ the new nlmem primitives. The mapping preserves the logical meaning of each step while remaining within the structural congruence framework, allowing the LMNtal runtime to treat proof‑net reduction as ordinary graph rewriting. Third, to demonstrate the generality of the extension, the authors encode the Ambient Calculus, a well‑known concurrency model that also relies on hierarchical box structures for mobility and restriction. The encoding shows that ambient operations (in, out, open) can be expressed using the same cloning and deletion primitives, confirming that the language is not limited to MELL but can serve as a broader workbench for hierarchical concurrent systems.

Implementation details are provided for integrating the new primitives into the LMNtal runtime. The authors modify the rule parser to recognize nlmem calls, extend the matching engine to handle bundles with arbitrary free links, and ensure that the state‑space generator respects LMNtal’s built‑in symmetry reduction (graph isomorphism modulo structural congruence). They validate the approach by running cut‑elimination on several example proof nets, visualizing each intermediate graph with the State Viewer, and checking LTL properties with the built‑in model checker. The experiments confirm that promotion boxes can be cloned an indefinite number of times without violating the link condition, and that deletion removes boxes cleanly without leaving orphaned edges.

In the discussion, the authors outline future work, including support for additional linear‑logic constructs such as dereliction and digging, performance optimizations through graph compression, and comparative studies with other concurrency formalisms like the π‑calculus or Petri nets. Overall, the paper bridges the gap between hierarchical graph rewriting languages and linear‑logic proof‑net semantics, delivering a practical, extensible platform that can be used both for formal proof‑net manipulation and for modeling a wide range of hierarchical concurrent systems.