Drag Rewriting

TL;DR

This work introduces the Drag Model, a powerful algebraic framework for graph rewriting based on drags—finite directed rooted labeled ordered multisgraphs with roots and sprouts. By defining a rich algebra with sum, product, and a wiring-based composition, the authors unify term rewriting, dag rewriting, and general graph rewriting, while enabling sharing and cloning and avoiding dangling edges. They develop a full theory of drag morphisms, coherence, decomposition into atomic drags, and a rewriting calculus with congruences, rewriting extensions, and root-maps, including a categorical interpretation and a detailed comparison with DPO and term-graph approaches. The framework offers a flexible, scalable alternative to traditional graph-rewriting formalisms, with potential impact on modeling sharing, pattern matching, and non-linear rewriting in a wide range of computational settings.

Abstract

We present a new and powerful algebraic framework for graph rewriting, based on drags, a class of graphs enjoying a novel composition operator. Graphs are embellished with roots and sprouts, which can be wired together to form edges. Drags enjoy a rich algebraic structure with sums and products. Drag rewriting naturally extends graph rewriting, dag rewriting, and term rewriting models.

Figures (11)

• Figure 1: Two drags with the same subdrag but different context drags.
• Figure 2: A monomorphism ($\hookrightarrow$) on the left and a morphism ($\rightarrow$) on the right. Incoming arrows at some vertex annotated with numbers stand for multiple roots.
• Figure 3: Additional examples of monomorphisms.
• Figure 4: Formation of cycles via composition, two versions.
• Figure 5: Maximally shared form of a drag.

Theorems & Definitions (143)

• Definition 2.1: Drag
• Definition 2.2: Linear; bare; ground; empty; disjoint
• Definition 2.3: Accessibility
• Definition 2.4: Connected component
• Definition 2.5: Predecessor; indegree
• Remark 2.6
• Remark 2.7
• Definition 3.1: Subdrag; context
• Lemma 3.2
• proof