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We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting with strong matching. Our approach, called PBPO + , allows more control over the embedding of the pattern in the host graph, which is important for a large class of rewrite systems. We argue that PBPO + can be considered a unifying theory in the general setting of quasitoposes, by demonstrating that PBPO + can define all rewrite relations definable by PBPO, AGREE and DPO, as well as additional ones. Additionally, we show that PBPO + is well suited for rewriting labeled graphs and some classes of attributed graphs, by introducing a lattice structure on the label set and requiring graph morphisms to be order-preserving.
The basic principle of graph rewriting is the stepwise replacement of subgraphs inside a host graph. A challenge in such replacement steps is the treatment of the patch graph, consisting of those edges of the host graph that touch the subgraph, but are not part of it. We introduce patch graph rewriting, a visual graph rewriting language with precise formal semantics. The language has rich expressive power in two ways. First, rewrite rules can flexibly constrain the permitted shapes of patches touching matching subgraphs. Second, rules can freely transform patches. We highlight the framework's distinguishing features by comparing it against existing approaches.
We provide a tutorial introduction to the algebraic graph rewriting formalism PBPO + . We show how PBPO + can be obtained by composing a few simple building blocks, and model the reduction rules for binary decision diagrams as an example. Along the way, we comment on how alternative design decisions lead to related formalisms in the literature, such as DPO. We close with a detailed comparison with Bauderon's double pullback approach.
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