We introduce techniques for proving uniform termination of graph transformation systems, based on matrix interpretations for string rewriting. We generalize this technique by adapting it to graph rewriting instead of string rewriting and by generalizing to ordered semirings. In this way we obtain a framework which includes the tropical and arctic type graphs of [5] and a new variant of arithmetic type graphs. These type graphs can be used to assign weights to graphs and to show that these weights decrease in every rewriting step in order to prove termination. We present an example involving counters and discuss the implementation in the tool Grez. 4
Graph transformation or graph rewriting has been developed for nearly 50 years and has become a mature and manifold formal technique. Basically, rewrite rules are used to manipulate graphs. These rules are given by a left-hand side and a right-hand side graph and the application comprises matching the left-hand side and replacing it with the right-hand side of the rule. In this contribution we give a tutorial on graph transformation that explains the so-called double-pushout approach to graph transformation in a rigorous, but non-categorical way, using a gluing construction. We explicate the definitions with several small examples. We also introduce attributes and attributed graph transformation in a lightweight form. The paper is concluded by a more extensive example on a leader election protocol, the description of tool support and pointers to related work.
We investigate three formalisms to specify graph languages, i.e. sets of graphs, based on type graphs. First, we are interested in (pure) type graphs, where the corresponding language consists of all graphs that can be mapped homomorphically to a given type graph. In this context, we also study languages specified by restriction graphs and their relation to type graphs. Second, we extend this basic approach to a type graph logic and, third, to type graphs with annotations. We present decidability results and closure properties for each of the formalisms. PreliminariesWe first introduce graphs and graph morphisms. In the context of this paper we use edge-labeled, directed graphs.Definition 1 (Graph). Let Λ be a fixed set of edge labels. A Λ-labeled graph is a tuple G = V, E, src, tgt , lab , where V is a finite set of nodes, E is a finite set of edges, src, tgt : E → V assign to each edge a source and a target node, and lab : E → Λ is a labeling function.
The paper develops an abstract (over-approximating) semantics for double-pushout rewriting of graphs and graph-like objects. The focus is on the socalled materialization of left-hand sides from abstract graphs, a central concept in previous work. The first contribution is an accessible, general explanation of how materializations arise from universal properties and categorical constructions, in particular partial map classifiers, in a topos. Second, we introduce an extension by enriching objects with annotations and give a precise characterization of strongest post-conditions, which are effectively computable under certain assumptions.Abstract interpretation [12] is a fundamental static analysis technique that applies not only to conventional programs but also to general infinite-state systems. Shape analysis [32], a specific instance of abstract interpretation, pioneered an approach for analyzing pointer structures that keeps track of information about the "heap topology", e.g., out-degrees or existence of certain paths. One central idea of shape analysis is materialization, which arises as companion operation to summarizing distinct objects that share relevant properties. Materialization, a.k.a. partial concretization, is also fundamental in verification approaches based on separation logic [6,5,25], where it is also known as rearrangement [28], a special case of frame inference. Shape analysis-construed in a wide sense-has been adapted to graph transformation [31], a general purpose modelling language for systems with dynamically evolving topology, such as network protocols and cyber-physical systems. Motivated by earlier work of shape analysis for graph transformation [33,4,1,2,30,29], we want to put the materialization operation on a new footing, widening the scope of shape analysis.A natural abstraction mechanism for transition systems with graphs as states "summarizes" all graphs over a specific shape graph. Thus a single graph is used as abstraction ⋆ Partially supported by AFOSR.⊳ A rewriting formalism for graph abstractions that lifts the rule-based rewriting from single graphs to abstract graphs; it is developed for (abstract) objects in a topos.⊳ We characterize the materialization operation for abstract objects in a topos in terms of partial map classifiers, giving a sound and complete description of all occurrences of right-hand sides of rules obtained by rewriting an abstract object.→ Sec. 3 ⊳ We decorate abstract objects with annotations from an ordered monoid and extend abstract rewriting to abstract objects with annotations. For the specific case of graphs, we consider global annotations (counting the nodes and edges in a graph), local annotations (constraining the degree of a node), and path annotations (constraining the existence of paths between certain nodes). → Sec. 4 ⊳ We show that abstract rewriting with annotations is sound and, with additional assumptions, complete. Finally, we even derive strongest post-conditions for the case of graph rewriting with annotations. → Sec. 5Related work: ...
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