Parameterized Verification of Graph Transformation Systems with Whole Neighbourhood Operations

Abstract

We introduce a new class of graph transformation systems in which rewrite rules can be guarded by universally quantified conditions on the neighbourhood of nodes. These conditions are defined via special graph patterns which may be transformed by the rule as well. For the new class for graph rewrite rules, we provide a symbolic procedure working on minimal representations of upward closed sets of configurations. We prove correctness and effectiveness of the procedure by a categorical presentation of rewrite rules as well as the involved order, and using results for well-structured transition systems. We apply the resulting procedure to the analysis of the Distributed Dining Philosophers protocol on an arbitrary network structure.

Well-structured graph transformation systems

“…We already introduced an extension of the presented framework with rules, which can uniformly change the entire neighborhood of nodes in [19]. In this case the computed set of predecessor graphs will be an over-approximation.…”

“…3 as well as a public-private server protocol, using the subgraph order. In addition the table contains a variant of the dining philosophers problem on an arbitrary graph structure modeled in [19]. It shows for each case the restricted graph set Q, the variant of the procedure used (for the minor ordering they coincide), the runtime and the number of minimal graphs representing all predecessors of error graphs (#EG).…”

From Well Structured Transition Systems to Program Verification

“…There exist many other illustrations of the power of WSTS to verify programs like hardware design, multithreaded programs, distributed systems. Let's quote programs with pointers and the use of graphs and orderings on graphs (subgraph ordering and minor ordering) to model the state of the memory [1], parameterized verification of distributed algorithms [8], programs with time constraints (timed Petri nets), cryptographic protocols [9], broadcast protocols,...etc.…”