Verification of Well-Structured Graph Transformation Systems
The aim of this thesis is the definition of a high-level framework for verifying concurrent and distributed systems. Verification in computer science is challenging, since models that are sufficiently expressive to describe real-life case studies suffer from the undecidability of interesting problems. This also holds for the graph transformation systems used in this thesis. To still be able to analyse these system we have to restrict either the class of systems we can model, the class of states we can express or the properties we can verify. In fact, in the framework we will present, all these limitations are possible and each allows to solve different verification problems. For modelling we use graphs as the states of the system and graph transformation rules to model state changes. More precisely, we use hypergraphs, where an edge may be incident to an arbitrary long sequence of nodes. As rule formalism we use the single pushout approach based on category theory. This provides us with a powerful formalisms that allows us to use a finite set of rules to describe an infinite transition system. To obtain decidability results while still maintaining an infinite state space we use the theory of well-structured transition systems (WSTS), the main source of decidability results in the infinite case. We need to equip our state space with a well-quasi-order (wqo) which is a simulation relation for the transition relation (this is also known as compatibility condition or monotonicity requirement). If a system can be seen as a WSTS and some additional conditions are satisfied, one can decide the coverability problem, i.e., the problem of verifying whether, from a given initial state one can reach a state that covers a final state, i.e. is larger than the final state with respect to a chosen order. This problem can be used for verification by giving a finite set of minimal error states that represent an infinite class of erroneous states (i.e. all larger states). By checking whether one of these minimal states is coverable, we verify whether an error is reachable. The theory of WSTS provides us with a generic backwards algorithm to solve this problem. For graphs we will introduce three orders, the minor ordering, the subgraph ordering and the induced subgraph ordering, and investigate which graph transformation systems form WSTS with these orders. Since only the minor ordering is a wqo on all graphs, we will first define so-called Q-restricted WSTS, where we only require that the chosen order is a wqo on the downward-closed class Q. We examine how this affects the decidability of the coverability problem and present appropriate classes Q such that the subgraph ordering and induced subgraph ordering form Q-restricted WSTS. Furthermore, we will prove the computability of the backward algorithm for these Q-restricted WSTS. More precisely, we will do this in the form of a framework and give necessary conditions for orders to be compatible with this framework. For the three mentioned orders we prove that they satisfy these conditions. Being compatible with different orders strengthens the framework in the following way: On the one hand error specifications have to be invariant wrt. the order, meaning that different orders can describe different properties. On the other hand, there is the following trade-off: coarser orders are wqos on larger sets of graphs, but fewer GTS are well-structured wrt. coarse orders (analogously the reverse holds for fine orders). Finally, we will present the tool Uncover which implements most of the theoretical framework defined in this thesis. The practical value of our approach is illustrated by several case studies and runtime results.
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