A General Framework for Well-Structured Graph Transformation Systems

Graph transformation systems (GTSs) can be seen as wellstructured transition systems (WSTSs), thus obtaining decidability results for certain classes of GTSs. In earlier work it was shown that wellstructuredness can be obtained using the minor ordering as a well-quasiorder. In this paper we extend this idea to obtain a general framework in which several types of GTSs can be seen as (restricted) WSTSs. We instantiate this framework with the subgraph ordering and the induced subgraph ordering and apply it to analyse a simple access rights management system.Comment: Extended version (including proofs) of a paper accepted at CONCUR 201

Lattice-extended Coloured Petri Net Rewriting for Adaptable User Interface Models

Adaptable user interfaces (UI) have shown a great variety of advantages in human computer interaction compared to classic UI designs. We show how adaptable UIs can be built by introducing coloured Petri nets to connect the UI’s physical representation with the system to be controlled. UI development benefits from formal modelling approaches regarding the derived close integration of creation, execution, and reconfiguration of formal UI models. Thus, adaptation does not only change the physical representation, but also the connecting Petri net. For the latter transformation, we enhance the DPO rewriting formalism by using an order on the set of labels and softening the label-preserving property of morphisms, i.e., an element can also be mapped to another element if the label is larger. We use lattices to ensure correctness and state application conditions of rewriting steps. Finally we define an order compatible with our framework for the use in our implementation

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

Construction of Pushout Complements in the Category of Hypergraphs

We describe a concrete construction of all pushout complements for two given morphisms f : A -> B, m: B -> D in the category of hypergraphs, valid also for the case where f, m are non-injective. It is based on the generation of suitable equivalence relations. We also give a combinatorial interpretation and show how well-known coefficients from combinatorics, such as the Bell numbers, can be recovered. Furthermore we present a formula that can be used to compute the number of pushout complements for two given morphisms

Construction of pushout complements in the category of hypergraphs

Abstract. We describe a concrete construction of all pushout complements for two given morphisms f : A → B, m : B → D in the category of hypergraphs, valid also for the case where f, m are non-injective. To our knowledge such a construction has not been discussed before in the literature. It is based on the generation of suitable equivalence relations. We also give a combinatorial interpretation and show how well-known coefficients from combinatorics, such as the Bell numbers, can be recovered

On the Decidability Status of Reachability and Coverability in Graph Transformation Systems

We study decidability issues for reachability problems in graph transformation systems, a powerful infinite-state model. For a fixed initial configuration, we consider reachability of an entirely specified configuration and of a configuration that satisfies a given pattern (coverability). The former is a fundamental problem for any computational model, the latter is strictly related to verification of safety properties in which the pattern specifies an infinite set of bad configurations. In this paper we reformulate results obtained, e.g., for context-free graph grammars and concurrency models, such as Petri nets, in the more general setting of graph transformation systems and study new results for classes of models obtained by adding constraints on the form of reduction rules