Various kinds of typed attributed graphs can be used to represent states of systems from a broad range of domains. For dynamic systems, established formalisms such as graph transformation can provide a formal model for defining state sequences. We consider the case where time may elapse between state changes and introduce a logic, called Metric Temporal Graph Logic (MTGL), to reason about such timed graph sequences. With this logic, we express properties on the structure and attributes of states as well as on the occurrence of states over time that are related by their inner structure, which no formal logic over graphs concisely accomplishes so far. Firstly, based on timed graph sequences as models for system evolution, we define MTGL by integrating the temporal operator until with time bounds into the well-established logic of (nested) graph conditions. Secondly, we outline how a finite timed graph sequence can be represented as a single graph containing all changes over time (called graph with history), how the satisfaction of MTGL conditions can be defined for such a graph and show that both representations satisfy the same MTGL conditions. Thirdly, we present how MTGL conditions can be reduced to (nested) graph conditions and show using this reduction that both underlying logics are equally expressive. Finally, we present an extension of the tool AutoGraph allowing to check the satisfaction of MTGL conditions for timed graph sequences, by checking the satisfaction of the (nested) graph conditions, obtained using the proposed reduction, for the graph with history corresponding to the timed graph sequence.

Today, software has become an intrinsic part of complex distributed embedded real-time systems. The next generation of embedded real-time systems will interconnect the today unconnected systems via complex software parts and the service-oriented paradigm. Therefore besides timed behavior and probabilistic behavior also structure dynamics, where the architecture can be subject to changes at run-time, e.g. when dynamic binding of service end-points is employed or complex collaborations are established dynamically, is required. However, a modeling and analysis approach that combines all these necessary aspects does not exist so far.To fill the identified gap, we propose Probabilistic Timed Graph Transformation Systems (PTGTSs) as a high-level description language that supports all the necessary aspects of structure dynamics, timed behavior, and probabilistic behavior. We introduce the formal model of PTGTSs in this paper and present a mapping of models with finite state spaces to probabilistic timed automata (PTA) that allows to use the PRISM model checker to analyze PTGTS models with respect to PTCTL properties. 5

Embedded real-time systems generate state sequences where time elapses between state changes. Ensuring that such systems adhere to a provided specification of admissible or desired behavior is essential. Formal model-based testing is often a suitable cost-effective approach. We introduce an extended version of the formalism of symbolic graphs, which encompasses types as well as attributes, for representing states of dynamic systems. Relying on this extension of symbolic graphs, we present a novel formalism of timed graph transformation systems (TGTSs) that supports the model-based development of dynamic real-time systems at an abstract level where possible state changes and delays are specified by graph transformation rules. We then introduce an extended form of the metric temporal graph logic (MTGL) with increased expressiveness to improve the applicability of MTGL for the specification of timed graph sequences generated by a TGTS. Based on the metric temporal operators of MTGL and its built-in graph binding mechanics, we express properties on the structure and attributes of graphs as well as on the occurrence of graphs over time that are related by their inner structure. We provide formal support for checking whether a single generated timed graph sequence adheres to a provided MTGL specification. Relying on this logical foundation, we develop a testing framework for TGTSs that are specified using MTGL. Lastly, we apply this testing framework to a running example by using our prototypical implementation in the tool AutoGraph.