Formal testing of timed graph transformation systems using metric temporal graph logic

Abstract

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.

Probabilistic Metric Temporal Graph Logic

“…Using the variation of symbolic graphs [15] from [17], we consider typed attributed graphs (short graphs) (such as G 0 in Figure 2b), which are typed over a type graph TG (such as TG in Figure 2a). In such graphs, attributes are connected to local variables and an Attribute Condition (AC) over a many sorted first-order attribute logic is used to specify the values for these variables.…”

“…Lastly, we denote monomorphisms (short monos) by m : G 1 G 2 . Graph Conditions (GCs) [17,6] of GL are used to state properties on graphs requiring the presence or absence of certain subgraphs in a host graph.…”

“…Cyber-physical systems often encompass complex concurrent behavior with timing constraints and probabilistic failures on demand [16,17]. Such behavior can then be captured in terms of probabilistic timed state sequences (or spaces) where time may elapse between successive states and where each step in such a sequence has a designated probability.…”

“…However, structural changes over time in the state space cannot always be directly specified using APs that are locally evaluated for each graph. To express such structural changes over time, we introduced MTGL [5,17] based on GL. Using MTGL conditions, an unbounded number of subgraphs can be tracked over timed graph transformation steps in a considered state sequence once bindings have been established for them via graph matching.…”

“…Moreover, we present a BMC approach for PTGTSs w.r.t. PMTGL properties by combining the PTCTL model checking approach for PTGTSs from [13] (which is based on a translation of PTGTSs into PTA) with the satisfaction checking approach for MTGL from [5,17]. In our approach, we just support bounded model checking since the binding capabilities of PMTGL conditions require non-local satisfaction checking taking possibly the entire history of a (finite) path into account as for MTGL conditions.…”