Metric Temporal Graph Logic over Typed Attributed Graphs

Abstract

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.

Optimistic and Pessimistic On-the-fly Analysis for Metric Temporal Graph Logic

“…, and (i G , α) |= φ, then G |= φ. This novel adaptation of the satisfaction relation (informally handled in [8]) allows to express quantification over attribute values as required for our on-thefly analysis procedure in Sect. 4.…”

“…For specifying desired system behavior such as the property P from above, we employ an extension of the Metric Temporal Graph Logic (MTGL) [8]. This logic permits to concisely express (a) state properties to specify single graphs in a TGS and (b) sequence properties relating graphs at different timepoints in a TGS by their attributes and inner structure.…”

“…The main contributions of this paper are as follows. Firstly, we integrate the metric-exists-new operator as a first-class citizen into MTGL, which was not explicitly done in [8]. It matches graph patterns as early as possible to fix a timepoint from which a deadline can be started as in our running example when a new task is matched.…”

“…The optimistic lower bound does not contain all true violations to handle TGSs that are continued in a way leading to satisfaction in the future whereas the pessimistic upper bound contains additional false violations to handle TGSs that are continued in a way not leading to satisfaction in the future. Returning the optimistic bound in addition to the pessimistic bound as computed in [8] results essentially in a three valued logic 1 where an intervention (e.g. by a user) may depend on whether a pessimistic violation is also an optimistic violation.…”

Modeling Adverse Conditions in the Framework of Graph Transformation Systems

“…In [10], a metric temporal logic over typed attributed graphs is introduced. The atoms are graph conditions.…”