The Pullback-Pushout approach to algebraic graph transformation

Some recent algebraic approaches to graph transformation include a pullback construction involving the match, that allows one to specify the cloning of items of the host graph. We pursue further this trend by proposing the Pullback-Pushout (pb-po) Approach, where we combine smoothly the classical modifications to a host graph specified by a rule (a span of graph morphisms) with the cloning of structures specified by another rule. The approach is shown to be a conservative extension of agree (and thus of the sqpo approach), and we show that it can be extended with standard techniques to attributed graphs. We discuss conditions to ensure a form of locality of transformations, and conditions to ensure that the attribution of transformed graphs is total

Double-Pullback Transitions and Coalgebraic Loose Semantics for Graph Transformation Systems

The classical algebraic approach to graph transformation is a mathematical theory based on categorical techniques with several interesting applications in computer science. In this paper, a new semantics of graph transformation systems (in the algebraic, double-pushout (DPO) approach) is proposed in order to make them suitable for the specification of concurrent and reactive systems. Classically, a graph transformation system comes with a fixed behavioral interpretation. Firstly, all transformation steps are intended to be completely specified by the rules of the system, that is, there is an implicit frame condition: it is assumed that there is a complete control about the evolution of the system. Hence, the interaction between the system and its (possibly unknown) environment, which is essential in a reactive system, cannot be modeled explicitly. Secondly, each sequence of transformation steps represents a legal computation of the system, and this makes it difficult to model systems with control. The first issue is addressed by providing graph transformation rules with a loose semantics, allowing for unspecified effects which are interpreted as activities of the environment. This is formalized by the notion of double-pullback transitions, which replace (and generalize) the well-known double-pushout diagrams by allowing for spontaneous changes in the context of a rule application. Two characterizations of double-pullback transitions are provided: the first one describes them in terms of extended direct DPO derivations, and the second one as incomplete views of parallel or amalgamated derivations. The issue of constraining the behavior of a system to transformation sequences satisfying certain properties is addressed instead by introducing a general notion of logic of behavioral constraints, which includes instances like start graphs, application and consistency conditions, and temporal logic constraints. The loose semantics of a system with restricted behavior is defined as a category of coalgebras over a suitable functor. Such category has a final object which includes all finite and infinite transition sequences satisfying the constraints