A Formal Semantics of UML State Machines Based on Structured Graph Transformation

On translating UML models into graph transformation systems

In this work we show that the reachability problem for graph transformation systems is in the complexity class XP when parameterized with respect to the depth of derivations and the cutwidth of the source graph. More precisely, we show that for any set \(\mathcal {R}\) of graph transformation rules, one can determine in time \(f(c,d)\cdot |G|\cdot |H|^{g(c,d)}\) whether a graph G of cutwidth c can be transformed into a graph H in depth at most d by the application of graph transformation rules from \(\mathcal {R}\). In particular, our algorithm runs in polynomial time when c and d are constants. On the other hand, we show that the problem becomes NP-hard if we allow \(c=O(|G|)\) and \(d=5\). In the case in which all transformation rules are monotone we get an algorithm running in time \(f(c,d)\cdot |G|^{O(c)}\cdot |H|\). To prove our main theorems we will establish an interesting connection between graph transformation systems and regular slice languages. More precisely, we show that if \(\mathcal {A}\) is a slice automaton representing a set \({\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})\) of graphs, then one can construct in time linear in \(|\mathcal {A}|\) a slice automaton \(\mathcal {N}(\mathcal {A})\) representing the set of all graphs that can be obtained from graphs in \({\mathcal {L}}_{{\mathcal {G}}}(\mathcal {A})\) by the application of one layer of transformation rules in \(\mathcal {R}\).

The main idea of graph grammars and graph transformation is the rule-based graph rewriting, where the application of every rule graph rule leads to a graph transformation step. Graph grammars can be used to generate graph languages which is similar to Chomsky grammars in formal language theory. Moreover, graphs can be used to model the states of all kinds of systems, which allows one to use graph transformation to model state changes in these systems. A rule-based transformation system may show two kinds of nondeterminism: (1) for each rule can exist several matches, and (2) several rules can be applicable. This paper concentrates on the second non-determinism, studying a method to detect the rules which are not applicable in graph transformation systems, proposing a backward graph transformation conflict detecting algorithm. At first the computation of conflict and dependency of two rules is available by use of AGG . Secondly the set of dependencies has to be tested by Warshall algorithm and the set of conflicts has to be computed by backward algorithm. At last, the termination of graph transformation systems and validity of rules can be detected as well.

Graph Transformation Systems (GTSs) provide visual and explicit semantics for dynamically evolving multi-process systems such as network programs and communication protocols. Existing symmetry reduction techniques that generate a reduced, bisimilar model for alleviating state explosion in model checking are not applicable to dynamic models such as those given by GTSs. We develop symmetry reduction techniques applicable to evolving GTS models and the programs that generate them. We also provide an on-the-fly algorithm for generating a symmetry-reduced quotient model directly from a set of graph transformation rules. The generated quotient model is GTS-bisimilar to the model under verification and may be exponentially smaller than that model. Thus, analysis of the system model can be performed by checking the smaller GTS-bisimilar model.

Model checking is one of the successful techniques in automated verification of software and hardware systems. However, employing this technique for verification of properties such as the safety and liveness requires that all possible states of a system are generated and then the given property is checked. In large and complex systems, generating all states causes the state space explosion problem. Hence, it is possible to refute such properties by searching the state space to find a state in which the given property is violated. Of course, it is possible that detecting such states in complex systems leads to the exhaustive exploration of the state space. Recent researches confirm that using meta-heuristic and evolutionary approaches to intelligently explore a portion of the state space can be a promising idea. Hence, in this paper, we propose an approach based on ant colony optimization algorithm for refuting the safety and liveness properties and also analyzing reachability ones in systems specified formally through graph transformation system. Refutation of liveness and analyzing reachability properties in systems modeled by graph transformations are the prominent advantages of this approach in comparison with the previous approaches. The proposed approach is implemented in GROOVE which is an open source toolset for designing and model checking graph transformation systems. Experimental results show that our proposed approach is faster and it generates shorter counterexamples in comparison with others.

In the framework of graph transformation systems with Negative Application Conditions (NACs) the classical notion of switch equivalence of derivations is extended to permutation equivalence, because there are intuitively equivalent derivations which are not switch-equivalent if NACs are considered. By definition, two derivations are permutation-equivalent, if they respect the NACs and disregarding the NACs they are switch equivalent. A direct analysis of permutation equivalence is very complex in general, thus we propose a much more efficient analysis technique. For this purpose, we construct a Place/Transition Petri net, called dependency net, which encodes the dependencies among rule applications of the derivation, including the inhibiting effects of the NACs. The analysis of permutation equivalence is important for analysing simulation runs within development environments for systems modelled by graph transformation. The application of the technique is demonstrated by a graph transformation system within the context of workflow modelling. We show the effectiveness of the approach by comparing the minimal costs of a direct analysis with the costs of the efficient analysis applied to a derivation of our example system.

The design of business or production systems frequently necessitates to simultaneously fulfill several logical and numerical constraints as requirements in order to deliver a functionally correct and optimal system. Such a problem can be typically formulated as a combined optimization and reachability analysis. In the current paper, we show how this problem can be formalized when the evolution of the system is captured by graph transformation systems (GTS) with a cost parameter attached to each graph transformation rule denoting the cost of firing the rule. Furthermore, we discuss how to solve such problems by combining guided state space exploration with algebraic techniques of Petri nets.

The paper presents a conceptual framework for a module concept for graph transformation systems from a software engineering as well as from a theoretical point of view. The basic idea is to reuse concepts, which are known within or without the graph grammar field, to structure large specifications. These are the concept of distributed graph transformation systems, the concept of inheritance of specifications, and the import-export-interface concept. All these concepts are presented in a uniform framework based on the syntactical notion of a graph class specification and its semantics given by a graph transformation system. This is the basis for an explicit integration of these concepts and a corresponding specification language, to be discussed in a subsequent paper.

In a Parallel Communicating Grammar System the components of a grammar system can also work in parallel (each having its own rewriting form) and communicate to each other in some way(sending the currently generated resultant graph to other components). In this paper we define parallel communicating edNCE graph grammar and parallel communicating ETPL(k) graph grammar. We present the condition to determine the parsing of a parallel communicating NCE graph grammar. PCedNCE graph grammar is found to be powerful than PCETPL(k) graph grammar.

Scenario-driven analysis of systems specified through graph transformations

In this paper a graph transformation system with the parallel derivation is used to model the process of distribution and adaptation for computer aided design. It is based on earlier research in formal language theory, especially graph grammars, and distributed models. The motivation for the ideas presented here is given and possible ways of application are described. A connection to the multi-agent model is also presented.

AGG is a general development environment for algebraic graph transformation systems which follows the interpretative approach. Its special power comes from a very flexible attribution concept. AGG graphs are allowed to be attributed by any kind of Java objects. Graph transformations can be equipped with arbitrary computations on these Java objects described by a Java expression. The AGG environment consists of a graphical user interface comprising several visual editors, an interpreter, and a set of validation tools. The interpreter allows the stepwise transformation of graphs as well as rule applications as long as possible. AGG supports several kinds of validations which comprise graph parsing, consistency checking of graphs and conflict detection in concurrent transformations by critical pair analysis of graph rules. Applications of AGG include graph and rule-based modeling of software, validation of system properties by assigning a graph transformation based semantics to some system model, graph transformation based evolution of software, and the definition of visual languages based on graph grammars.

A rich concurrent semantics has been developed along the years for graph transformation systems, often generalizing in non-trivial ways concepts and results fist introduced for Petri nets. Besides the theoretical elegance, the concurrent semantic has potential applications in verification, e.g. in partial order reduction or in the use of finite prefixes of the unfolding for model checking. In practice (graph) transformation systems are often equipped with Negative Application Conditions, that describe forbidden contexts for the application of a rule. The talk will summarize some recent results showing that if the NACs are sufficiently simple (“incremental”) the concurrent semantics lifts smoothly to systems with NACs, but the general case requires original definitions and intuitions. Joint work with Reiko Heckel, Frank Hermann, Susann Gottmann and Nico Nachtigall

The new approach of distributed graphs and graph transformation developed in this article allows us to use structured graph transformation on two abstraction levels: the network and the local level. The network level contains the description of topological structures of a system. The local level covers the description of states and their transitions in local systems. Local state transitions may depend on others using suitable synchronization mechanisms. The main distribution concepts of categorical graph grammars presented by Schneider are combined with the algebraic approach to distributed graph transformation introduced by Ehrig et al. Modeling of distributed systems by this new kind of distributed graph transformation offers a clear and elegant description of dynamic networks, distributed actions as well as communication and synchronization using a graphical notation. Moreover, distributed graph transformation offers the possibility to describe splitting and joining of local graphs as well as parallel transformations in local systems. The formalization of distributed graph transformation is performed by means of category theory. A distributed graph is formalized by a diagram in the category GRAPH of graphs and total graph morphisms. A distributed transformation step is characterized by a double-pushout in the category DISTR(GRAPH) of distributed graphs and distributed graph morphisms. A pushout over distributed graph morphisms cannot always be constructed componentwise in each local part. But especially the componentwise construction of a distributed transformation reflects distributed computations best. Thus, we present the necessary conditions for componentwise construction of pushouts in DISTR(GRAPH) and show that they are also sufficient. These conditions are summarized in the distributed gluing condition. Moreover, these conditions are needed for componentwise construction of pushout complements which are used to characterize the first step of a distributed graph transformation.

Applying an attributed graph transformation rule to a given object graph always implies some kind of constraint solving. In many cases, the given constraints are almost trivial to solve. For instance, this is the case when a rule describes a transformation G ⇒ H, where the attributes of H are obtained by some simple computation from the attributes of G. However there are many other cases where the constraints to solve may be not so trivial and, moreover, may have several answers. This is the case, for instance, when the transformation process includes some kind of searching. In the current approaches to attributed graph transformation these constraints must be completely solved when defining the matching of the given transformation rule. This kind of early binding is well-known from other areas of Computer Science to be inadequate. For instance, the solution chosen for the constraints associated to a given transformation step may be not fully adequate, meaning that later, in the search for a better solution, we may need to backtrack this transformation step. In this paper, based on our previous work on the use of symbolic graphs to deal with different aspects related with attributed graphs, including attributed graph transformation, we present a new approach that, based on the new notion of narrowing graph transformation rule, allows us to delay constraint solving when doing attributed graph transformation, in a way that resembles lazy computation. For this reason, we have called lazy this new kind of transformation. Moreover, we show that the approach is sound and complete with respect to standard attributed graph transformation. A running example, where a graph transformation system describes some basic operations of a travel agency, shows the practical interest of the approach.