Double Pushout Research Articles

Using Matrix Graph Grammars for the Analysis of Behavioural Specifications: Sequential and Parallel Independence

In this paper we present a new approach for the analysis of rule-based specification of system dynamics. We model system states as simple digraphs, which can be represented with boolean matrices. Rules modelling the different state changes of the system can also be represented with boolean matrices, and therefore the rewriting is expressed using boolean operations only.The conditions for sequential independence between pair of rules are well-known in the categorical approaches to graph transformation (e.g. single and double pushout). These conditions state when two rules can be applied in any order yielding the same result. In this paper, we study the concept of sequential independence in our framework, and extend it in order to consider derivations of arbitrary finite length. Instead of studying one-step rule advances, we study independence of rule permutations in sequences of arbitrary finite length. We also analyse the conditions under which a sequence is applicable to a given host graph. We introduce rule composition and give some preliminary results regarding parallel independence. Moreover, we improve our framework making explicit the elements which, if present, disable the application of a rule or a sequence.

Subobject Transformation Systems

Subobject transformation systems STS are proposed as a novel formal framework for the analysis of derivations of transformation systems based on the algebraic, double-pushout (DPO) approach. They can be considered as a simplified variant of DPO rewriting, acting in the distributive lattice of subobjects of a given object of an adhesive category. This setting allows a direct analysis of all possible notions of dependency between any two productions without requiring an explicit match. In particular, several equivalent characterizations of independence of productions are proposed, as well as a local Church–Rosser theorem in the setting of STS. Finally, we show how any derivation tree in an ordinary DPO grammar leads to an STS via a suitable construction and show that relational reasoning in the resulting STS is sound and complete with respect to the independence in the original derivation tree.

Construction and Properties of Adhesive and Weak Adhesive High-Level Replacement Categories

As presented in Ehrig et al. (Fundamentals of Algebraic Graph Trans- formation, EATCS Monographs, Springer, 2006), adhesive high-level replacement (HLR) categories and systems are an adequate framework for several kinds of trans- formation systems based on the double pushout approach. Since (weak) adhesive HLR categories are closed under product, slice, coslice, comma and functor category constructions, it is possible to build new (weak) adhesive HLR categories from existing ones. But for the general results of transformation systems, as additional properties initial pushouts, binary coproducts compatible with a special morphism class M and a pair factorization are needed to obtain the full theory. In this paper, we analyze under which conditions these additional properties are preserved by the categorical constructions in order to avoid checking these properties explicitly.

Adding Recursion to Graph Transformation

In this paper we define recursive rules in the double pushout approach (DPO) to graph transformation. Classical DPO rules are extended with a base case condition and a recursion condition. Mechanisms are provided to pass the match from both conditions to the rule's left hand side, and also between two consecutive steps in the recursion. The approach is useful when recursive structures (such as inheritance hierarchies, nested component hierarchies, networks of functional blocks, etc.) have to be processed. Although we present the recursion for DPO, it can also be adapted to other approaches to graph and model transformation. We present examples for model transformation, model simulation and model optimization in different application domains.

Event-driven grammars: relating abstract and concrete levels of visual languages

In this work we introduce event-driven grammars, a kind of graph grammars that are especially suited for visual modelling environments generated by meta-modelling. Rules in these grammars may be triggered by user actions (such as creating, editing or connecting elements) and in their turn may trigger other user-interface events. Their combination with triple graph transformation systems allows constructing and checking the consistency of the abstract syntax graph while the user is building the concrete syntax model, as well as managing the layout of the concrete syntax representation. As an example of these concepts, we show the definition of a modelling environment for UML sequence diagrams. A discussion is also presented of methodological aspects for the generation of environments for visual languages with multiple views, its connection with triple graph grammars, the formalization of the latter in the double pushout approach and its extension with an inheritance concept.

Termination of Algebraic Rewriting with Inhibitors

We proceed with the study of termination properties in the double pushout approach to algebraic rewriting, and show a concrete termination criterion for rewriting systems with inhibitors. Inhibitors prevent elements in an algebra to participate in rule matches, so that termination depends only on whether new possibilities for matches are created. The notion of inhibitor can be extended to considering different levels of inhibition, by which the ability of an element to participate in a match is progressively reduced. We illustrate the approach by considering some application contexts in model transformation.

Deriving bisimulation congruences in the DPO approach to graph rewriting with borrowed contexts

Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to address this problem in the DPO (double-pushout) approach to graph rewriting. Unlike the case with previous approaches, we consider graphs as objects, rather than arrows, of the category under consideration. This allows us to present a very simple way of deriving labelled transitions (called rewriting steps with borrowed context), which integrates smoothly with the DPO approach, has a very constructive nature and requires only a minimum of category theory. The core part of this paper is the proof that the bisimilarity based on graph rewriting with borrowed contexts is a congruence relation. We will also introduce some proof techniques and compare our approach with the derivation of labelled transitions via relative pushouts.

Cloning and Expanding Graph Transformation Rules for Refactoring

Refactoring is a software engineering technique that aims at enhancing the structure of object-oriented software while preserving its behavior. Several authors have studied how graph transformation can be used to specify refactoring, because such specifications are more precise and can thus, in principle, easier be verified to preserve a program's behavior. It has turned out that “standard” ways of graph transformation do not suffice to define refactoring: their expressive power must be increased if they shall be useful in this application area. Two mechanisms have been proposed so far: one for cloning, and one for expanding nodes by graphs. However, the mechanisms and notations needed are rather complex. In this paper we provide, in the context of double pushout graph transformation, a more elegant and intuitive description. It is based on a notion of rule instantiation, where the instantiation transforms rule schemes into rule instances by cloning and expansion. The power of the technique is demonstrated by an application to two well-known refactoring operations.

The uniqueness condition for the double pushout transformation of algebras

The double pushout approach to the algebraic graph transformation of hypergraphs was invented 30 years ago and it has been generalized since then to more general objects, like for instance relational systems or total and partial unary algebras. We have recently introduced the double pushout transformation of partial and partly total algebras over an arbitrary signature. In this paper we study the uniqueness condition for these rewriting formalisms, which turns out to be given by a suitable generalization to partial algebras of the well known congruence extension property for total algebras.

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.

A Bi-Categorical Axiomatisation of Concurrent Graph Rewriting

In this paper the concurrent semantics of double-pushout (DPO) graph rewriting, which is classically defined in terms of shift-equivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bi-category. In contrast to a previous attempt based on 2-categories, the use of bi-categories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.

Termination of Graph Rewriting is Undecidable

It is shown that it is undecidable in general whether a graph rewriting system (in the “double pushout approach”) is terminating. The proof is by a reduction of the Post Correspondence Problem. It is also argued that there is no straightforward reduction of the halting problem for Turing machines or of the termination problem for string rewriting systems to the present problem.

Double-pushout hypergraph rewriting using total conformisms

The algebraic transformation of hypergraphs, under the so-called double-pushout (DPO) approach, was invented more than two decades ago, and thoroughly developed since then. We introduce in this paper a new approach to DPO algebraic transformation of hypergraphs and, more in general, of unary partial algebras, which generalizes the aforementioned “classical” DPO approach to hypergraph transformation. While the classical approach was based on the (usual) homomorphisms of hypergraphs, our new approach is based on the total conformisms, a type of morphisms of hypergraphs imported from the theory of partial algebras, which can be described, roughly speaking, as those mappings between hypergraphs that “reflect” the structure of the target object. In this paper we give both an algebraic and an operational characterization of this new DPO transformation, first for unary partial algebras and then, as a particular case, for hypergraphs. We also study its abstract properties related to parallelism and concurrency through the determination of the HLR conditions it satisfies with respect to several natural classes of morphisms.

Locating the contractum in the double pushout approach

Double pushout (algebraic) graph rewriting, which works by first removing the part of the graph to be regarded as garbage, and then gluing in the new part of the graph, is contrasted with term graph rewriting, which works by first gluing in the new part of the graph (the contractum) and performing redirections, and then removing garbage. It is shown that in the algebraic framework these two strategies can be reconciled. This is done by finding a natural analogue of the contractum in the algebraic framework, which requires the reformulation of the customary double pushout construction. The new and old algebraic constructions coexist within a pushout cube. In this, the usual “outward” form of the double pushout appears as the two rear squares, and the alternative “inward” formulation as the two front squares. The two formulations are entirely equivalent in the world of double pushout graph rewriting.