This special issue of Mathematical Structures in Computer Science is dedicated to structure transformation and contains contributions to two main research areas: extending the constructions that have been developed over nearly forty years for graph transformations to more general structures; and employing these transformations to model the operational and abstract semantics of distributed and mobile systems.

We generalise both the notion of a non-sequential process and the unfolding construction (which was previously developed for concrete formalisms such as Petri nets and graph grammars) to the abstract setting of (single pushout) rewriting of objects in adhesive categories. The main results show that processes are in one-to-one correspondence with switch-equivalent classes of derivations, and that the unfolding construction can be characterised as a coreflection, that is, the unfolding functor arises as the right adjoint to the embedding of the category of occurrence grammars into the category of grammars.

As the unfolding represents potentially infinite computations, we need to work in adhesive categories with ‘well-behaved’ colimits of ω-chains of monos. Compared with previous work on the unfolding of Petri nets and graph grammars, our results apply to a wider class of systems, which is due to the use of a refined notion of grammar morphism.

Finitary $\mathcal{M}$-adhesive categories are $\mathcal{M}$-adhesive categories with finite objects only, where $\mathcal{M}$-adhesive categories are a slight generalisation of weak adhesive high-level replacement (HLR) categories. We say an object is finite if it has a finite number of $\mathcal{M}$-subobjects. In this paper, we show that in finitary $\mathcal{M}$-adhesive categories we not only have all the well-known HLR properties of weak adhesive HLR categories, which are already valid for $\mathcal{M}$-adhesive categories, but also all the additional HLR requirements needed to prove classical results including the Local Church-Rosser, Parallelism, Concurrency, Embedding, Extension and Local Confluence Theorems, where the last of these is based on critical pairs. More precisely, we are able to show that finitary $\mathcal{M}$-adhesive categories have a unique $\mathcal{E}$-$\mathcal{M}$ factorisation and initial pushouts, and the existence of an $\mathcal{M}$-initial object implies we also have finite coproducts and a unique $\mathcal{E}$′-$\mathcal{M}$ pair factorisation. Moreover, we can show that the finitary restriction of each $\mathcal{M}$-adhesive category is a finitary $\mathcal{M}$-adhesive category, and finitarity is preserved under functor and comma category constructions based on $\mathcal{M}$-adhesive categories. This means that all the classical results are also valid for corresponding finitary $\mathcal{M}$-adhesive transformation systems including several kinds of finitary graph and Petri net transformation systems. Finally, we discuss how some of the results can be extended to non-$\mathcal{M}$-adhesive categories.

In this paper we focus on the synthesis of labelled transition systems (LTSs) for process calculi using Mobile Ambients (MAs) as a testbed. Our proposal is based on a graphical encoding: a process is mapped into a graph equipped with interfaces such that the denotation is fully abstract with respect to the standard structural congruence. Graphs with interfaces are amenable to the synthesis mechanism based on borrowed contexts (BCs), which is an instance of relative pushouts (RPOs). The BC mechanism allows the effective construction of an LTS that has graphs with interfaces as states and labels, and such that the associated bisimilarity is a congruence. We focus here on the analysis of an LTS over processes as graphs with interfaces: we use the LTS on graphs to recover an LTS directly defined over the structure of MA processes and define a set of SOS inference rules capturing the same operational semantics.

Amalgamation is a well-known concept for graph transformations that is used to model synchronised parallelism of rules with shared subrules and corresponding transformations. This concept is especially important for an adequate formalisation of the operational semantics of statecharts and other visual modelling languages, where typed attributed graphs are used for multiple rules with nested application conditions. However, the theory of amalgamation for the double-pushout approach has so far only been developed on a set-theoretical basis for pairs of standard graph rules without any application conditions.

For this reason, in the current paper we present the theory of amalgamation for $\mathcal{M}$-adhesive categories, which form a slightly more general framework than (weak) adhesive HLR categories, for a bundle of rules with (nested) application conditions. The two main results are the Complement Rule Theorem, which shows how to construct a minimal complement rule for each subrule, and the Multi-Amalgamation Theorem, which generalises the well-known Parallelism and Amalgamation Theorems to the case of multiple synchronised parallelism. In order to apply the largest amalgamated rule, we use maximal matchings, which are computed according to the actual instance graph. The constructions are illustrated by a small but meaningful running example, while a more complex case study concerning the firing semantics of Petri nets is presented as an introductory example and to provide motivation.

Nested application conditions generalise the well-known negative application conditions and are important for several application domains. In this paper, we present Local Church–Rosser, Parallelism, Concurrency and Amalgamation Theorems for rules with nested application conditions in the framework of $\mathcal{M}$-adhesive categories, where $\mathcal{M}$-adhesive categories are slightly more general than weak adhesive high-level replacement categories. Most of the proofs are based on the corresponding statements for rules without application conditions and two shift lemmas stating that nested application conditions can be shifted over morphisms and rules.

In model transformations, where source models are automatically translated into target models or code, termination is necessary for the transformation to be well defined. There are a number of specific termination criteria that can be used when specifying model transformations by graph transformation, though termination is undecidable in general. Unfortunately, and particularly for large and heterogeneous specifications, it is often not possible to use a single termination criterion. In this paper, we propose an approach that applies different criteria to suitable subsets of rules so that termination can be shown locally using the most suitable technique for each subset. Global termination then follows if certain causal dependencies between rules in different subsets are acyclic. The theory is developed at the level of typed attributed graphs, and is motivated and illustrated by a case study translating UML activity diagrams to CSP.

Triple graph grammars (TGGs) are a well-established concept for the specification and execution of bidirectional model transformations within model driven software engineering. Their main advantage is an automatic generation of operational rules for forward and backward model transformations, which simplifies specification and enhances usability as well as consistency. In this paper we present several important results for analysing model transformations based on the formal categorical foundation of TGGs within the framework of attributed graph transformation systems.

Our first main result shows that the crucial properties of correctness and completeness are ensured for model transformations. In order to analyse functional behaviour, we generate a new kind of operational rule, called a forward translation rule. We apply existing results for the analysis of local confluence for attributed graph transformation systems. As additional main results, we provide sufficient criteria for the verification of functional behaviour as well as a necessary and sufficient condition for strong functional behaviour. In fact, these conditions imply polynomial complexity for the execution of the model transformation. We also analyse information and complete information preservation of model transformations, that is, whether a source model can be reconstructed (uniquely) from the target model computed by the model transformation. We illustrate the results for the well-known model transformation example from class diagrams to relational database models.

$\mathcal{M}$-adhesive categories provide an abstract framework for a large variety of specification frameworks for modelling distributed and concurrent systems. They extend the well-known frameworks of adhesive and weak adhesive HLR categories and integrate high-level constructs such as attribution as in the case of typed attributed graphs.

In the current paper, we investigate $\mathcal{M}$-adhesive transformation systems including negative application conditions (NACs) for transformation rules, which are often used in applications. For such systems, we propose an original equivalence on transformation sequences, called permutation equivalence, that is coarser than the classical switch equivalence. We also present a general construction of deterministic processes for $\mathcal{M}$-adhesive transformation systems based on subobject transformation systems. As a main result, we show that the process obtained from a transformation sequence identifies its equivalence class of permutation-equivalent transformation sequences. Moreover, we show how the analysis of this process can be reduced to the analysis of the reachability graph of a generated Place/Transition Petri net. This net encodes the dependencies between rule applications of the transformation sequence, including the inhibiting effects of the NACs.