Fundamenta Informaticae

The introduction of adhesive categories revived interest in the study of properties of pushouts with respect to pullbacks that started over thirty years ago for the category of graphs. Adhesive categories – of which graphs are the “archetypal” example – are defined by a single property of pushouts along monos that implies essential lemmas and central theorems of double pushout rewriting such as the local Church-Rosser Theorem. The present paper shows that a strictly weaker condition on pushouts suffices to obtain essentially the same results: it suffices to require pushouts to be hereditary, i.e. they have to remain pushouts when they are embedded into the associated category of partial maps. This fact however is not the only reason to introduce partial map adhesive categories as categories with pushouts along monos (of a certain stable class) that are hereditary. There are two equally important motivations: first, there is an application relevant example category that cannot be captured by the more established variations of adhesive categories; second, partial map adhesive categories are “conceptually similar” to adhesive categories as the latter can be characterized as those categories with pushout along monos that remain bi-pushouts when they are embedded into the associated bi-category of spans. Thus, adhesivity with partial maps instead of spans appears to be a natural candidate for a general rewriting framework.

Graph transformation systems have been studied extensively and applied to several areas of computer science like formal language theory, the modeling of databases, concurrent or distributed systems, and visual, logical, and functional programming. In most kinds of applications it is necessary to have the possibility of restricting the applicability of rules. This is usually done by means of application conditions. In this paper, we continue the work of extending the fundamental theory of graph transformation to the case where rules may use arbitrary (nested) application conditions. More precisely, we generalize the Embedding theorem, and we study how local confluence can be checked in this context. In particular, we define a new notion of critical pair which allows us to formulate and prove a Local Confluence Theorem for the general case of rules with nested application conditions. All our results are presented, not for a specific class of graphs, but for any arbitrary ℳ-adhesive category, which means that our results apply to most kinds of graphical structures. We demonstrate our theory on the modeling of an elevator control by a typed graph transformation system with positive and negative application conditions.

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.

Model-driven development (MDD) has become a promising trend in software engineering. The model-driven development of highly complex software systems may lead to large models which deserve a modularization concept to enable their structured development in larger teams. Graphs are a natural way to represent the underlying structure of visual models. Typed graphs with inheritance and containment structures are well suited to describe the essentials of models based on the Eclipse Modeling Framework (EMF). Composite graphs can specify the logical distribution of EMF models and therefore, can form the conceptual basis for composite modeling in model-driven development. This is done based on the formal foundation of distributed graphs. Moreover, this category-theoretical foundation allows for the precise definition of consistent composite graph transformations satisfying all inheritance and containment conditions.

GP (for Graph Programs) is an experimental nondeterministic programming language for solving problems on graphs and graph-like structures. The language is based on graph transformation rules, allowing visual programming at a high level of abstraction. In particular, GP frees programmers from dealing with low-level data structures. In this paper, we present a Hoare-style proof system for verifying the partial correctness of (a subset of) graph programs. The pre- and post-conditions of the calculus are nested graph conditions with expressions, a formalism for specifying both structural graph properties and properties of labels. We show that our proof system is sound with respect to GP's operational semantics and give examples of its use.

Quantified μ-calculi combine the fix-point and modal operators of temporal logics with (existential and universal) quantifiers, and they allow for reasoning about the possible behaviour of individual components within a software system. In this paper we introduce a novel approach to the semantics of such calculi: we consider a sort of labeled transition systems called counterpart models as semantic domain, where states are algebras and transitions are defined by counterpart relations (a family of partial homomorphisms) between states. Then, formulae are interpreted over sets of state assignments (families of partial substitutions, associating formula variables to state components). Our proposal allows us to model and reason about the creation and deletion of components, as well as the merging of components. Moreover, it avoids the limitations of existing approaches, usually enforcing restrictions of the transition relation: the resulting semantics is a streamlined and intuitively appealing one, yet it is general enough to cover most of the alternative proposals we are aware of. The paper is rounded up with some considerations about expressiveness and decidability aspects.