Harald König

A categorical framework for the transformation of object-oriented systems: Models and data

Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access those data. Data and programs all have to be migrated in a consistent manner such that their semantics does not change. This paper addresses the data part of the problem and introduces a model for object-oriented structures, describing the schema level with classes, associations, and inheritance as well as the instance level with objects and links. Positive Horn formulas based on predicates are used to formulate constraints to be obeyed by the schema and instance level, in order to reflect object-oriented structures. Homomorphisms are used for the typing of the instance level as well as for the description of refactorings which specify the addition, folding, and unfolding of schema elements. A categorial framework is presented which allows us to derive instance migrations from schema transformations in such a way that instances of the old schema are automatically migrated into instances of the new schema. The natural use of the pullback functor for unfolding is followed by an initial semantics approach: Instance migration is completed with the help of a co-adjoint functor on arrow categories.

Model transformation and induced instance migration: a universal framework

Software restructuring and refactoring facilitate the use of models as primary artifacts. Model evolution becomes agile if consistency between evolving models and depending artifacts is spontaneously maintained. In this paper we study endogenous model transformations at medium or fine granularity with impact on data structures and objects. We propose a formal framework in which transformation rules for class models can be formulated, whose application induces automatic migration of corresponding data structures. The main contribution is a correctness criterion for rule-induced instance migration based on initial semantics.

Algebraic Graph Transformations with Inheritance

In this paper, we propose a new approach to inheritance in the context of algebraic graph transformation by providing a suitable categorial framework which reflects the semantics of class-based inheritance in software engineering. Inheritance is modelled by a type graph T that comes equipped with a partial order. Typed graphs are arrows with codomain T which preserve graph structures up to inheritance. Morphisms between typed graphs are “down typing” graph morphisms: An object of class t can be mapped to an object of a subclass of t. We prove that this structure is an adhesive HLR category, i.e. pushouts along extremal monomorphisms are “well-behaved”. This infers validity of classical results such as the Local Church-Rosser Theorem, the Parallelism Theorem, and the Concurrency Theorem.

Categorical Framework for the Transformation of Object-Oriented Systems: Operations and Methods

Refactoring of information systems is hard, for two reasons. On the one hand, large databases exist which have to be adjusted. On the other hand, many programs access that data. These programs all have to be migrated in a consistent manner such that their semantics does not change. It cannot be relied upon, however, that no running processes exist during such a migration. Consequently, a refactoring of an information system needs to take care of the migration of data, programs, and processes. This paper extends the model described in [SLK10] by operations, messages, and methods, which allows to model complete object-oriented systems. Methods are expressed by special double-pushout graph transformations. Homomorphisms are used for the typing of the instance level as well as for the description of refactorings which specify the addition, folding, and unfolding of schema elements. Finally, a categorical framework is presented which allows to derive instance migrations from schema transformations in such a way that programs and processes to the old schema are correctly migrated into programs and processes to the new schema.

Incremental Consistency Checking of Heterogeneous Multimodels

The local approaches to global consistency checking (GCC) of heterogeneous multimodels strive to reduce the model merging and matching workload within GCC. The paper’s contribution to such approaches is a framework allowing the user to do matching incrementally: to build the match required for checking the multimodel w.r.t. a new constraint, the user employs matches produced in previous GCC sessions.

Fibred Amalgamation, Descent Data, and Van Kampen Squares in Topoi

Reliable semantics for software systems has to follow the semantics-as-instance principal (fibred semantics) rather than the semantics-as-interpretation principal (indexed semantics). While amalgamation of interpretations is simple and nearly always possible, amalgamation of instances is very much involved and not possible in many cases. A condition when two compatible instances (a span of pullbacks) are amalgamable, is presented for presheaves, i.e. functor categories SET𝒮. Based on this individual condition we prove further a total condition for amalgamation which simultaneously yields a necessary and sufficient condition for pushouts to be Van Kampen squares. As a necessary and adequate basis to achieve these results we provide a full revision and adaption of the theory of descent data in topoi for applications in diagrammatic specifications including graph transformations. Especially, we characterize Van Kampen squares in arbitrary topoi by pullbacks of categories of descent data.

Polymorphic Single-Pushout Graph Transformation

The paper extends single-pushout graph transformation by polymorphism, a key concept in object-oriented design. The notions sub-rule and remainder, well-known in single-pushout rewriting, are applied in order to model dynamic rule extension and type dependent rule application. This extension mechanism qualifies graph transformation as a modelling technique for extendable frameworks. Therefore, it contributes to the applicability of graph transformation in software engineering.

Graph Operations and Free Graph Algebras

We introduce a concept of graph algebra that generalizes the traditional concept of algebra in the sense that (1) we use graphs rather than sets as carriers, and (2) we generalize algebraic operations to diagrammatic operations over graphs, which we call graph operations.

Toward Product Lines of Mathematical Models for Software Model Management

We present a general view on theoretical aspects of model synchronization and consistency management, and discuss technical challenges in making it sound, and cultural challenges in bringing it to practice.

Van Kampen Squares for Graph Transformation

This paper demonstrates the benefits of a recent result by the authors, proving a necessary and sufficient condition for a pushout of two morphisms to be a Van Kampen Square, even if both morphisms are not monomorphisms. The theorem can be applied in categories that are based on graph structure signatures. We discuss its value in the context of general views on co-transformations and illustrate an application in a software co-evolution scenario.