Christoph Schulz

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.

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.

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.

Refactoring Informations Systems

We present our formal framework for the refactoring of complete information systems, i.e., the data model and the data itself. It is described using general and abstract notions of category theory and can handle addition, renaming and removal of model objects as well as folding and unfolding within complete and partial object compositions.

Composition of model transformations: a categorical framework

Consistency management in evolving 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 synchronized in a consistent manner. It cannot be relied upon, however, that no running processes exist during such a migration. Consequently, a restructuring of an information system needs to take care of the migration of object-oriented systems comprising data, programs, and processes. This co-evolution together with the notion of model transformation and instance migration has been introduced in earlier papers. This paper builds upon this exploratory work and analyses under which circumstances composed model transformations are compatible with composed instance migrations. We develop the notion of shortcut paths and show that composition is possible if shortcut paths are reflected by the underlying model transformations.