A Category Theoretical Approach to the Concurrent Semantics of Rewriting : Adhesive Categories and Related Concepts
This thesis studies formal semantics for a family of rewriting formalisms that have arisen as category theoretical abstractions of the so-called algebraic approaches to graph rewriting. The latter in turn generalize and combine features of term rewriting and Petri nets. Two salient features of (the abstract versions of) graph rewriting are a suitable class of categories which captures the structure of the objects of rewriting, and a notion of independence or concurrency of rewriting steps – as in the theory of Petri nets. Category theoretical abstractions of graph rewriting such as double pushout rewriting encapsulate the complex details of the structures that are to be rewritten by considering them as objects of a suitable abstract category, for example an adhesive one. The main difficulty of the development of appropriate categorical frameworks is the identification of the essential properties of the category of graphs which allow to develop the theory of graph rewriting in an abstract framework. The motivations for such an endeavor are twofold: to arrive at a succint description of the fundamental principles of rewriting systems in general, and to apply well-established verification and analysis techniques of the theory of Petri nets (and also term rewriting systems) to a wide range of distributed and concurrent systems in which states have a "graph-like" structure. The contributions of this thesis thus can be considered as two sides of the same coin: on the one side, concepts and results for Petri nets (and graph grammars) are generalized to an abstract category theoretical setting; on the other side, suitable classes of "graph-like" categories which capture the essential properties of the category of graphs are identified. Two central results are the following: first, (concatenable) processes are faithful partial order representations of equivalence classes of system runs which only differ w.r.t. the rescheduling of causally independent events; second, the unfolding of a system is established as the canonical partial order representation of all possible events (following the work of Winskel). Weakly ω-adhesive categories are introduced as the theoretical foundation for the corresponding formal theorems about processes and unfoldings. The main result states that an unfolding procedure for systems which are given as single pushout grammars in weakly ω-adhesive categories exists and can be characetrised as a right adjoint functor from a category of grammars to the subcategory of occurrence grammars. This result specializes to and improves upon existing results concerning the coreflective semantics of the unfolding of graph grammars and Petri nets (under an individual token interpretation). Moreover, the unfolding procedure is in principle usable as the starting point for static analysis techniques such as McMillan’s finite complete prefix method. Finally, the adequacy of weakly ω-adhesive categories as a categorical framework is argued for by providing a comparison with the notion of topos, which is a standard abstraction of the categories of sets (and graphs).