DPO Rewriting and Abstract Semantics via Opfibrations

AbstractThe classical DPO graph rewriting construction is re-expressed using the opfibration approach introduced originally for term graph rewriting. Using a skeleton category of graphs, a base of canonical graphs-in-context, with DPO rules as arrows, and with categories of redexes over each object in the base, yields a category of rewrites via the discrete Grothendieck construction. The various possible ways of combining rules and rewrites leads to a variety of functors amongst the various categories formed. Categories whose arrows are rewriting sequences have counterparts where the arrows are elementary event structures, and an event structure semantics for arbitrary graph grammars emerges naturally

Minimisation of event structures

Event structures are fundamental models in concurrency theory, providing a representation of events in computation and of their relations, notably concurrency, conflict and causality. In this paper we present a theory of minimisation for event structures. Working in a class of event structures that generalises many stable event structure models in the literature, (e.g., prime, asymmetric, flow and bundle event structures) we study a notion of behaviour-preserving quotient, taking hereditary history preserving bisimilarity as a reference behavioural equivalence. We show that for any event structure a uniquely determined minimal quotient always exists. We observe that each event structure can be seen as the quotient of a prime event structure, and that quotients of general event structures arise from quotients of (suitably defined) corresponding prime event structures. This gives a special relevance to quotients in the class of prime event structures, which are then studied in detail, providing a characterisation and showing that also prime event structures always admit a unique minimal quotient