Church-Rosser for Borrowed Context Rewriting

Modelling distributed and mobile systems at a suitable level of abstraction maybe considered the main application area of process calculi and graph transformation systems. Analysis and verification methods for the resulting models abound. Here we focus on two lines of research: on the one hand the work following the influential theory of Reactive Systems (rs) [6] (originally developed for process calculi), and on the other hand the classical concurrency theory of the double pushout approach (dpo) to graph transformation[4, 7]. Recall the idea of the theory of rs: one derives from a given set of reaction rules a labelled transition system (lts) such that the induced bisimulation relation is a congruence. This powerful technique has been adapted to dpo transformation over graphs [3] and even to rewriting in any adhesive category [8]. This generalization is known as dpo with borrowed contexts (dpobc) and it is the main object of study in this paper. The question we ask is whether the natural notion of true concurrency of dpo rewriting, which is in contrast to the “interleaving only” semantics of process calculi, carries over to dpobc. In other words, we set out to develop a dpostyle parallelism theory for dpobc. Below we illustrate how borrowed context rewriting faithfully models the concurrency aspects of distributed and mobile systems. As a proof of concept we present the local Church-Rosser theorem for dpobc. A reader which is not familiar with dpobc might skim the main ideas from the following model of an interactive system. We have only one reaction rule ( ◦→◦) − (◦ ◦) − (◦→◦ ), which models the dispatching of the message from one network node to the other using a channel of unit capacity between them. Now suppose we have the network ⊕⇆⊙, consisting of two nodes ⊕ and ⊙ which are connected by two complementary channels of unit capacity. However we do not want the channels themselves to be visible, but only the “access points” ⊕ and ⊙. This system (state) is succinctly modelled by the inclusion (⊕ ⊙) − (⊕⇆⊙), which we also write as ֌ ⊕ ⊙