Deriving bisimulation congruences in the DPO approach to graph rewriting with borrowed contexts

Abstract

Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to address this problem in the DPO (double-pushout) approach to graph rewriting. Unlike the case with previous approaches, we consider graphs as objects, rather than arrows, of the category under consideration. This allows us to present a very simple way of deriving labelled transitions (called rewriting steps with borrowed context), which integrates smoothly with the DPO approach, has a very constructive nature and requires only a minimum of category theory. The core part of this paper is the proof that the bisimilarity based on graph rewriting with borrowed contexts is a congruence relation. We will also introduce some proof techniques and compare our approach with the derivation of labelled transitions via relative pushouts.

Borrowed Contexts for Attributed Graphs

“…Unfortunately, as a counter-example in Sect. 4 shows, bisimilarity for symbolic graphs, as defined in [5], does not coincide with bisimilarity for attributed graphs. Hence, in this work we define a new notion of symbolic bisimilarity, which is also a congruence, and we show that it coincides with attributed graph bisimilarity.…”

“…In [5] Ehrig and König they show how a notion of bisimilarity could be defined for graph transformation systems. In particular, they introduced borrowed context graph transformation as a simple and powerful technique that allow us to derive labelled transitions and bisimulation congruences for graph transformation systems or, in general, for process calculi that can be defined in terms of graph transformation systems (e.g.…”

“…Moreover, in [11], Rangel, König and Ehrig showed how to use these techniques for the verification of bisimilarity. Unfortunately, the approach to verification introduced informally in [5] and studied in detail in [11] does not work when dealing with attributed graphs. This is especially unfortunate, because one of the motivations for [11] is being able to show that model transformations (and, in particular, refactorings) preserve behavioural equivalence, and in these cases we work with attributed graphs.…”

“…Actually, since symbolic graphs form an adhesive HLR category, all the results in [5] apply to this class of graphs. Unfortunately, as a counter-example in Sect.…”

Towards a General Theory of Barbs, Contexts and Labels

“…Hence, the ingenuity of the researcher focussed on devising suitable labelled semantics from reduction ones, cases at hand being the calculus of mobile ambients (MAs) [7] and the core calculus for higher-order concurrency (HoCore) [10]. Indeed, a series of techniques [9,17,8,1,16] have been proposed to address the automatic derivation of an LTS starting from a reduction semantics, in order to distill behavioural equivalences that are congruences. Among these proposals, the most renowned one is Leifer and Milner's theory of reactive systems (RSs): the underlying intuition is that the labels of the derived transition system (called IPO LTS) are the "minimal" contexts that allow for a reduction to be performed, where minimality is captured by the categorical notion of relative pushout [11].…”