Deriving Bisimulation Congruences in the DPO Approach to Graph Rewriting

Abstract

Abstract. Motivated by recent work on the derivation of labelled transitions and bisimulation congruences from unlabelled reaction rules, we show how to solve this problem in the DPO (double-pushout) approach to graph rewriting. Unlike in previous approaches, we consider graphs as objects, instead of 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 smoothly integrates 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 sketch that the bisimilarity based on rewriting with borrowed contexts is a congruence relation.

Confluence of Graph Rewriting with Interfaces

“…This variant has emerged in several research threads, including rewriting with borrowed contexts [19], encodings of process calculi [24,5], connecting DPO rewriting systems with computads in cospans categories [25,44] and, more recently, for checking the equivalence of terms of symmetric monoidal theories [4]. Our key observation is that for DPO rewriting with interfaces, the Knuth-Bendix property holds and therefore confluence of a terminating system can be decided by checking whether its critical pairs are joinable.…”

Transition systems, link graphs and Petri nets

“…Ehrig [15] has investigated these links further. This has led to paper by Ehrig and König [16] in which the RPO technique is transferred to graph-embedding categories.…”

Labels from Reductions: Towards a General Theory

“…This appears to make a poor use of the generality, elegance and succinctness of theory of relative pushouts. Ideally, the rule should be expressed parametrically, as in 1 a.1 | a.2 −→ 1 | 2 and the labels should be derivable for open terms with universal property imposed both on the contexts and the parameters. A label should thus consist both of a smallest context and the most general parameter which makes a reduction possible; for example…”

“…Remarkably, such formalisation supports the central 'congruence theorem' that bisimulation on the labelled transition systems derived following the theory is a congruence, i.e., it is closed under all contexts. Due to the generality of the framework, such a result has already been applied to a variety of different models [1,3,9,[12][13][14] This paper's original contribution concerns our initial ideas on the treatment of open terms and parametric rules in the above framework.…”