Confluence up to Garbage

Abstract

The transformation of graphs and graph-like structures is ubiquitous in computer science. When a system is described by graphtransformation rules, it is often desirable that the rules are both terminating and confluent so that rule applications in an arbitrary order produce unique resulting graphs. However, there are application scenarios where the rules are not globally confluent but confluent on a subclass of graphs that are of interest. In other words, non-resolvable conflicts can only occur on graphs that are considered as "garbage". In this paper, we introduce the notion of confluence up to garbage and generalise Plump's critical pair lemma for double-pushout graph transformation, providing a sufficient condition for confluence up to garbage by non-garbage critical pair analysis. We apply our results to language recognition by backtracking-free graph reduction, showing how to establish that a graph language can be decided by a system which is confluent up to garbage. We present two case studies with backtracking-free graph reduction systems which recognise a class of flow diagrams and a class of labelled series-parallel graphs, respectively. Both systems are non-confluent but confluent up to garbage.

Monadic second-order incorrectness logic for GP 2

“…This section introduces GP 2, a nondeterministic rule-based programming language for manipulating graphs. The language is implemented by a compiler that generates C code [26,27], and is available to download online 1 .…”

String Diagram Rewrite Theory III: Confluence with and without Frobenius

“…For DPO rewriting, despite the undecidability result recalled before, local confluence has been shown to hold with respect to a stronger notion of joinability for critical pairs [43], and confluence is decidable whenever all critical pairs satisfy a syntactic condition, coverability [44]. More recently, we mention the work on confluence up-to garbage, whose intuition is to check if the rewriting system is confluent on a subclass of graphs that are of interest [14]. And, on an similar note, the work on confluence for DPO with applications conditions [20] seems also relevant for our investigation.…”

String diagram rewrite theory III: Confluence with and without Frobenius

“…For DPO rewriting, despite the undecidability result recalled before, local confluence has been shown to hold with respect to a stronger notion of joinability for critical pairs (Plump 1993), and confluence is decidable whenever all critical pairs satisfy a syntactic condition, coverability (Plump 2010). More recently, we mention the work on confluence up-to garbage, whose intuition is to check if the rewriting system is confluent on a subclass of graphs that are of interest (Campbell and Plump 2020). And, on a similar note, the work on confluence for DPO with applications conditions (Ehrig et al 2010) seems also relevant for our investigation.…”