Confluence up to Garbage

The transformation of graphs and graph-like structures is ubiquitous in computer science. When a system is described by graph-transformation 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.