Abstract
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 in two case studies about efficient language recognition: we present 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. We also give a critical pair condition for subcommutativity up to garbage which, together with closedness, implies confluence up to garbage even in non-terminating systems.
Highlights
Rule-based graph transformation and graph grammars date back to the late 1960s
We present two case studies with backtracking-free graph reduction systems which recognise a class of labelled series-parallel graphs and a class of flow diagrams, respectively
We review some terminology for binary relations, the DPO approach to graph transformation, graph languages, and confluence checking
Summary
Rule-based graph transformation and graph grammars date back to the late 1960s. The most developed theoretical framework is the so-called double-pushout (DPO) approach to graph transformation [1,2]. We present two case studies with backtracking-free graph reduction systems which recognise a class of labelled series-parallel graphs and a class of flow diagrams, respectively. Both systems are non-confluent but confluent up to garbage. We give a second version of our generalised critical pair lemma in this setting, showing how critical pair analysis can be used to check for subcommutativity up to garbage This property implies confluence up to garbage even in non-terminating systems, provided that non-garbage is closed under reduction. This is relevant for applications because confluence up to garbage in such systems implies that non-garbage graphs can be reduced to at most one irreducible graph
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