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

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Summary

Introduction

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

Abstract reduction systems
Labelled graphs and morphisms
Double-pushout graph transformation
Graph languages
Confluence checking
Closedness and confluence up to garbage
Confluence and subcommutativity up to garbage
Confluence and subcommutativity modulo garbage
Generalised critical pair lemma
Subgraph closure and subgraph closed languages
Generalising the critical pair lemma
Generation of non-garbage critical pairs
Checking for strong joinability
Summary
Backtracking-free language recognition
Backtracking-free specifications
Backtracking-free specification of series-parallel graphs
Backtracking-free specification of extended flow diagrams
Subcommutativity
Generalisations
Related work
Future work
Full Text
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