Abstract
We investigate the integration of graph constraints into graph grammars and consider the filter problem: Given a graph grammar and a graph constraint, does there exist a “goal-oriented” grammar that generates all graphs of the original graph language satisfying the constraint. We solve the filter problem for specific graph grammars and specific graph constraints. As an intermediate step, we construct a constraint automaton accepting exactly the graphs in the graph language that satisfy the constraint.
This work is partly supported by the German Research Foundation (DFG), Grants HA 2936/4-2 and TA 2941/3-2 (Meta-Modeling and Graph Grammars: Generating Development Environments for Modeling Languages).
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Notes
- 1.
For definition & existence of pushouts in the category of graphs see e.g. [EEPT06].
- 2.
A rule set \(\mathcal {R}\) is terminating if there is no infinite transformation \(G_0\mathop {\Rightarrow }\limits _{\mathcal {R}} G_1\mathop {\Rightarrow }\limits _{\mathcal {R}}G_2\ldots \).
- 3.
For a rule \(\varrho =\langle p,\mathrm {ac}\rangle \), \(\varrho '=\langle \varrho ,\mathrm {ac}'\rangle \) denotes the rule \(\langle p,\mathrm {ac}\wedge \mathrm {ac}'\rangle \).
- 4.
A pair \((a',b')\) is jointly surjective if, for each \(x\in C'\), there is a preimage \(y\in P'\) with \(a'(y)=x\) or \(z\in C\) with \(b'(z)=x\).
- 5.
For a rule \(p=\langle {L}\hookleftarrow {K}\hookrightarrow {R}\rangle \), \(p^{-1}=\langle {R}\hookleftarrow {K}\hookrightarrow {L}\rangle \) denotes the inverse rule. For \(L'\Rightarrow _p R'\) with intermediate graph \(K'\), \(\langle {L'}\hookleftarrow {K'}\hookrightarrow {R'}\rangle \) is the derived rule.
- 6.
Karl-Heinz Pennemann. Generalized constraints and application conditions for graph transformation systems. Diploma thesis, University of Oldenburg, 2004.
- 7.
A rule \(\varrho \) is \(\mathcal {R}\) -restricting if \(\varrho '\) is \(\varrho \)-restricting for some \(\varrho \in \mathcal {R}\).
- 8.
The match of the rule is marked in a blue color.
- 9.
Two application conditions \(\mathrm {ac}\) and \(\mathrm {ac}'\) are disjoint if the sets \([\![\mathrm {ac}]\!]\) and \([\![\mathrm {ac}']\!]\) are disjoint. \([\![\mathrm {ac}]\!]=\{g\mid g\,\models \, \mathrm {ac}\}\) denotes the semantics of the application condition \(\mathrm {ac}\).
- 10.
- 11.
A binary relation \(\preceq \) defined on a set Q is a quasi-ordering [Din92] if it is reflexive and transitive. A sequence \(q_1 ,q_2,\ldots \) of members of Q is called a good sequence (with respect to \(\preceq \)) if there exist \(i < j\) such that \(q_i\preceq q_j\). It is a bad sequence if otherwise. We call \((Q,\preceq )\) a well-quasi-ordering (or a wqo) if there is no infinite bad sequence.
- 12.
Let T be a transition system with a preorder \(\preceq \) defined on its states. T is monotone wrt. to \(\preceq \) if, for any states \(c_1, c_2\) and \(c_3\), with \(c_1 \preceq c_2\) and \(c_1 \rightarrow c_3\), there exists a state \(c_4\) such that \(c_3 \preceq c_4\) and \(c_2 \rightarrow c_4\).
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Acknowledgements
We are grateful to Jan Steffen Becker, Berthold Hoffmann, Jens Kosiol, Nebras Nassar, Christoph Peuser, Lina Spiekermann, and Gabriele Taentzer and the anonymous reviewers for their helpful comments to this paper.
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Habel, A., Sandmann, C., Teusch, T. (2018). Integration of Graph Constraints into Graph Grammars. In: Heckel, R., Taentzer, G. (eds) Graph Transformation, Specifications, and Nets. Lecture Notes in Computer Science(), vol 10800. Springer, Cham. https://doi.org/10.1007/978-3-319-75396-6_2
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