Designing and Learning Substitutable Plane Graph Grammars

Abstract

Though graph grammars have been widely investigated for 40 years, few learning results exist for them. The main reasons come from complexity issues that are inherent when graphs, and a fortiori graph grammars, are considered. The picture is however different if one considers drawings of graphs, rather than the graphs themselves. E.g., it has recently been proved that the isomorphism and pattern searching problems could be solved in polynomial time for plane graphs, that is, planar embedding of planar graphs. In this paper, we introduce the Plane Graph Grammars (PGG) and detail how they differ to usual graph grammar formalisms while at the same time they share important properties with string context-free grammars. In particular, the parsing of a graph with a given PGG is polynomial for languages with appropriate restrictions. We demonstrate that PGG are well-shaped for learning: we show how recent results on string grammars can be extended to PGG by providing a learning algorithm that identifies in the limit the class of substitutable plane graph languages. Our algorithm runs in polynomial time assuming the same restriction used for polynomial parsing, and the amount of data needed for convergence is comparable to the one required in the case of strings.