Monadic second-order definable graph transductions: a survey

Abstract

Formulas of monadic second-order logic can be used to specify graph transductions, i.e., multi-valued functions from graphs to graphs. We obtain in this way classes of graph transductions, called monadic second-order definable graph transductions (or, more simply, definable transductions) that are closed under composition and preserve the two known classes of context-free sets of graphs, namely the class of hyperedge replacement (HR) and the class of vertex replacement (VR) sets. These two classes can be characterized in terms of definable transductions and recognizable sets of finite trees, independently of the rewriting mechanisms used to define the HR and VR grammars. When restricted to words, the definable transductions are strictly more powerful than the rational transductions such that the image of every finite word is finite; they do not preserve context-free languages. We also describe the sets of discrete (edgeless) labelled graphs that are the images of HR and VR sets under definable transductions: this gives a version of Parikh's theorem (i.e., the characterization of the commutative images of context-free languages) which extends the classical one and applies to HR and VR sets of graphs