Characterizations of Monadic Second Order Definable Context-Free Sets of Graphs

TL;DR

This paper characterizes graph sets that are both definable in Counting Monadic Second Order Logic and context-free, establishing their equivalence with recognizable, parsable, and transducible sets, linking logical and algebraic graph properties.

Contribution

It provides a new characterization of CMSO-definable, context-free graph sets through algebraic, parsable, and transduction-based frameworks, connecting key results in graph logic and decomposition.

Findings

Equivalence of CMSO-definable, context-free graph sets with recognizable sets of bounded tree-width.

Recognition of these sets via finitely generated subalgebras of HR-algebra.

Representation of graph sets as images of recognizable tree sets under definable transductions.

Abstract

We give a characterization of the sets of graphs that are both definable in Counting Monadic Second Order Logic (CMSO) and context-free, i.e., least solutions of Hyperedge-Replacement (HR) grammars introduced by Courcelle and Engelfriet. We prove the equivalence of these sets with: (a) recognizable sets (in the algebra of graphs with HR-operations) of bounded tree-width; we refine this condition further and show equivalence with recognizability in a finitely generated subalgebra of the HR-algebra of graphs; (b) parsable sets, for which there is a definable transduction from graphs to a set of derivation trees labelled by HR operations, such that the set of graphs is the image of the set of derivation trees under the canonical evaluation of the HR operations; (c) images of recognizable unranked sets of trees under a definable transduction, whose inverse is also definable. We rely on a novel connection between two seminal results, a logical characterization of context-free graph languages in terms of tree-to-graph definable transductions, by Courcelle and Engelfriet and a proof that an optimal-width tree decomposition of a graph can be built by an definable transduction, by Bojanczyk and Pilipczuk.