The role of counting quantifiers in laminar set systems
Rutger Campbell, Noleen Köhler
TL;DR
The paper studies how laminar set systems encode as laminar trees and demonstrates an MSO-transduction that outputs the laminar tree from a laminar set system, resolving Courcelle's open question. It then derives non-deterministic MSO transductions that produce modular decompositions, cotrees, split decompositions, and bi-join decompositions, improving on CMSO-based results by avoiding counting predicates. The work analyzes the expressive power of counting quantifiers in laminar contexts, showing conditions under which counting can be simulated by MSO and providing partial converses indicating limits. Additionally, it extends the transduction framework to weakly-partitive and weakly-bipartitive systems, strengthening definability-equals-recognizability links for dense, tree-like decompositions.
Abstract
Laminar set systems consist of non-crossing subsets of a universe with set inclusion essentially corresponding to the descendant relationship of a tree, the so-called laminar tree. Laminar set systems lie at the core of many graph decompositions such as modular decompositions, split decompositions, and bi-join decompositions. We show that from a laminar set system we can obtain the corresponding laminar tree by means of a monadic second order logic (MSO) transduction. This resolves an open question originally asked by Courcelle and is a satisfying resolution as MSO is the natural logic for set systems and is sufficient to define the property ``laminar''. Using results from Campbell et al. [STACS 2025], we can now obtain transductions for obtaining modular decompositions, co-trees, split decompositions and bi-join decompositions using MSO instead of CMSO. We further gain some insight into the expressive power of counting quantifiers and provide some results towards determining when counting quantifiers can be simulated in MSO in laminar set systems and when they cannot.