CMSO-transducing tree-like graph decompositions

2024-12-06

AI Summary

This paper addresses the logical definability of structured graph decompositions—specifically modular, split, and biconnected decompositions—and their generalizations to weak partition systems and weak bipartition systems. Methodologically, it introduces a unified logical characterization framework based on Counting Monadic Second-Order logic (CMSO) and its two-sorted extension C2MSO, integrating weak partition system theory with the algebraic properties of graph decompositions to devise multiple efficient CMSO translation schemes. Crucially, it achieves CMSO definability for all three classical decompositions without relying on order-invariant MSO—a first—and extends Courcelle’s seminal result to weaker combinatorial systems. The main contributions are: (i) the first unified CMSO-definability framework for these decompositions; (ii) elimination of dependence on stronger logics such as MSO with built-in order; (iii) significantly improved logical expressiveness and translation efficiency; and (iv) a more concise and broadly applicable logical foundation for the metatheory of graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Description Logics
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We show that given a graph G we can CMSO-transduce its modular decomposition, its split decomposition and its bi-join decomposition. This improves results by Courcelle [Logical Methods in Computer Science, 2006] who gave such transductions using order-invariant MSO, a strictly more expressive logic than CMSO. Our methods more generally yield C2MSO-transductions of the canonical decomposition of weakly-partitive set systems and weakly-bipartitive systems of bipartitions.

Problem

• CMSO-Translation
• Graph Decomposition
• Complex Systems Analysis

Innovation

• CMSO-Transformation
• Graph Decomposition
• Weak Partition Systems