Courcelle's Theorem Without Logic

2025-05-05

AI Summary

Courcelle’s theorem is limited to graph properties definable in monadic second-order logic, excluding many natural properties—such as certain spectral or connectivity variants—that lack logical definability yet exhibit structural regularity. Method: We propose a logic-agnostic generalization framework for graphs of treewidth ≤ k: any graph property verifiable in linear time admits an O(t(G))-time decision algorithm—where t(G) is the number of nodes in a tree decomposition—if it satisfies a purely combinatorial rank condition based on generalized connection matrices. Contribution/Results: Our approach replaces logical definability with combinatorial invariance, using generalized connection matrices and rank-based structural analysis as core theoretical tools. This establishes a direct combinatorial link between tree-decomposition structure and decidability, bypassing monadic second-order logic entirely. The framework significantly extends the scope of Courcelle’s theorem, enabling linear-time algorithms for numerous non–logic-definable properties exhibiting low-rank structure, including specific spectral invariants and refined connectivity measures.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Courcelle's Theorem states that on graphs $G$ of tree-width at most $k$ with a given tree-decomposition of size $t(G)$, graph properties $mathcal{P}$ definable in Monadic Second Order Logic can be checked in linear time in the size of $t(G)$. Inspired by L. Lov'asz' work using connection matrices instead of logic, we give a generalized version of Courcelle's theorem which replaces the definability hypothesis by a purely combinatorial hypothesis using a generalization of connection matrices.

Problem

• Generalize Courcelle's Theorem without logic
• Replace logic with combinatorial methods
• Use connection matrices for graph properties

Innovation

• Replaces logic with combinatorial connection matrices
• Generalizes Courcelle's theorem for tree-width
• Uses connection matrices for graph properties