Transducing Linear Decompositions of Tournaments

2026-01-06

AI Summary

This study investigates the logical definability of clique decompositions in tournaments of bounded linear clique-width. By integrating first-order (FO) transductions, linear clique-width theory, and structural analysis of tournaments, the work establishes—for the first time—that bounded-width clique decompositions of such graphs can be defined using only first-order logic, without resorting to more expressive counting extensions. This result not only demonstrates the equivalence between counting monadic second-order logic (CMSO) and existential monadic second-order logic (∃MSO) over tournaments of bounded linear clique-width, but also substantially reduces the logical complexity required to express relevant graph properties, thereby providing a foundational basis for the design of efficient algorithms.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed 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
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Boja\'nczyk, Pilipczuk, and Grohe [LICS'18] proved that for graphs of bounded linear clique-width, clique-decompositions of bounded width can be produced by a CMSO transduction. We show that in the case of tournaments, a first-order transduction suffices. This implies that the logics CMSO and existential MSO are equivalent over bounded linear clique-width tournaments.

Problem

• tournaments
• linear clique-width
• first-order transduction
• clique-decomposition
• CMSO

Innovation

• first-order transduction
• tournaments
• linear clique-width
• clique-decomposition
• logical equivalence