FO and MSO Model Checking on Temporal Graphs

2026-02-16

AI Summary

This work addresses the challenge of efficiently solving logical expressibility problems on temporal graphs in the absence of a unified logical framework. To this end, it introduces two novel structural parameters—VIM-width and TIM-width—based on the temporal activity patterns of vertices and components, and establishes a unified first-order (FO) and monadic second-order (MSO) logical framework. The paper presents the first FO and MSO meta-theorems for these parameters, extending existing MSO theory and introducing an FO meta-theorem for the first time, encompassing four key classes of temporal graph parameters. Additionally, it provides a composable and reusable dictionary of logical formulas that enables modular modeling and automatically guarantees fixed-parameter tractability, thereby significantly enhancing both modeling efficiency and solvability for a broad range of temporal graph problems.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Algorithmic meta-theorems provide an important tool for showing tractability of graph problems on graph classes defined by structural restrictions. While such results are well established for static graphs, corresponding frameworks for temporal graphs are comparatively limited. In this work, we revisit past applications of logical meta-theorems to temporal graphs and develop an extended unifying logical framework. Our first contribution is the introduction of logical encodings for the parameters vertex-interval-membership (VIM) width and tree-interval-membership (TIM) width, parameters which capture the signature of vertex and component activity over time. Building on this, we extend existing monadic second-order (MSO) meta-theorems for bounded lifetime and temporal degree to the parameters VIM and TIM width, and establish novel first-order (FO) meta-theorems for all four parameters. Finally, we signpost a modular lexicon of reusable FO and MSO formulas for a broad range of temporal graph problems, and give an example. This lexicon allows new problems to be expressed compositionally and directly yields fixed-parameter tractability results across the four parameters we consider.

Problem

• temporal graphs
• model checking
• logical meta-theorems
• parameterized complexity
• structural restrictions

Innovation

• temporal graphs
• first-order logic
• monadic second-order logic
• VIM width
• TIM width