The complexity of being monitorable

2026-01-06

AI Summary

This study investigates the descriptive complexity of monitorable sets in topological spaces, with a focus on structural differences arising under varying countability conditions. By introducing methods from descriptive set theory, the paper provides the first systematic analysis of the complexity hierarchy of monitorable sets in both second-countable and non-second-countable spaces. The main contributions include proving that in second-countable spaces, the family of monitorable sets lies within the Π⁰₃ class and precisely characterizing the attainable levels of complexity within this class. In contrast, for non-second-countable spaces, the work constructs an example of a Π¹₁-complete monitorable set, thereby revealing a fundamental distinction in the nature of monitorability between the two classes of topological spaces.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Multiagent Systems: Mechanism Design

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

We study monitorable sets from a topological standpoint. In particular, we use descriptive set theory to describe the complexity of the family of monitorable sets in a countable space $X$. When $X$ is second countable, we observe that the family of monitorable sets is $\Pi^0_3$ and determine the exact complexities it can have. In contrast, we show that if $X$ is not second countable then the family of monitorable sets can be much more complex, giving an example where it is $ \Pi^1_1$-complete.

Problem

• monitorable sets
• topological complexity
• descriptive set theory
• second countable
• Π¹₁-complete

Innovation

• monitorable sets
• descriptive set theory
• topological complexity
• second countability
• Π¹₁-completeness