Technology Category
Knowledge Representation and Reasoning: Computational Complexity of ReasoningReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyMultiagent Systems: Mechanism Design
The complexity of being monitorable
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
monitorable sets
topological complexity
descriptive set theory
second countable
Π¹₁-complete
Innovation
Methods, ideas, or system contributions that make the work stand out.
monitorable sets
descriptive set theory
topological complexity
second countability
Π¹₁-completeness
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