Common Knowledge Always, Forever

2026-02-14

AI Summary

This study addresses the effective representation and reasoning about common knowledge and its generalizations within a topological semantics, with a focus on the finite model property in multi-topological spaces. To this end, the paper proposes a polytopological propositional dynamic logic (polytopological PDL), which for the first time incorporates common knowledge into a multi-topological logical framework. The main contribution lies in establishing that this logic enjoys the finite model property over closure spaces, while it fails to have this property over Cantor derivative spaces. The latter result is demonstrated by constructing a counterexample via an embedding of linear temporal logic augmented with a “past” operator. These findings significantly deepen our understanding of the expressive power and inherent limitations of topological epistemic logics.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

There has been an increasing interest in topological semantics for epistemic logic, which has been shown to be useful for, e.g., modelling evidence, degrees of belief, and self-reference. We introduce a polytopological PDL capable of expressing common knowledge and various generalizations and show it has the finite model property over closure spaces but not over Cantor derivative spaces. The latter is shown by embedding a version of linear temporal logic with `past', which does not have the finite model property.

Problem

• common knowledge
• topological semantics
• finite model property
• epistemic logic
• closure spaces

Innovation

• polytopological PDL
• common knowledge
• finite model property
• closure spaces
• Cantor derivative spaces