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Common Knowledge Always, Forever

Martín Diéguez, David Fernández-Duque

TL;DR

The paper develops a polytopological PDL framework for multi-agent common knowledge in topological semantics, establishing an effective finite model property for closure-space variants while showing a lack of FMP in Cantor-derivative spaces via an embedding of PLTL. It demonstrates decidability for restricted derivative-space classes and provides a reduction from PLTL to topological PDL to illustrate fundamental limits. The work also explores translations and constructions that connect topological and Kripke perspectives, and it sketches future directions, including alternative derivative notions and distributed knowledge operators. Overall, it clarifies the algorithmic boundaries of common-knowledge logics in topological settings and opens paths for further decidability and expressivity results. The results have implications for formal epistemology and the modeling of evidence and self-reference in topological terms.

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.

Common Knowledge Always, Forever

TL;DR

The paper develops a polytopological PDL framework for multi-agent common knowledge in topological semantics, establishing an effective finite model property for closure-space variants while showing a lack of FMP in Cantor-derivative spaces via an embedding of PLTL. It demonstrates decidability for restricted derivative-space classes and provides a reduction from PLTL to topological PDL to illustrate fundamental limits. The work also explores translations and constructions that connect topological and Kripke perspectives, and it sketches future directions, including alternative derivative notions and distributed knowledge operators. Overall, it clarifies the algorithmic boundaries of common-knowledge logics in topological settings and opens paths for further decidability and expressivity results. The results have implications for formal epistemology and the modeling of evidence and self-reference in topological terms.

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.
Paper Structure (8 sections, 8 theorems, 5 equations, 1 figure)

This paper contains 8 sections, 8 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Layout
  3. Derivative spaces
  4. Syntax and Semantics
  5. Decidability Results
  6. Embedding $\sf LTL$
  7. Concluding remarks
  8. Acknowledgements

Key Result

Theorem 4.1

$\mathcal{L}^*$ has the effective finite model property over the class of closure spaces and hence the validity problem for this class is decidable.

Figures (1)

  • Figure 1: The bitopological representation of a bijective frame. Numbers in boldface represent the points where the variable whole holds. Dashed arrows illustrate the successor function, while red and blue lines represent the equivalence relations associated to agents $a$ and $b$, respectively.

Theorems & Definitions (23)

  • Definition 2.1
  • Example 2.2: topological closure spaces
  • Example 2.3: topological derivative spaces
  • Example 2.4: weakly transitive Kripke frames
  • Example 2.5: Alexandroff closure spaces as ${\sf S4}$ Kripke frames
  • Example 2.6: Alexandroff derivative spaces as irreflexive $\sf wK4$ frames
  • Example 2.7: Monadic spaces
  • Theorem 4.1
  • proof : Proof sketch
  • Theorem 4.2
  • ...and 13 more