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Logic of (Common or Distributed) Knowledge

Chenwei Shi

TL;DR

Addresses reasoning about complex group knowledge by unifying common and distributed knowledge into a lattice-structured epistemic logic. Defines a language of terms with meet and join operations, presents S5 LCDK and S4 LCDK axiomatizations, and proves completeness for regular frames. Extends the static logic with semi-public reading and arbitrary reading event dynamics, providing reduction laws and showing how updates propagate through the lattice structure. The framework enables precise analysis of how knowledge states evolve under information-sharing events and suggests avenues for exploring richer lattice properties and interpretations.

Abstract

In this paper, we generalize epistemic logic so that it can help reason about ways of combining common knowledge and distributed knowledge such as "common distributed knowledge", "distributed common knowledge", "distributed common distributed knowledge" and so on. Moreover, we study the logic of its dynamic update by arbitrary reading events. We axiomatize these logics and prove their soundness and completeness.

Logic of (Common or Distributed) Knowledge

TL;DR

Addresses reasoning about complex group knowledge by unifying common and distributed knowledge into a lattice-structured epistemic logic. Defines a language of terms with meet and join operations, presents S5 LCDK and S4 LCDK axiomatizations, and proves completeness for regular frames. Extends the static logic with semi-public reading and arbitrary reading event dynamics, providing reduction laws and showing how updates propagate through the lattice structure. The framework enables precise analysis of how knowledge states evolve under information-sharing events and suggests avenues for exploring richer lattice properties and interpretations.

Abstract

In this paper, we generalize epistemic logic so that it can help reason about ways of combining common knowledge and distributed knowledge such as "common distributed knowledge", "distributed common knowledge", "distributed common distributed knowledge" and so on. Moreover, we study the logic of its dynamic update by arbitrary reading events. We axiomatize these logics and prove their soundness and completeness.

Paper Structure

This paper contains 14 sections, 26 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Syntax, Semantics and Axiomatization
  3. Semi-public Reading Events
  4. Arbitrary reading events
  5. Conclusion and Future Work
  6. Soundness and Completeness
  7. Step (1) Fischer-Ladner closure and atoms
  8. Step (2): Finite Canonical Model over $\Sigma$
  9. Step (3): Transitive FCM
  10. Step (4): Quasi-regularity of TFCM
  11. Step (5): Building S5-Regular Model Step by Step
  12. Proving the completeness of $S5\mathsf{LCDK}$
  13. Proving the completeness of $S4\mathsf{LCDK}$
  14. Acknowledgements

Key Result

Theorem 2.1

$S5\mathsf{LCDK}$ is sound and weakly complete with repsect to the class of S5-regular frames. $S4\mathsf{LCDK}$ is sound and weakly complete with respect to the class of S4-regular frames.

Theorems & Definitions (52)

  • Example 1.1
  • Definition 2.1: Language $\mathcal{L}$
  • Definition 2.2: Regular model
  • Definition 2.3: Truth condition
  • Definition 2.4: Axiom system $S5\mathsf{LCDK}$
  • Theorem 2.1
  • Theorem 3.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 42 more