Common Belief Revisited

2026-02-17

AI Summary

This work resolves a long-standing open problem in multi-agent systems concerning the complete axiomatization of common belief when individual beliefs are modeled by the KD45 logic. Through a careful semantic analysis within modal logic and the construction of an appropriate axiomatic system, the study demonstrates that, in addition to the shift-reflexivity axiom, a novel axiom dependent on the number of agents is required. This leads to the first complete axiomatization of common belief over KD45, thereby establishing its precise logical characterization and settling a key question in the field.

Technology Category

• Knowledge Representation and Reasoning: Reasoning with Beliefs
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Contrary to common belief, common belief is not KD4. If individual belief is KD45, common belief does indeed lose the 5 property and keep the D and 4 properties -- and it has none of the other commonly considered properties of knowledge and belief. But it has another property: $C(Cφ\rightarrow φ)$ -- corresponding to so-called shift-reflexivity (reflexivity one step ahead). This observation begs the question: is KD4 extended with this axiom a complete characterisation of common belief in the KD45 case? If not, what \emph{is} the logic of common belief? In this paper we show that the answer to the first question is ``no'': there is one additional axiom, and, furthermore, it relies on the number of agents. We show that the result is a complete characterisation of common belief, settling the open problem.

Problem

• common belief
• KD45
• axiomatization
• multi-agent logic
• shift-reflexivity

Innovation

• common belief
• KD45
• axiomatization
• shift-reflexivity
• multi-agent logic