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Agent-Knowledge Logic for Alternative Epistemic Logic

Yuki Nishimura

TL;DR

The paper addresses the limitation of standard epistemic logic in handling agent-specific propositions and inter-agent relations by introducing agent-knowledge logic (AK), a two-dimensional fusion of Facebook logic and the Logic of Hide and Seek. AK is built as a product structure with separate agent- and knowledge-proposition spaces, enabling expressions such as "one of my friends knows $p$" and providing nominals to name agents and epistemic alternatives. It establishes a core embedding result showing that every epistemic logic formula can be translated into AK via $T(K_i\varphi)=@_{T(i)}\Box_K T(\varphi)$, and develops a terminating tableau calculus $\mathbf{T}_{AK}$ with a finite-model property, proving decidability. The framework supports an explicit semantic separation between agent-centric and knowledge-centric information, yielding richer expressiveness and natural language interpretations, while also pointing toward Hilbert-style axiomatizations and a PSPACE-completeness result for AK satisfiability. Overall, AK offers a flexible alternative to standard epistemic logic with potential applications in multi-agent reasoning and natural language semantics.

Abstract

Epistemic logic is known as a logic that captures the knowledge and beliefs of agents and has undergone various developments since Hintikka (1962). In this paper, we propose a new logic called agent-knowledge logic by taking the product of individual knowledge structures and the set of relationships among agents. This logic is based on the Facebook logic proposed by Seligman et al. (2011) and the Logic of Hide and Seek Game proposed by Li et al. (2021). We show two main results; one is that this logic can embed the standard epistemic logic, and the other is that there is a proof system of tableau calculus that works in finite time. We also discuss various sentences and inferences that this logic can express.

Agent-Knowledge Logic for Alternative Epistemic Logic

TL;DR

The paper addresses the limitation of standard epistemic logic in handling agent-specific propositions and inter-agent relations by introducing agent-knowledge logic (AK), a two-dimensional fusion of Facebook logic and the Logic of Hide and Seek. AK is built as a product structure with separate agent- and knowledge-proposition spaces, enabling expressions such as "one of my friends knows " and providing nominals to name agents and epistemic alternatives. It establishes a core embedding result showing that every epistemic logic formula can be translated into AK via , and develops a terminating tableau calculus with a finite-model property, proving decidability. The framework supports an explicit semantic separation between agent-centric and knowledge-centric information, yielding richer expressiveness and natural language interpretations, while also pointing toward Hilbert-style axiomatizations and a PSPACE-completeness result for AK satisfiability. Overall, AK offers a flexible alternative to standard epistemic logic with potential applications in multi-agent reasoning and natural language semantics.

Abstract

Epistemic logic is known as a logic that captures the knowledge and beliefs of agents and has undergone various developments since Hintikka (1962). In this paper, we propose a new logic called agent-knowledge logic by taking the product of individual knowledge structures and the set of relationships among agents. This logic is based on the Facebook logic proposed by Seligman et al. (2011) and the Logic of Hide and Seek Game proposed by Li et al. (2021). We show two main results; one is that this logic can embed the standard epistemic logic, and the other is that there is a proof system of tableau calculus that works in finite time. We also discuss various sentences and inferences that this logic can express.
Paper Structure (16 sections, 11 theorems, 27 equations, 3 figures)

This paper contains 16 sections, 11 theorems, 27 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Epistemic Logic
  4. Facebook Logic
  5. Logic of Hide and Seek Game
  6. Agent-Knowledge Logic
  7. Agent-Knowledge Model
  8. Examples
  9. Embedding Epistemic Logic into Agent-Knowledge Logic
  10. Proof System
  11. Tableau Calculus
  12. Termination and Completeness
  13. Future Work and Perspective
  14. Seeking More Usage
  15. Hilbert-Style Axiomatization
  16. ...and 1 more sections

Key Result

Proposition 3.5

The formula $@_a \Box_K p_K \rightarrow p_K$ is valid with respect to the class of $\mathcal{M}_\text{AK}$ where all of $S_x$ are equivalence relations.

Figures (3)

  • Figure 1: An agent-knowledge model.
  • Figure 2: The condition $\beta_A$ satisfies (id$_A$ is the identity on $\mathbf{A}$).
  • Figure 3: The rules of $\mathbf{T}_\text{AK}$

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Proposition 3.5
  • ...and 30 more