Depth-bounded Epistemic Logic

TL;DR

This paper introduces DBEL, a logic for modeling agents with limited reasoning depth about beliefs, extending standard epistemic logic with explicit depth reasoning, and explores its properties, extensions, and applications.

Contribution

The paper develops DBEL, a novel epistemic logic that models agents with bounded reasoning depth, including axiomatizations, extensions, and complexity analysis.

Findings

DBEL effectively models bounded-depth reasoning in epistemic scenarios.

Extensions support public announcements and reveal properties like amnesia and knowledge leakage.

Application to the muddy children problem demonstrates practical reasoning bounds.

Abstract

Epistemic logics model how agents reason about their beliefs and the beliefs of other agents. Existing logics typically assume the ability of agents to reason perfectly about propositions of unbounded modal depth. We present DBEL, an extension of S5 that models agents that can reason about epistemic formulas only up to a specific modal depth. To support explicit reasoning about agent depths, DBEL includes depth atoms Ead (agent a has depth exactly d) and Pad (agent a has depth at least d). We provide a sound and complete axiomatization of DBEL. We extend DBEL to support public announcements for bounded depth agents and show how the resulting DPAL logic generalizes standard axioms from public announcement logic. We present two alternate extensions and identify two undesirable properties, amnesia and knowledge leakage, that these extensions have but DPAL does not. We provide axiomatizations of these logics as well as complexity results for satisfiability and model checking. Finally, we use these logics to illustrate how agents with bounded modal depth reason in the classical muddy children problem, including upper and lower bounds on the depth knowledge necessary for agents to successfully solve the problem.