Depth-bounded epistemic logic

TL;DR

This paper introduces DBEL, a new epistemic logic for modeling agents with limited reasoning depth, providing axiomatizations, extensions, and complexity results, and demonstrating its application to the muddy children problem.

Contribution

It presents DBEL, the first logic explicitly modeling agents with bounded modal depth, along with axiomatizations, extensions, and complexity analysis.

Findings

DBEL effectively models bounded-depth agent reasoning.

Extensions support public announcements and generalize standard logic axioms.

Bounded-depth reasoning impacts agents' success in the muddy children problem.

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 $E_a^d$ (agent $a$ has depth exactly $d$) and $P_a^d$ (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.