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.

DBEL是S5的扩展，用来模拟深度有限的代理人进行关于知识公式的推理，同时添加了深度原子Ead和Pad，扩展了公开声明逻辑的公理，解决了遗忘和知识泄漏等问题，并用这些逻辑来解决经典的泥泞孩子问题。

深度有限的认知逻辑