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.

翻译：认知逻辑用于建模智能体如何推理自身的信念以及其他智能体的信念。现有逻辑通常假设智能体具备对无界模态深度命题进行完美推理的能力。本文提出DBEL（S5的扩展），用于建模仅能推理特定模态深度以内的认知公式的智能体。为支持对智能体深度的显式推理，DBEL包含深度原子公式Ead（智能体a精确深度为d）和Pad（智能体a深度至少为d）。我们给出了DBEL的可靠且完备的公理化系统。进一步将DBEL扩展至支持面向有界深度智能体的公开宣告，并展示由此生成的DPAL逻辑如何泛化公开宣告逻辑的标准公理。我们提出两种替代扩展方案，并识别出这些方案具有而DPAL不具备的两个不良性质——记忆缺失与知识泄露。本文给出了这些逻辑的公理化系统及其可满足性与模型检验问题的复杂度结果。最后，我们运用这些逻辑阐释了经典泥孩子问题中有界模态深度智能体的推理过程，包括智能体成功解决问题所需深度知识的上界与下界。