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.

翻译：认知逻辑模拟了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能够完美推理无界模态深度的命题。我们提出DBEL，它是S5的扩展，用于建模只能推理特定模态深度内认知公式的智能体。为支持对智能体深度的显式推理，DBEL包含深度原子$E_a^d$（智能体$a$的准确深度为$d$）和$P_a^d$（智能体$a$的深度至少为$d$）。我们给出了DBEL的可靠且完备的公理化系统。我们将DBEL扩展以支持针对有界深度智能体的公开宣告，并展示由此产生的DPAL逻辑如何泛化公开宣告逻辑中的标准公理。我们提出了两种备选扩展，并识别出这些扩展具有而DPAL不具备的两个不良性质：遗忘和知识泄露。我们还给出了这些逻辑的公理化系统，以及可满足性和模型检验的复杂度结果。最后，我们利用这些逻辑阐述了在有界模态深度条件下，智能体如何在经典的泥孩子问题中进行推理，包括智能体成功解决问题所需深度知识的上界和下界。