  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.
