  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.
