Autoepistemic logic is a nonmonotonic logic for modeling the beliefs of an ideally rational agent who reflects on his own beliefs. Autoepistemic logic has been lacking a decision procedure to answer the question: given a set of premises describing the beliefs of an agent does the agent believe a given formula (is the given formula derivable from the given set of premises). Two derivability relations are defined for propositional autoepistemic logic corresponding to the agent's own point of view and the point of view of an external observer of the beliefs of the agent. Decision procedures based on analytic tableaux are then given and proved correct for both of the relations.