  Awareness structures by Fagin and Halpern (1988) (FH) feature a syntactic
awareness correspondence and accessibility relations modeling implicit
knowledge. They are a flexible model of unawareness, and best interpreted from
a outside modeler's perspective. Unawareness structures by Heifetz, Meier, and
Schipper (2006, 2008) (HMS) model awareness by a lattice of state-spaces and
explicit knowledge via a possibility correspondence. They can be interpreted as
providing the subjective views of agents. Open questions include (1) how
implicit knowledge can be defined in HMS structures, and (2) in which way FH
structures can be extended to model the agents' subjective views. In this
paper, we address (1) by showing how to derive implicit knowledge from explicit
knowledge in HMS models. We also introduce a variant of HMS models that instead
of explicit knowledge, takes implicit knowledge and awareness as primitives.
Further, we address (2) by introducing a category of FH models that are modally
equivalent relative to sublanguages and can be interpreted as agents'
subjective views depending on their awareness. These constructions allow us to
show an equivalence between HMS and FH models. As a corollary, we obtain
soundness and completeness of HMS models with respect to the Logic of
Propositional Awareness, based on a language featuring both implicit and
explicit knowledge.
