  We show that a history-based variant of alternating bisimulation with
imperfect information allows it to be related to a variant of Alternating-time
Temporal Logic (ATL) with imperfect information by a full Hennessy-Milner
theorem. The variant of ATL we consider has a common knowledge semantics, which
requires that the uniform strategy available for a coalition to accomplish some
goal must be common knowledge inside the coalition, while other semantic
variants of ATL with imperfect information do not accommodate a Hennessy-Milner
theorem. We also show that the existence of a history-based alternating
bisimulation between two finite Concurrent Game Structures with imperfect
information (iCGS) is undecidable.
