  We compare two approaches for modelling imperfect information in infinite
games by using finite-state automata. The first, more standard approach views
information as the result of an observation process driven by a sequential
Mealy machine. In contrast, the second approach features indistinguishability
relations described by synchronous two-tape automata.
  The indistinguishability-relation model turns out to be strictly more
expressive than the one based on observations. We present a characterisation of
the indistinguishability relations that admit a representation as a
finite-state observation function. We show that the characterisation is
decidable, and give a procedure to construct a corresponding Mealy machine
whenever one exists.
