  We consider a class of systems over finite alphabets with linear internal
dynamics, finite-valued control inputs and finitely quantized outputs. We
motivate the need for a new notion of observability and propose three new
notions of output observability, thereby shifting our attention to the problem
of state estimation for output prediction. We derive necessary and sufficient
conditions for a system to be output observable, algorithmic procedures to
verify these conditions, and a construction of finite memory output observers
when certain conditions are met. We conclude with simple illustrative examples.
