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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.Index Terms-Output observability, systems over finite alphabets, quantized outputs, finite memory observers.

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