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We motivate the need for a new notion of observability for systems over finite alphabets and propose three new notions of output observability, thereby shifting our attention to the problem of state estimation for output prediction. We then consider a class of systems over finite alphabets with linear internal dynamics, finite-valued control inputs, and finitely quantized outputs. We derive a set of sufficient conditions and a set of necessary conditions for these systems to be output observable, propose an algorithmic procedure to verify one of these conditions, and construct finite memory output observers when certain conditions are met. [ABSTRACT FROM AUTHOR]

We consider discrete-time systems with linear dynamics and a finite input alphabet. We propose a sufficient condition for the existence of finite uniform bisimulations, and an algorithm that terminates when the condition holds and constructs the corresponding equivalence classes. We derive a stronger sufficient condition for the construction of arbitrarily many equivalence classes, together with a corresponding constructive algorithm. Finally, we derive a necessary condition for the existence of finite uniform bisimulations. All our conditions are topological in nature. [ABSTRACT FROM AUTHOR]