Natural states and past-determinism of general time systems

The concepts of natural state and past-determinism are given for general time systems, i.e., systems whose input-output pairs are ordered pairs of abstract functions defined on a linearly ordered set. It is shown that every general time system has a causal realization; however, only the past-determined systems have causal realizations such that the state can be identified by observing the past input and output over a sufficient inteval of time. Such identifiable states, the natural states, are shown to be unique up to an isomorphism and characterized by equivalence classes of input-output pairs.