DISCRETE-TIME SYSTEMS THAT ADMIT FINITELY MANY CAUSAL STATE-REPRESENTATIONS
TOSHITAMI MATSUMOTO · International Journal of General Systems · 1999
In this paper, relationship between causal discrete-time systems and the number of their causal state-representations is studied using the metric structure defined on the input and output sets of the systems, which enables us to introduce some topological notions to the investigation. For any given causal discrete-time system, an equivalence relation is defined on the input set, where any two inputs are equated if they equally distinguish all possible causal states of the system, and the quotient set of the input set with respect to the equivalence relation is studied in detail. The main results include: A discrete-time system admits essentially only a finite number of causal state-representations if and only if it is finitely observable. As an application, an algorithm to decide if a finite-state automaton has only finitely many automata having the same behavior as that of the original automaton is obtained.