Behavior is more fundamental than representations

An input-output system is a relation between two function spaces. The classical input-output framework treats a system as a map between function spaces. The graph of this map, which is the collection of all compatible input-output pairs, constitutes the behavior of the system. The behavior of a system can sometimes admit a behavioral equation representation such as a kernel representation or a difference equation representation. Such a representation, when it exists, may not be unique. Given a representation with a certain structure it is usually easily shown that the represented behavior has a corresponding property. However, if the behavior has a property (say, non-anticipation), representations of the behavior may not have the corresponding structure (lower triangularity). Therefore, representations are of secondary importance to behaviors. It is the behavior that is fundamental, not its representation