On Decomposition Theory: Generalized Dependence

Although system decomposition is a fundamental tool of systems theory, no theory exists that unifies its many manifestations. The beginnings of such a theory are proposed here. The goal of this theory is to understand the structure of decompositions of large complex systems. In particular, it is to uncover those structural features that are implicit in each specific method of decomposition. The key assumption is that any system decomposition is based, either explicitly or implicitly, on some concept of dependence. Therefore, study of decomposition becomes enmeshed with the study of dependence. Three aspects of system decomposition are emphasized. First, various kinds of dependence are employed in system decomposition. Second, refinement of a decomposition to obtain a new one is an important part of decomposition. Third, shifts in point of view and level of detail can lead to different decompositions. No attempt is made to cover all aspects of decomposition theory. Rather, attention is focused on developing some properties of generalized dependences. A number of examples are presented showing the applicability of the ideas presented here to a variety of systems. These include relational data bases, finite-state machines, cognitive maps, Petri nets, and vector processor computation.