Topological graph theory foundations of design and control in intelligent discrete systems

Large-scale management information systems, decision support systems, knowledge-based systems, neural systems, control systems, manufacturing systems and others are becoming extremely sophisticated. It is possible to consider finite or locally finite sets or systems in some integer analogs of the Euclidean spaces with only integer coordinates and parametrization. Thus all the properties of such systems could be described using finite sequences of integers, and they could be saved in computer memory for computer aided systems analysis, control and design. All such systems are finitely connected, and they can be considered as digital topological graphs and trees, which are subspaces of some integer product spaces, Alexandroff spaces, primitively derived spaces, or other subclass of primitively path connected spaces. Connectivity, local connectivity and order relation in such spaces are the major topics of this paper. Digital topological graphs and trees are especially important because of their widespread applications.