Constraint Theory, Part II: Model Graphs and Regular Relations

The foundations of a "constraint theory" whose goal is the systematic analysis of consistency and computability in heterogeneous mathematical models of very high dimension were established in a previous paper [1]. The eventual objective of this theory is to automate the automatic determination of whether a complex mathematical model and its required computations are "well posed." This part concentrates on the topological properties of the bipartite model graph defined in [1] and the application of these properties to the location of intrinsic constraint in large mathematical models composed of "regular" relations. In particular, the model graph concepts of connected components, trees, circuits, circuit rank, circuit index, and constraint potential are defined with sufficient precision to allow automatic computation. Regular relations, the most commonly employed for scientific models, are defined and the sources of constraint are identified with the "basic nodal square," a special subgraph embedded within the total model graph. A procedure is then developed which uses the topological properties developed earlier to locate the basic nodal squares within a large complex model graph. The ultimate use of the sources of intrinsic constraint is to check the consistency of the model and the allowability of the computations put to it.