Constraint Theory, Part I: Fundamentals

The purpose of this paper is to develop an analytic foundation for the determination of whether a mathematical model and its desired computations are "well-posed" in order to help alleviate the software problems associated with the simulation of complex large-scale systems by heterogeneous mathematical models involving several hundred dimensions. The problem is approached by providing a rigorous basis for the commonplace notion of constraint. Four distinct viewpoints of the mathematical model are established: 1) the set theoretic relation space; 2) the family of submodels; 3) the bipartite graph, which provides topological insight; and 4) the constraint matrix. Fundamental definitions of mathematical model consistency, computational allowability, and extrinsic and intrinsic constraint are established on a set theory basis. Correspondences are proved between the topological properties of a model's graph and its constraint properties. Variables located in different connected components of a graph are always mutually consistent, but computations performed on them are never allowable. If a model graph of universal relations has a tree structure, then all its variables are mutually consistent. Detailed treatment of special relation classes will be given in Parts II and III.