Constraint theory: an overview

The purpose of this paper is to develop an analytic foundation—called Constraint Theory—for the systematic determination of mathematical model consistency and computational allowability. Constraint Theory's primary application is the more efficient construction and use of heterogeneous, multidimensional mathematical models, especially when interdisciplinary technical teams, system analysts, and managers are involved. A rigorous basis for the formerly ill-defined notion of ‘ constraint ’ is established and four interrelated viewpoints of a mathematical model are defined : (a) the set-theoretic relation space, (b) the submodel family, (c) the bipartite graph, and (d) the constraint matrix. The first two viewpoints represent complete models ; the second two represent metamodels. Many 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, all its variables are mutually consistent. If a model graph of regular relations has a tree structure, all its variables are consistent and, moreover, none can be point constrained. The circuit-cluster portions of the model graph are the only possible locations for point constraint and overconstraint to occur in a set of regular relations. In these cases, point constraint can always be identified by a ‘ basic nodal square ’. Topological concepts such as circuit rank, circuit index, and constraint potential are applied to extract the basic nodel squares from a large, complex circuit-cluster portion of a model graph. Several examples applying the principles of constraint theory are provided.