Showing papers on "Binary constraint published in 1969"
TL;DR: 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.
Abstract: 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.
40 citations
TL;DR: 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.
Abstract: 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.
12 citations
TL;DR: Parts I and II of this three-part paper provided the fundamental concepts underlying constraint theory whose goal is the systematic determination of whether a mathematical model and its computations are well posed.
Abstract: Parts I and II of this three-part paper provided the fundamental concepts underlying constraint theory whose goal is the systematic determination of whether a mathematical model and its computations are well posed. In addition to deriving results for the general relation, special relations defined as universal and regular were treated. This concluding part treats two more special relations: inequality and discrete. Employing the axiom of transitivity for inequalities, results relating to the consistency of a mathematical model of inequalities in terms of its model graph are derived. Rules for the simultaneous propagation of four types of constraint, over, point, interval, and slack, through a heterogeneous model graph are established. In contrast to other relation types, discrete relations point constrain every relevant variable, so that finding intrinsic constraint sources is trivial. A general procedure is provided to determine the allowability of requested computations on a discrete model.
12 citations
2 citations