Constraint Theory

by FRIEDMAN, GEORGE J.

• ISBN13: 9780387234182
• ISBN10: 0387234187
• Format: Hardcover
• Copyright: 2005-05-30
• Publisher: Springer Nature

Summary

The enormous potential of digital computation to manage new complex systems is impeded by exponential increases in complexity. As the model's dimensionality increases from hundreds to thousands of variables, and as submodels constructed by diverse technical teams are integrated into the total model, the model is likely to become inconsistent and even more likely, the computational requests on the model become unallowable. This text analyzes the way constraint theory employs bipartite graphs and constraint matrices to detect and correct these well-posed problems. It also presents the process of locating the "kernel of constraint", literally trillions of times faster than a random search, determining consistency and compatibility within seconds. This text is an invaluable reference to all engineers, mathematicians and managers concerned with modeling.

Table of Contents

CHAPTER 1 MOTIVATIONS	1	(24)
WHAT IS CONSTRAINT THEORY AND WHY IS IT IMPORTANT?
1.1 TRENDS AND PROBLEMS IN SYSTEM TECHNOLOGIES
1.2 AN EXAMPLE OF LOW DIMENSION
1.3 THE MANAGER AND ANALYST CONTINUE THEIR DIALOGUE
1.4 PRELIMINARY CONCLUSIONS
1.5 A LITTLE WINDOW INTO FUTURE CHAPTERS
1.6 PROBLEMS FOR THE CURIOUS READER
CHAPTER 2: THE FOUR-FOLD WAY	25	(24)
HOW TO PERCEIVE COMPLEX MATHEMATICAL MODELS AND WELL-POSED PROBLEMS
2.1 PROLOGUE: THE MANAGER AND ANALYST DISCUSS THE ORIGINS OF MULTIDIMENSIONAL MODELS AND WELLPOSEDNESS
2.2 THE FIRST VIEW: SET THEORETIC
2.3 THE SECOND VIEW: FAMILY OF SUBMODELS
2.4 THE THIRD VIEW: THE BIPARTITE GRAPH
2.5 THE FOURTH VIEW: THE CONSTRAINT MATRIX
2.6 MODEL CONSISTENCY AND COMPUTATIONAL ALLOWABILITY
2.7 THE MANAGER AND ANALYST CONTINUE THEIR DIALOGUE
2.8 CHAPTER SUMMARY
2.9 PROBLEMS FOR THE INTERESTED STUDENT
CHAPTER 3: GENERAL RESULTS	49	(12)
FROM PROTOMATH TO MATH TO METAMATH
3.1 LANGUAGE AND MATHEMATICS
3.2 MOST GENERAL TRUSTWORTHY RESULTS
3.3 CLASSES OF RELATIONS
3.4 MANAGER AND ANALYST REVISITED
3.5 CHAPTER SUMMARY
3.6 PROBLEMS FOR THE GENERAL STUDENT
CHAPTER 4: REGULAR RELATIONS	61	(50)
SEARCHING FOR THE KERNELS OF CONSTRAINT
4.1 COGNITIVE BARRIERS TO CIRCUITS
4.2 NODE, KNOT AND BASIC NODAL SQUARE SANCTIFICATION
4.3 USEFUL PROPERTIES OF BAPARTITE GRAPHS
4.4 CORNERING THE CULPRIT KERNELS; TEN EASY PIECES
4.5 CONTINUING THE PURSUIT INSIDE THE CIRCUIT CLUSTERS (CC)
4.6 LOCATING THE BNSS IN MINUTES, NOT UNIVERSE LIFETIMES
4.7 COMPARISON OF COMPUTATIONAL COMPLEXITY; TRILLIONS AND TRILLIONS OF TIMES FASTER
4.8 ZERO CONSTRAINT ALL ALONG THE COMPUTATIONAL PATH
4.9 RECAPITULATION OF COMPUTATIONAL FLOW
4.10 GENERAL PROCEDURE FOR DETERMINING CONSISTENCY AND ALLOWABILITY IN A MODEL OF REGULAR RELATIONS
4.11 SUMMARY OF CHAPTER AND CONSTRAINT THEORY TOOLKIT
4.12 QUERIES FOR THE REGULAR STUDENT
CHAPTER 5: DISCRETE AND INTERVAL RELATIONS	111	(18)
THE DIMINISHED UTILITY OF METAMODELS
5.1 METAMODEL ISSUES AND PERSPECTIVES
5.2 THE GENERAL TAXONOMY AND PRIMARY PROPERTY OF DISCRETE RELATIONS
5.3 BOLEAN RELATIONS
5.4 TOPOLOGICAL IMPLICATIONS
5.5 ALLOWABILITY OF DISCRETE COMPUTATIONS
5.6 INEQUALITY RELATIONS
5.7 SUMMARY
5.8 PROBLEMS FOR THE DISCRETE STUDENT
CHAPTER 6: THE LOGICAL STRUCTURE OF CONSTRAINT THEORY	129	(8)
A COMPACT SUMMARY
6.1 OVERVIEW
6.2 POSTULATES AND PHILOSOPHICAL ASSUMPTIONS
6.3 DEFINITIONS
6.4 THEOREMS
6.5 GRAPHS OF THE LOGICAL STRUCTURE OF CONSTRAINT THEORY
6.6 COMPLETENESS
CHAPTER 7: EXAMPLES OF CONSTRAINT THEORY APPLIED TO REAL-WORLD PROBLEMS	137	(20)
7.1 APOLOGIES NOT REQUIRED
7.2 COST AS AN INDEPENDENT VARIABLE (CAIV)
7.3 THE KINEMATICS OF FREE-FALL WEAPONS
7.4 THE DEFLECTION OF AN EARTH-THREATENING ASTEROID EMPLOYING MASS DRIVERS
CHAPTER 8: MANAGER AND ANALYST MEET AGAIN	157	(6)
GISTS AND SCHIZOPHRENIA
APPENDICES
APPENDIX A: COMPUTATIONAL REQUEST DISAPPOINTMENTS; RESULTS OF THE USC ALLOWABILITY PROJECT	163	(8)
APPENDIX B: GRAPH THEORY OVERVIEW	171	(4)
APPENDIX C: THE LOGIC OF "IF" AND "IF AND ONLY IF"	175	(2)
APPENDIX D: VECTOR SPACES APPLIED TO GRAPH THEORY	177	(4)
REFERENCES	181	(2)
INDEX	183