Constraint Theory (eBook)

Fronticepiece 7
Preface 8
Acknowledgments 9
Introduction 10
A Warmup Problem in Complexity 14
About the Author 16
Contents 17
Chapter 1 MOTIVATIONS 20
What is Constraint Theory and why is it important? 20
1.1 TRENDS AND PROBLEMS IN SYSTEM TECHNOLOGIES 20
1.2 AN EXAMPLE OF LOW DIMENSION 23
1.3 THE MANAGER AND ANALYST CONTINUE THEIR DIALOGUE 35
1.4 PRELIMINARY CONCLUSIONS 39
1.5 A LITTLE WINDOW INTO FUTURE CHAPTERS 41
1.6 PROBLEMS FOR THE CURIOUS READER 42
Chapter 2 THE FOUR-FOLD WAY 43
How to Perceive Complex Mathematical Models and Well-Posed Problems 43
2.1 PROLOGUE: THE MANAGER AND ANALYST DISCUSS THE ORIGINS OF MULTIDIMENSIONAL MODELS AND WELL-POSEDNESS 43
2.2 THE FIRST VIEW: SET THEORETIC 46
2.3 THE SECOND VIEW: FAMILY OF SUBMODELS 53
2.4 THE THIRD VIEW: THE BIPARTITE GRAPH 56
2.5 THE FOURTH VIEW: THE CONSTRAINT MATRIX 57
2.6 MODEL CONSISTENCY AND COMPUTATIONAL ALLOWABILITY 58
2.7 THE MANAGER AND ANALYST CONTINUE THEIR DIALOGUE 58
2.8 CHAPTER SUMMARY 64
2.9 PROBLEMS FOR THE INTERESTED STUDENT 65
Chapter 3 GENERAL RESULTS 67
From Protomath to Math to Metamath 67
3.1 LANGUAGE AND MATHEMATICS 67
3.2 MOST GENERAL TRUSTWORTHY RESULTS 69
3.3 CLASSES OF RELATIONS 74
3.4 MANAGER AND ANALYST REVISITED 77
3.5 CHAPTER SUMMARY 78
3.6 PROBLEMS FOR THE GENERAL STUDENT 78
Chapter 4 REGULAR RELATIONS 79
Searching for the Kernels of Constraint 79
4.1 COGNITIVE BARRIERS TO CIRCUITS 79
4.2 NODE, KNOT AND BASIC NODAL SQUARE SANCTIFICATION 81
4.3 USEFUL PROPERTIES OF BIPARTITE GRAPHS 90
4.4 CORNERING THE CULPRIT KERNELS TEN EASY PIECES
4.5 CONTINUING THE PURSUIT INSIDE THE CIRCUIT CLUSTERS ( cc) 105
4.6 LOCATING BNSs IN MINUTES, NOT UNIVERSE LIFETIMES 108
4.7 COMPARISON OF COMPUTATIONAL COMPLEXITY TRILLIONS AND TRILLIONS OF TIMES FASTER
4.8 ZERO CONSTRAINT ALL ALONG THE COMPUTATIONAL PATH 114
4.9 RECAPITULATION OF COMPUTATIONAL FLOW 116
4.10 GENERAL PROCEDURE FOR DETERMINING CONSISTENCY AND ALLOWABILITY IN A MODEL OF REGULAR RELATIONS 118
4.11 SUMMARY OF CHAPTER AND CONSTRAINT THEORY TOOLKIT 122
4.12 QUERIES FOR THE REGULAR STUDENT 127
Chapter 5 DISCRETE AND INTERVAL RELATIONS 129
The Diminished Utility of Metamodels 129
5.1 Metamodel Issues and Perspectives 129
5.2 The General Taxonomy and Primary Property of Discrete Relations 130
5.3 Boolean Relations 130
5.4 Topological Implications 136
5.5 Allowability of Discrete Computations 138
5.6 Inequality Relations 140
5.7 Summary 143
5.8 PROBLEMS FOR THE DISCRETE STUDENT 145
Chapter 6 THE LOGICAL STRUCTURE OF CONSTRAINT THEORY 146
A Compact Summary 146
6.1 Overview 146
6.2 Postulates and Philosophical Assumptions 146
6.3 Definitions 147
6.4 Theorems 147
6.5 Graphs of the Logical Structure of Constraint Theory 148
6.6 Completeness 148
Chapter 7 EXAMPLES OF CONSTRAINT THEORY APPLIED TO REAL- WORLD PROBLEMS 153
7.1 Apologies Not Required 153
7.2 Cost as an independent variable (CAIV) 153
7.3 The Kinematics of Free-Fall Weapons 161
7.4 The Deflection of an Earth-Threatening Asteroid Employing Mass Drivers 166
Chapter 8 MANAGER AND ANALYST MEET AGAIN 173
Gists and Schizophrenia 173
Appendix A COMPUTATIONAL REQUEST DISAPPOINTMENTS RESULTS OF THE USC ALLOWABILITY PROJECT
Appendix B GRAPH THEORY OVERVIEW 186
Why was the Bipartite Graph Chosen? 186
Appendix C THE LOGIC OF "IF" AND "IF AND ONLY IF" 190
Appendix D VECTOR SPACES APPLIED TO GRAPH THEORY 192
References 195
Index 197