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Über dieses Buch

Packed with new material and research, this second edition of George Friedman’s bestselling Constraint Theory remains an invaluable reference for all engineers, mathematicians, and managers concerned with modeling. As in the first edition, this text analyzes the way Constraint Theory employs bipartite graphs and presents the process of locating the “kernel of constraint” trillions of times faster than brute-force approaches, determining model consistency and computational allowability. Unique in its abundance of topological pictures of the material, this book balances left- and right-brain perceptions to provide a thorough explanation of multidimensional mathematical models. Much of the extended material in this new edition also comes from Phan Phan’s PhD dissertation in 2011, titled “Expanding Constraint Theory to Determine Well-Posedness of Large Mathematical Models.”

Praise for the first edition:

"Dr. George Friedman is indisputably the father of the very powerful methods of constraint theory." --Cornelius T. Leondes, UCLA

"Groundbreaking work. ... Friedman's accomplishment represents engineering at its finest. ... The credibility of the theory rests upon the formal proofs which are interspersed among the illuminating hypothetical dialog sequences between manager and analyst, which bring out distinctions that the organization must face, en route to accepting Friedman's work as essential to achieve quality control in developing and applying large models." --John N. Warfield



Chapter 1. Motivations

What is Constraint Theory and why is it Important?

Chapter 1 provides an example of low dimension, showing how problems of consistency and computational allowability can arise in even simple situations. The reader is introduced to the two main characters of this book – an experienced manager and an analyst – whose dialogue will hopefully illuminate this book’s many concepts. The bipartite graph is introduced, as are a few simple rules. However, the analyst argues that, in order to expand the tools to models of very high dimension, and in order to trust the reliability of these tools, the theory must be based on a more rigorous foundation.

George J. Friedman, Phan Phan

Chapter 2. The Four-Fold Way

How to Perceive Complex Mathematical Models and Well-Posed Problems

Chapter 2 begins to establish the rigorous foundation by defining four “views” of a mathematical model: (1) set theoretic, (2) submodel family, (3) bipartite graph, and (4) constraint matrix. The first two views are full models; the last two views are metamodels. Then, rigorous definitions of consistency and computational allowability are made in the context of these views.

George J. Friedman, Phan Phan

Chapter 3. General Results

From Protomath to Math to Metamath

Chapter 3 discusses the similarities between language and mathematics and provides some general consistency and computability results with respect to any class of relation. In order to provide a basis for the next three chapters, three classes of exhaustive and mutually exclusive relations are defined: discrete, continuum, and interval.

George J. Friedman, Phan Phan

Chapter 4. Regular Relations

Searching for the Kernels of Constraint

Chapter 4 addresses the constraint theoretic properties of regular relations, the most important type within the continuum class and the most often employed in the development of multidimensional math models. The rules presented in Chapter 1 are rigorously proved employing the foundations of Chapters 2 and 3. The topological properties of the bipartite graph are analyzed to provide key conclusions of the model’s consistency and computational properties. The Basic Nodal Square (BNS), is identified as the “kernel of intrinsic constraint” and is accused of being the culprit in model inconsistency and unallowable computability. Trivially easy computations on the bipartite graph – such as circuit rank and constraint potential – are shown to have enormous utility in locating the BNSs which hide in tangled circuit clusters. Additionally, the step-by-step algorithm for locating BNSs within a model graph are provided.

George J. Friedman, Phan Phan

Chapter 5. Model Consistency and Computational Allowability

Chapter 5 discusses the general issue of constraint propagation through a connected model graph of regular relations. A detailed procedure for determining model consistency and computational allowability in such a model is introduced. In addition, techniques to detect overlapping BNSs, relieving over- and under-constraint, expanding resultant constraint domains, and processing computational requests made on a model are demonstrated. Finally, Section 5.8 provides a constraint theory toolkit to employ the rules and theorems in an orderly manner,and which can find BNSs trillions of times faster than brute force approaches.

George J. Friedman, Phan Phan

Chapter 6. Discrete and Interval Relations

The Diminished Utility of Metamodels

Chapter 6 addresses the constraint properties of discrete and interval functions such as those from Boolean algebra, logic, and inequalities.

George J. Friedman, Phan Phan

Chapter 7. The Logical Structure of Constraint Theory

A Compact Summary

Chapter 7 provides a compact structure of constraint theory. All postulates, definitions, and theorems are listed and their logical interrelationships are displayed in the form of quasi-bipartite graphs

George J. Friedman, Phan Phan

Chapter 8. Examples of Constraint Theory Applied to Real-World Problems

Chapter 8 presents detailed examples of the application of constraint theory to the areas of operations analysis, kinematics of free-fall weapon delivery systems, and the dynamics of deflecting asteroids with mass drivers.

George J. Friedman, Phan Phan

Chapter 9. Manager and Analyst Meet Again

Gists and Schizophrenia

Chapter 9 summarizes the book and provides the manager and analyst a final opportunity to dialogue and discuss their common background.

George J. Friedman, Phan Phan


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