The Mathematical Principles of Natural Languages

The First Course in Physical Linguistics

Isaac Newton (1642-1727) published his Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) in 1687. Three hundred and twenty years later, this book extended Newton’s Laws from physical world to spiritual world. To set up the mathematics of minds is the goal of this textbook.
Computational Verb Theory(CVT) was invented by Tao Yang in 1997 in the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. In 2002, based on CTV, Professor Yang invented the first measurable linguistics known as Physical Linguistics(PL). PL is the theoretical basis for the coming cognitive industry in the next fifty years when services will be personalized. To make the spiritual facets of cognition measurable, in 2004 Professor Yang set up the Theory of the Unicogse(TOU). The Unicogse is the super being containing the Universe and the Cognition as a pair of dual beings. TOU predicts that natural languages are interfaces between the Universe-Cognition dual. By applying the Universe-Cognition duality, all spiritual facets of minds become measurable in physical world. Furthermore, a measurable paradigm of cognition can be set up for building robots with human minds. TOU and PL point out the way of implementing human-machine co-evolution in many centuries to come.
PL is the only theory to bridge the huge gap between natural sciences and social sciences. TOU and PL paved the way of measuring cognitions in social sciences such that all experimental methodologies developed in natural sciences can be easily applied to social sciences. With the help of TOU, in PL the concepts such as meanings and feelings can be defined quantitatively based on the mass of a piece of information and the work done by it upon natural languages. By using similar methods provided in TOU, PL makes natural languages the most complicated and advanced mathematical theories.
In the last decade, CVT has been growing up into a multidisciplinary scientific field attracting attentions of researchers from information sciences, linguistics, economics, biology, psychology, physics and computer sciences. CVT is now taught in many Universities and Institutions worldwide. Many commercial products have been developed based on PL for applications such as intelligent transportation systems(ITS), stock trading, visual flame detectors, CCTV surveillance, intelligent controllers, image understanding engines, image search engines and many others. PL is the initiative of the next industrial revolution called cognitive industry in which personalized services will be the next wave following the history of personalizing transportation, computation and communication.
Written by the founding father of CVT, PL and TOU, this is a lucid, solid and timely textbook for professionals, scientists, academic researchers and students in information sciences, cognitive sciences, cognitive engineering, robotics, linguistics, fuzzy logic, computer sciences and control engineering.
(December 28, 2007, Tucson, Arizona, USA.)

Preface[pdf]
Table of Contents[pdf]
Index[pdf]
Computational Nouns[pdf]
Computational Verbs[pdf]
Computational Verb Collapses and Computational VerbExtension Principles[pdf]

Book Infos

Tao Yang, The Mathematical Principles of Natural Languages --The First Course in Physical Linguistics, Dec. 2007. USD $297.00
Preface
Exactly ten years ago, in September 1997 I wrote two technical reports[79, 80] in which the issues of modeling verbs in natural languages by using nonlinear dynamical systems were addressed. Both reports evidenced the beginning of the computational verb theory. In these ten years, the following milestones marked the most important events in the history of computational verb theory: 1. In 2002, I published the book [96] and the jargon "physical linguistics" was officially assigned the meaning of "a measurable linguistics". 2. In 2004, I published the book [100] in which the Theory of the Unicogse was set up as the foundation of measuring cognition. 3. In 2002, the home company of computational verb, Yang's Scientific Research Institute, L.L.C.(Yang's Scientific), was set up. Since then, this company has been providing all financial supports to the research of computational verb theory. It these ten years, the research of computational verb theory is an epic voyage of one man, one family and one company. Since computational verb theory is brand new to both scientific and engineering communities, it was fateful for me to work alone as the only pioneer in this new line of research. On the one hand, I was lucky to be the chosen one to witness the growth of such a beautiful theory and on the other hand, I was unlucky to shoulder the tremendous pressures of pioneering such a staggeringly wonderful land of promise. And yet, if the time could rewind back ten years, I would like to get a Ph.D degree and become a professor in any university to enjoy a plain and boring life by following whatever mainstream researches and publish around ten research papers in mainstream journals each year. Unfortunately and luckily, the history cannot be rewritten. Boosted by the commercial successes of computational verb theory in Yang's Scientific, more and more resources have been put into the researches of computational verb theory. On the one hand, more and more students and researchers around the world want to learn computational verb theory. On the other hand, I have been a professor in Xiamen University since this July and have chances to teach my young students the computational verb theory. Therefore, a comprehensive textbook becomes the immediate demand for pushing the researches of computational verb theory further. The outline of this book was drafted in the spring breeze in 2007 when I was a visiting professor in the Department of Physics, University of Cagliari, Italy. The first draft was finished in the hot summer of 2007 when I stayed with my family in Shanghai and waited for the birth of my daughter Yang Yang. Many rounds of revision were mainly undertaken in my apartment in Xiamen University. Special Styles of the Book Some special styles are adopted to help the reader browser through the text quickly. For theorems, lemmas, definitions and etc., the symbols £ are used to terminate statements. For remarks, examples and proofs, the symbols ¨, F and ¥ are used to show the ends of statements. To fit its style as a textbook, the scopes of this book will cover as broad as possible and yet to make it as self-contained as possible as well. The students of this course will only need some basics of calculus to understand most of the chapters. Some chapters are prepared for advanced readers and can be skipped by undergraduates. Organization of the Book In Chapter 1 the purpose, history, scope and blueprints of long-term applications of computational verb theory will be addressed. Computational verb theory will function as the very basis for building the cognitive engineering and as an important building block for personalizing services in the next fifty years. In Chapter 2 the Theory of the Unicogse will be presented as the basis for building a measuring mechanism into cognition. In the Unicogse , the Universe-Cognition duality functions as the bridge between minds and physical world such that the "spiritual" quantities can be measured indirectly in the physical world. The Theory of the Unicogse guarantees that we can study many social sciences by using the methodologies used in the study of natural sciences. In Chapter 3 the Cognition mechanics will be presented as the basis for studying the concepts of infor- mation, truth, and meaning for a measurable platform. In this theory we know that the meanings of a piece of information is the work done by it upon a Cognition medium. The inner energy of a piece of information is the temperature of this piece of information. Many concepts in mechanics can be extended into the Cognition and be used in the study of minds. In Chapter 4 the computational nouns are defined based on attributes and attribute values. Attribute values can be modeled by using pdf's and fuzzy membership functions. Adjectives are modeled as modifiers to attribute values. In Chapter 5 computational verbs are defined as dynamical systems. Computational verbs can also be viewed as action values of nouns if we treat nouns as the only primary components in natural languages. Adverbs and preverbs are modeled by using the modifiers to dynamical systems. In Chapter 6 the computational verb collapses and computational verb extension principles are presented. A computational verb collapses into an adjective when its dynamics are extracted away from it. In a reverse process, when we add dynamics into an adjective, we can generate different verbs from an adjective. This kind of relation between adjectives and verbs can be generalized into the logic operations between adjectives and verbs. Computational verb collapses and computational verb extension principles provide basic tools for extending conclusions in conventional disciplines into physical linguistics and vice versa. In Chapter 7 the concept of computational verb set will be presented. All elements of a given verb set share the same dynamical properties or the same computational verb collapse. The basic properties and computations of computational verb sets will be presented. In Chapter 8 computational verb numbers are studied as special kinds of computational verbs. The computations between computational verb numbers are presented. We define the computations between verb numbers based on their evolving functions or their computational verb collapses. In Chapter 9 the canonical forms of computational verbs will be presented. The canonical forms of computational verbs reduce the number of computational verbs dramatically when we need to implement computational verbs in a computer system. The methods of finding and defining different kinds of canonical forms will be presented as well. In Chapter 10 the distances and similarities between computational verbs will be defined and studied. The similarity between computational verbs is a very important tool for reasoning via computational verbs. In this chapter, many different ways of calculating the distances and similarities of computational verbs will be presented. In Chapter 11 we will introduce different kinds of uncertainties caused by computational verbs. The ambiguities in the states and parameters of computational verbs are calculated based on the ambiguities in verb collapses. Different ways of measuring ambiguities in computational verbs are presented. In Chapter 12 the cross correlations and correlations among computational verbs will be studied. Verb correlations can be used to find the cause-effect relations between different dynamical processes and therefore they are efficient tools of discovering knowledge from dynamical records. In Chapter 13 we will study computation verb logic that is the basis for representing knowledge in the form of verb rules. The liar's paradox will be studied based on verb logic. In Chapter 14 the inference based on computational verb rules will be presented. Since verb rules are the most important tools to encode dynamical experiences, the reasoning based on verb rules plays the most important role in applying computational verb theory to engineering problems. In this chapter the reasoning based on adjectives will also be studied. In Chapter 15 we will present the method of constructing computational verb decision trees from databases of historical records. Verb decision trees extend the data ming and knowledge discovery from static data into dynamical data. Computational verb entropy will be used to extend ID3 learning algorithm into data-mining historical records. In Chapter 16 we will construct dynamical systems based on verb rules and verb reasoning. We will show that simple verb rules can generate complex behaviors. The tremendous flexibility of modeling dynamics based on natural languages will be shown. In Chapter 17 the structures, designs and stabilities of computational verb controllers will be studied. As an important application of verb reasoning to engineering problems, a verb controller can be designed based on partial knowledge of the dynamics of control plants. Verb controllers are typical examples of industrial applications of computational verb theory. In Chapter 18 the applications of computational verb theory to image processing will be presented. Based on the semantic network of cognitive features in images and spatial computational verbs, real-time and effective image processing algorithms can be developed for applications such as image search engine and real-time object recognition. In Chapter 19 the method of finding the probability of dynamical events based on conventional events will be given. The dynamics of an event can change the probability in three different ways. Based on the probability of a verb event, we can find the probabilities of verb events modified by using different verb modifiers. In Chapter 20 we will apply computational verb theory to model stock markets and find trading patterns of stocks. We will present the principles of studying economical systems based on cognition-based modeling methods. In Chapter 21 we will present a systematic way of recovering feelings from texts for building a new generation of man-machine interface based on the concept of cognitive engineering. The next generation of robots for personalizing services will be built based on this systematic method. Xiamen, September 2007 Tao Yang(杨涛)
Table of Contents
Preface : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : VI 1. A Beautiful Theory of Natural Languages : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 1.1 The Beauty of a Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Evolve Intelligent Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 A Brief History of Computational Verb Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Symbol Grounding Problems in the Simulation of Life System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.5 From Measurement-based to Cognition-based Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.6 The Scientific and Technical Foundations for the Revolution of Cognitive Engineering . . . . . . . . . 8 1.7 The Long-term Influences of Computational Verb Theory on Sciences and Technologies . . . . . . . . 9 1.8 Applications of Computational Verb Theory to Different Industrial Sections . . . . . . . . . . . . . . . . . . 11 2. The Theory of the Unicogse : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Universe -Cognition Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 Energy, Matter and Truth Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Truth Velocity and Cognitive Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.5 Truth Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.1 Truth Distance in Boolean Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5.2 Truth Distance in Multivalued Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Vacuum and Medium Velocities of Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.1 Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.2 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.3 Mathematics of Truth Reflection and Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.7 The Big Bang Theory of the Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.8 Look into the Cognition from the Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2.9 Temperature in the Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9.1 Formal Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.9.2 Proposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.10 Conventional Engineering Versus Cognitive Engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 3. Cognition Mechanics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 31 3.1 The Cognition Properties of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.1 Cognition Mass, Volume and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.2 Temperature of Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.1.3 Communication Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2 Cognition Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.1 Cognition Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.2 Cognition Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.2.3 Vectors in the Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.4 Shape and Size in the Cognition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.3 Cognition Force and Cognition Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 Information, Meanings and Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.5 Newton's First Law and Its Unicogse Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.6 Newton's Second Law and Its Unicogse Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.7 Newton's Third Law and Its Unicogse Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 Newton's Law of Gravitation and its Unicogse Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.9 Potential Truth and Kinetic Truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10 Fraction and Its Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10.1 Fuzzification and Defuzzification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.10.2 Image Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.11 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4. Computational Nouns : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 41 4.1 Nouns as Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.2 Characteristic Functions as Attribute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.3 Fuzzy Membership Functions as Attribute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.4 Attribute Values in Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5 Logic Operations Among Attribute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5.1 Probabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.5.2 Possibilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.6 Modifying Attribute Values by Using Adverbs of Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.6.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.6.2 Possibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.6.3 Fuzzy Linguistic Hedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.6.4 Probability Linguistic Hedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.7 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5. Computational Verbs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 51 5.1 Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1.2 Types of Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.2 Qualitative Behaviors of 3-Dimensional Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.3 Definitions of Computational Verb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 5.4 Modifiers of Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.4.1 Topologically Equivalent Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.4.2 T-homomorphic Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.4.3 Modifying <=-Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 5.4.4 Adverbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4.5 Preverbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.5 Computational Verbs as Action Values of Nouns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 6. Computational Verb Collapses and Computational Verb Extension Principles: : : : : : : : : : : 73 6.1 Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.2 Computational Verb Collapses of Monotonic Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.1 Sigmoid Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.2 Uniform Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.3 Linear Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 6.2.4 Reciprocal Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.2.5 Exponential Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2.6 Bell-shaped Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 6.3 Computational Verb Extension Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.3.1 Stage 1: Induce ª From One Attribute Value '= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 6.3.2 Stage 2: Construct Dynamics From Template Computational Verbs . . . . . . . . . . . . . . . . . . . 86 6.4 Extending Logic Expressions Between Two Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.4.1 Fuzzy ORing Between Two Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.4.2 Fuzzy ANDing Between Two Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.5 Trimmed Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.6 Extend a Single Computational Verb Collapse to Get More than One Computational Verb . . . . . 94 6.7 Extend Fuzzy Inference into Computational Verb Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 6.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 7. Computational Verb Sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 101 7.1 From Crisp and Fuzzy Sets to Computational Verb Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 7.2 Computational Verb Sets Induced by Using Computational Verb Collapses . . . . . . . . . . . . . . . . . . 103 7.3 Properties of Computational Verb Sets Induced by Using Computational Verb Collapses . . . . . . . 105 7.4 Computational Verb Sets Constructed from Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7.5 Computational Verb Sets Whose Elements Have the Same Computational Verb Collapse . . . . . . 107 7.6 Computational Verb Collapses of Computational Verb Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 7.6.1 Computational Verb Collapse of Computational Verb Set of the Same Type . . . . . . . . . . . 115 7.6.2 Computational Verb Collapse of Computational Verb Set of the Same Dynamics . . . . . . . 116 8. Computational Verb Numbers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 117 8.1 Definition of Computational Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 8.2 Basics of Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.2.1 Categories of Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.2.2 Basic Definitions for Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 8.3 Verb Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.3.1 Rules of Verb Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.3.2 Examples of Verb Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 8.4 Computational Verb Collapses of Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 8.5 Measurement of Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.5.1 V as Metric Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 8.5.2 Distances Between Verb Numbers Based on Computational Verb Collapses . . . . . . . . . . . . 130 8.6 Computational Verb Extension Principle of Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 8.6.1 Extension Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.6.2 An Example of Extending Verb Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 8.7 Computation Between Verb Numbers with Preserved Qualitative Properties of Dynamics . . . . . 133 8.7.1 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 8.7.2 Substraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 9. Canonical Forms of Computational Verbs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 137 9.1 How to Program All Computational Verbs in English . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 9.1.1 Decompose Computational Verbs Based on Dynamics Near Hyperbolic Equilibria . . . . . . 138 9.1.2 Piecewise Linear Composition of Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 9.2 Piecewise Linear Decomposition of Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 9.3 Segmenting Trends of Smooth Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 9.4 Canonical Forms in Become . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.4.1 Definition and Existence of Become . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 9.4.2 Become's with Two States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.4.3 Become's with One State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 9.4.4 Become's with No State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.4.5 Become's of Unchanged . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.5 Clustering Centers of Computational Verb Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 9.6 Compose Computational Verb Using Canonical Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . 158 9.7 Use Perceptron to Classify Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 10. Distances and Similarities of Computational Verbs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 163 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 10.2 Distances and Similarities of Saturated Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.2.1 Saturate Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 10.2.2 Definitions of Distances and Similarities of Saturated Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . 166 10.2.3 Distances and Similarities for Computational Verbs with Different Life Spans . . . . . . . . . . 167 10.2.4 Effects of Saturate Functions on Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . 170 10.3 Distances of Computational Verbs Based on Saturated Evolving Functions . . . . . . . . . . . . . . . . . . . 171 10.3.1 Cases of Continuous-Time Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 10.3.2 Cases of Discrete-Time Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 10.4 Similarities of Computational Verbs Based on Saturated Evolving Functions . . . . . . . . . . . . . . . . . 178 10.4.1 Cases of Continuous-Time Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 10.4.2 Cases of Discrete-Time Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 10.5 Distances and Similarities Defined in Attribute Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 10.5.1 Verb Distances for Cases of Having Fuzzy Dynamical Systems as Evolving Functions . . . . 182 10.5.2 Discrete-Time Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 10.6 Distances and Similarities Based on Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . 185 10.7 Computational Verb Similarity Based on Minkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 10.7.1 Cases with p = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 10.7.2 Cases with p = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 10.7.3 Properties of Computational Verb Similarities Based on Minkowski Inequality . . . . . . . . . . 190 10.7.4 Examples of Balanced Computational Verb Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 11. Uncertainties in Computational Verbs : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 195 11.1 The Measures of Uncertainty Based on Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . 196 11.1.1 Measures of Fuzziness in Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 11.2 The Measures of Uncertainties in Similarities of Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 11.3 Measures of Ambiguity of Computational Verbs Based on Computational Verb Collapses . . . . . . 200 11.4 Ambiguity of Decrease in a RC Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.4.1 Ambiguity in Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 11.4.2 Ambiguity in States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.5 Ambiguities of Computational Verb Sets with Normal Distributions as Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 11.5.1 Linear Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 11.5.2 Nonlinear Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 11.5.3 Ambiguity in the States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 11.6 Ambiguity of Computational Verb Sets with Exponential Distributions as Computational Verb Collapses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 11.6.1 Ambiguity in the States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 12. Cross Correlations and Correlations of Computational Verbs : : : : : : : : : : : : : : : : : : : : : : : : : : : 213 12.1 Computational Verb Cross-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.1.1 Example 1: Cross-correlation between Increase and Decrease . . . . . . . . . . . . . . . . . . . . . . . . . . 213 12.1.2 Example 2: Cross-correlation between Increase and Stay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12.2 Cognitively Correct Computational Verb Cross-correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 12.2.1 Eliminate Sensitivity to Initial States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 12.2.2 Eliminate Sensitivity to Absolute Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 12.3 Normalized Computational Verb Cross-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 12.4 Computational Verb Cross-correlations between Computational Verbs with Different Life Spans 216 12.5 Computational Verb Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.5.1 Discrete-time Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.5.2 Continuous-time Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 12.5.3 Properties of Computational Verb Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 12.5.4 Other Computational Verb Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 12.6 Applications to Knowledge Discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.6.1 Results Based on Verb Cross-correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 12.6.2 Results Based on Verb Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 13. Computational Verb Logic : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 221 13.1 Verb Logic Operations in Attribute Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 13.2 New Interpretation of the Paradox of Liar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 13.3 Verb Liar's Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 14. Inference Based on Computational Verb Rules: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 231 14.1 Inferences with Qualitative Invariant Adjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 14.1.1 Inference with Single Adjective Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 14.1.2 Inference with Adjective Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 14.2 Computational Verb Inference with Qualitatively Invariant Computational Verbs . . . . . . . . . . . . . 235 14.2.1 Inference with Single Computational Verb Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 14.2.2 Inference with Computational Verb Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 14.3 Verb Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14.3.1 Verb Inference with Single Verb Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 14.3.2 Verb Inference with Verb Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 14.3.3 Reconstruct Computational Verbs from Similarity Functions: Deverbi¯cation . . . . . . . . . . . 245 15. Computational Verb Decision Trees : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 247 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 15.2 Structure of Computational Verb Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 15.2.1 Disease Breakouts Depending on Dynamics of Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 249 15.2.2 Decision Trees for Controlling Greenhouses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 15.3 Building Computational Verb Decision Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 15.3.1 The Dynamics Involved in Classification Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 15.3.2 The Induction of Verb Decision Trees by Using Verb Entropy . . . . . . . . . . . . . . . . . . . . . . . . 256 16. Computational Verb Systems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 261 16.1 Rule-wise Computational Verb Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 16.2 Definitions of Rule-wise Computational Verb Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 16.2.1 Discrete-time Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 16.2.2 Continuous-time Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 16.3 First-Order Rule-wise Linear Computational Verb Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 16.3.1 Implementing Rule (16.27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 16.3.2 Implementing Rule (16.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 16.3.3 From Order to Chaotic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 16.3.4 Effects of Parameter L on Stabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 16.3.5 Effects of Parameter a on Stabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 16.4 Stability of Discrete-time Rule-wise Linear Computational Verb Systems . . . . . . . . . . . . . . . . . . . . 269 16.4.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 16.5 Robust Stability of Rule-wise Linear Computational Verb Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 273 16.5.1 Continuous-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 16.5.2 Discrete-time Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 16.6 Controlling Discrete-time Rule-wise Linear Computational Verb System with Disturbance . . . . . 275 16.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 17. Computational Verb Controllers : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 281 17.1 Design Verb PID Controllers by Generalizing Fuzzy Gain Schedulers . . . . . . . . . . . . . . . . . . . . . . . . 281 17.2 Basics of PID Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 17.2.1 Fuzzy Gain Schedulers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 17.2.2 Generalize Fuzzy Control Rules into Verb Control Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 17.2.3 Implementing Verb P-controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 17.2.4 Implementing P-controller for a Second-order Plant: Example . . . . . . . . . . . . . . . . . . . . . . . . 293 17.2.5 Implementing Verb PD-controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 17.2.6 Implementing Verb PID-controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 17.3 Design of Stable Computational Verb Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 17.4 Designing Computational Verb Control Laws. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 17.4.1 An Example of Second-order Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 17.4.2 Conventional Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 17.4.3 The Computational Verb Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 17.5 Verb Control of Chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 17.5.1 The Rule-wise Linear System Model for Chua's Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 17.6 The Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 17.7 Synchronization of Henon Maps by Using Verb Controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 17.7.1 The Computational Verb Control Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 17.7.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 18. Computational Verb Image Processing : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 331 18.1 Effective and Realtime Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 18.1.1 What is Effective and Realtime Image Understanding? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 18.2 Spatial Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 18.2.1 Constructing Canonical Spatial Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 18.2.2 Examples of Spatial Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 18.2.3 Composed Spatial Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 18.2.4 Similarities between Spatial Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 18.2.5 Verb Similarity for Canonical Forms of Spatial Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 18.3 Image Processing Using Verb Similarities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338 18.3.1 Vertical Texture Enhancement and Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 18.3.2 Robust Card Detection with Virtual Transparency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 18.4 Semantic Decomposition of Cognitive Features in Images by Using Computational Verbs . . . . . . 340 18.4.1 Case Study: Eyes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 18.4.2 Case Study: Skin Tones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 18.5 Image Understanding at the Physical Linguistic Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 18.6 Cognitive Image Search Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 18.6.1 The Blind-Men-and-the-Elephant Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 18.6.2 The PicSeer PL Image Search Engine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 18.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 19. Computational Verb Statistics : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 359 19.1 Basic Knowledge of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 19.2 Computational Verb Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 19.3 Computational Verb Event Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 19.3.1 Computational Verbs with Exponential Evolving Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 364 19.3.2 Different Types of Verb Event Modifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 19.4 Effects of Computational Verb Modifiers on Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 19.4.1 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 19.4.2 Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 19.4.3 Copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 19.5 Discrete Verb Probability Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 19.6 Verb Modifiers with Saturated Evolving Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 20. Cognitive Economics: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 369 20.1 Cognitive Stock Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 20.2 Modeling Stock Market Using Computational Verbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 20.2.1 Get the Feelings Out of the Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 370 20.2.2 Cognitive Stock Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 20.2.3 Calculating Cognitive Measures in Cognitive Stock Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 20.3 The Effects of Russell's Annual Index Reconstitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 20.4 Modeling Russell Reconstitution Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 20.4.1 Computational Verb Modules for Modeling Stock Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 20.4.2 Find the Computational Verb Similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 20.5 Understanding Russell Rebalance Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 20.5.1 Before Russell Rebalance Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 20.5.2 After Russell Rebalance Date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 20.5.3 Summary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 21. Feeling Retrieval from Texts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 387 21.1 Introduction: What is the "Feeling" in Natural Languages? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387 21.2 Differences Among Individual Brains and Digital Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 21.2.1 Differences Among Human Brains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389 21.2.2 Differences Among Human Brains and Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390 21.3 How to Mail Human Minds? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 21.4 Mathematics of Natural Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 21.4.1 Mathematical Functions for Noun Phrases and Adjectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 21.4.2 Mathematical Functions for Verb Phrases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 21.4.3 Mathematical Models for Adverbs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 21.5 Get the Feeling Out! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 21.5.1 Translate a Sentence into Mathematical Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 21.5.2 Translate a Set of Sentences into Mathematical Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 21.5.3 Measurements + Grounded Symbols = Feelings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 408 Index : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 415
Index
Lp-metric, 129 T-homomorphic operator, 70 F, VII ¨, VII ¥, VII V, 117, 120, 121, 129 ¾-¯eld, 359, 359 _, 34 ^, 34 s-norm, 242, 334, 335 s-plane, 306 t-norm, 242, 334, 335 (energy, matter)-(verb, noun) duality, 15, 17 (energy, matter)-pair, 15 (verb, noun)-pair, 15 [bit], 31{33, 35, 387 Truth , 28{42, 101 kinematic, 29, 38 kinetic, 38 potential, 30, 38 pure, 29, 30, 32 Cognition , 13{30, 41, 42, 101, 102, 117 Cognition Law of Inertial, 36 Cognition distance, 32{33 Cognition fraction, 32 Cognition gas, 33 Cognition gravitational constant, 37 Cognition heat, 39 Cognition inertial, 37 Cognition liquid, 33 Cognition mass, 31 measure, 31 Cognition medium, VIII, 32, 32, 36 Cognition particle, 38 Cognition reference point, 33 Cognition resultant, 36, 36 Cognition shape, 33{34 Cognition size, 33{34 Cognition solid, 33 Cognition space, 32 Cognition temperature, 39 Cognition vector, 33 Cognition volume, 31 Being, 391 Being, 4, 15{19, 23, 26, 28, 387, 388, 391, 392, 400, 406 Truth, 15{18 Truth distance, 19{21, 23 Truth of energy, 17 Truth of matter, 17 Truth ray, 23 Truth velocity, 19, 20, 22, 23, 26, 28 Unicogse, 15 Universe-Cognition duality , VII, 1, 29, 31, 38, 65 Unicogse , 13{30 Unicogse set, 101 Universe , 13{30 Universe +Cognition , 13 Universe -Cognition duality, 15 not fuzzy, 48 become, 150, 149{151 center, 150 no state, 151 one state, 150{151 saddle, 150 stable focus, 150 stable node, 150 two states, 150 unchanged, 151 unstable focus, 150 unstable node, 150 be-event, 365 absolute truth, 17 absolute zero, 44 Cognition , 29, 30 abstract algebra, 51 action, 37 action value, VIII, 2, 3, 71, 71{73, 151 actuator, 11, 14, 406 adenosine, 29 adjective, 14 adjective rule, 231 adverb, VIII, 4, 9, 14, 61{69, 131, 149, 240{243, 246, 282, 297, 359, 392, 397{399 a little, 363 before, 149 casually, 68 extremely, 48 fairly, 48 fast through 0, 97 fast, 68, 196, 239{242, 251, 288, 298, 366, 399 from above, 196 from blow, 196 here, 56 in greedy, 373 in panic, 373 late, 402 more or less, 397 more than, 397 more than, 49 more-or-less, 48 more or less, 48 near, 65, 118 negatively, 48 normally, 68 not, 48, 397 now, 56 often, 65, 241 positively, 48 probably, 118, 149 right now, 360 roughly, 48 self, 56 slightly less, 48 slightly more, 48 slightly, 397 slowly, 68, 196, 239{241, 251, 287, 288, 302, 365, 399 slow, 405 smoothly, 196 somewhat, 48 sort of, 397 through 0, 95, 97 unconditionally, 241 very fast, 149, 239, 399 very much, 363 very slowly, 239 very soon, 360 very well, 66 very fast, 240, 241 very slowly, 241 very well, 66 very, 43, 47, 48, 241, 397{399 very slowly, 240 very very, 48 very soon, 365 well, 66, 67 will, 366 yesterday, 65 exponentially, 209 linearly, 207 nonlinearly, 208 polynomially, 208 probably, 119 very, 405 equivalently, 239 fast, 298 late, 405 slowly, 298 cognitive, 65{66 ¯ring, 239, 297, 300, 301, 303, 308 measurable, 14 of degree, 47{49, 85, 132 operator, 241 set, 240, 241, 246, 290{291 similarity, 240, 241, 243, 244 spatial, 63, 65 temporal, 65, 68, 129 transformation, 241 adverb composition, 242, 243 adverb of degree, 397 a little bit, 85 more or less, 85 most, 85 very, 85 adverb similarity, 240 adverbial, 131, 149, 359, 362 right now, 362 very soon, 362, 363 temporal, 129, 362 AI, 390 strong, 391, 407 strong , 407 algebra, 359, 359 algebraic preordering, 64 algebraic structure, 51 ambiguity, VIII, 10, 11, 29, 33, 41, 133, 195{197, 225, 333, 369 computational verb, 200{211 in dynamics, 202{204 in state, 205 natural language, 3 analytic grammar, 29 anchor point, 338 AND verb, 221 ANN, 6, 6, 7 AntiBeing, 26, 387, 388, 406 antimatter, 28 arrow of time, 53, 64 asymptotical stability, 311 asymptotically stable equilibrium, 53 attribute, 42, 42 categorical, 248 numerical, 248 ordered, 248 time-dependent, 248 attribute space, 181{185 attribute transform, 64 attribute tree, 43 attribute value, VIII, 2, 42, 43{49, 55{57, 59, 71, 73{75, 77, 84{87, 101{103, 181, 182, 184, 221, 248, 254, 257, 342, 393{396, 400, 402{404 1/2, 342 1/4, 342 3/4, 342 Big, 252 Medium, 252 NO, 251, 254 No, 259 Small, 252 Very Big, 252 YES, 251, 254 Yes, 259 Zero, 252 a little bit around 0, 97 anxious, 406 around 0, 94, 97, 98 around 23, 397 around 35, 397 around p, 98 bigger, 290, 293 bigger than current, 266 big, 149, 285, 287, 288, 290, 293, 298, 301, 302, 314 close, 342 cloudy, 254 cold, 394, 395 cool, 254 current less than ten, 149 current, 149, 265, 314, 400 fast, 399 foggy, 402 heavy, 394, 395, 402 high, 149, 254, 257, 259, 396, 398 hot, 254 less than current, 149 less than current, 265 low, 396 medium, 288, 290, 298, 301, 302, 314, 396 mild, 254 near one, 291 near zero, 291 negative-big, 284 negative-medium, 284 negative-small, 284 negative, 284 normal, 254, 259 positive-big, 284 positive-medium, 284 positive-small, 284 positive, 284 rainy, 254 rare, 394 red, 395 satisfied, 396 smaller, 290, 293 small, 149, 285, 288, 290, 293, 298, 301, 302, 311, 314, 405 strong, 254 sunny, 254 unhappy, 402 very big, 290 very heavy, 402 very high, 396 very low, 396 very small, 290 weak, 254 wide opened, 401 zero, 288, 290, 301, 302, 314 dynamical, 247 fuzzy, 73, 74, 78, 257, 405 pdf, 73, 74, 78, 103 set, 101 space, 71{73 time-variant, 248 automorphism, 51 average brain, 388 balance factor, 191, 264{267, 292, 293, 315, 319 Banach space, 53 bandwidth, 53 BarSeer, 5 baryon, 28 BE event, 360, 362{364 beautiful theory, 1{12 behavior-based robotics, 6 benchmark evolving function, 98 benchmark implementation, 97 Bessel function of the ¯rst kind, 173 bias, 158 bifurcation Hopf, 241 period-doubling, 241 Big Bang, 15, 26, 28 Big Bang theory, 26, 28 binary verb relation, 238 black hole Cognition , 31 blind-men-and-the-elephant method, 349 body, 36{38 Cognition , 31, 36{38 Universe , 31 dynamics, 31 idealized, 31 information, 31{34, 36, 38, 39 motion, 31 mutually interacted, 37 body-mind duality, 1 Boolean logic, 2, 14, 18, 20{22, 29, 30, 32{35, 387{391, 407 Bose-Einstein condensate, 33 bottom-up, 340 branch, 248 brightness pro¯le function, 340 canonical form in become, 149{151 canonical form of computational verb, 137 canonical forms in become, 150 canonical verb, 137 capacitor, 57 card, 357 card counter, 339, 356 Cartesian product, 107, 238 cascade, 53 Cauchy-Schwarz inequality, 217, 218 CCTV signal, 5 CD disk, 32 cdf, 45 cellular phone cam, 347 chaos, 3, 5, 269, 317, 369 chaotic attractor, 153, 167, 268, 269, 313, 321, 324 chaotic behavior, 265, 279, 317 chaotic dynamics theory, 369, 370 chaotic event, 118 chaotic ¯lter, 69 chaotic mathematic theory, 369 chaotic state, 317 chaotic system, 1, 128, 152, 311, 316, 317, 324, 394 controlled, 315 Du±ng, 312{314, 316 chaotic waveform, 155 characteristic function, 43, 44, 101, 360, 363 Chi distribution, 173 Chua's circuit, 67, 69, 152{154, 157, 158, 317, 318, 320{322 closed linguistic world, 16 closed-loop control system, 276, 281, 304, 311, 318, 320 closed-loop controlled system, 313 clustering algorithm, 86, 151, 152, 196, 253 fuzzy c-mean, 154 clustering center, 151{158 clustering criterion, 196 CMOS, 7 CNN, 7 coefficient of kinetic friction, 38 coefficient of sliding friction, 38 coefficient of static friction, 38 cognition measurable, 14, 15 cognitive medium, 19 cognitive adverb, 65 cognitive architecture symbolic, 6 cognitive engineering, 30 cognitive feature, IX, 333, 340{344, 348{350, 352, 355, 372, 374{376 cognitive quantum, 18, 17{19, 23, 26 cognitive stock chart, 369{376 collapse bell-shaped, 83 exponential, 81{83 left, 81 right, 82 linear, 79 left, 79 right, 79 reciprocal, 80{81 left, 80 right, 81 sigmoid, 79 left, 79 right, 79 uniform, 79 collective verb, 163 color tone, 344 communication channel, 32 commutative monoid, 52 complement, 359 complex number, 117 compression algorithm, 31 image, 39 compression factor, 405 computational noun, 42 computational verb, 55 T-homomorphic, 63, 63{64 reverse, 64 ·-, 64, 64 approaches, 311 approach, 103, 131, 132, 195 approximates, 342 arrives at, 402 arrive, 64, 195 be above, 343 become, 72, 102, 209, 251, 254, 259, 265, 271, 288, 290{292, 294, 301, 302, 312, 314, 319, 360{362, 364, 365, 398, 402, 404 be, 3, 117, 231, 238, 254, 259, 281, 288, 359, 398 be summer, 362 buy, 372 can, 399 change, 251 climb, 258 collect, 375 decreases, 239, 242, 265, 272, 279, 285, 287{289, 311 decrease, 56, 64, 74, 75, 77, 145, 213, 231, 235, 236, 242, 258, 267, 272, 318, 319, 325, 377, 396 depart, 64 distribute, 375 do not understand, 363 drop, 57, 258 fluctuate, 258 give, 137, 140 go away, 140 grow up, 72 hear, 66, 68, 69 hesitate, 258, 279, 373, 374 increases, 239, 242, 265, 279, 288, 289, 311 increase, 57, 64, 72, 86, 145, 195, 213, 231, 236, 237, 242, 258, 267, 318, 319, 325, 367, 377, 396, 399 is becoming, 360 is leaving, 287 is, 402 jump, 258 keeps, 311 keep, 251, 254, 257, 259 leaves, 311 live, 65 look at, 400, 401 love, 241, 396 meet, 65 must, 399 observed, 292, 301{303 observe, 283, 403, 405 peak, 377 push, 375 rain, 59 reach, 129 receive, 137 reduce, 129 remain, 359 sell, 372, 375 send out, 137 speak, 66, 68, 69 stays at, 311 stays, 288, 289 stay, 145, 231, 258, 287, 290, 318, 319, 325, 359, 377 throw, 71 understand, 65, 66, 361, 363 valley, 377 will arrive at, 402 will become, 402, 404, 405 will be, 402 will, 399, 404 evolving function, 55 fuzzy singleton, 58 same type, 108 simpli¯ed, 58 spatial, 334{338 Computational verb collapse simpli¯ed, 77 computational verb collapse, 73, 73 discrete-time, 78 orbit set, 73 same type, 108 computational verb controller, 281{327 chaos, 317{327 stable, 303{316 computational verb correlation, 217{219 computational verb cross-correlation, 213{217 computational verb cross-covariance, 213{217 computational verb distance, 163{193 computational verb extension principle, 73, 84, 84{100 computational verb logic, 221{229 computational verb relation, 107, 238 computational verb set, 102 Computational verb set induced by computational verb collapse, 103 computational verb similarity, 163{193 computational verb system, 261{279 chaotic, 279 rule-wise linear, 261{279 continuous-time, 273 discrete-time, 274 ¯rst-order, 265 robust Stability, 273 computational verb theory, 2{5, 9{11, 14, 71, 73, 100, 151, 163, 164, 238, 251, 261, 279, 369, 370, 376, 377, 385, 392, 397 conjugate discrete-time computational verb, 60, 60 conjunction, 21, 46 operator, 34 verb, 221 continuous-time dynamical system, 53 continuum Truth , 137 dynamical behavior, 246 dynamics, 137 verb space, 61 control rule, 281, 284 fuzzy, 281, 284 local, 281 static, 281 verb, 286, 304, 311, 312, 314{316, 322{325 conventional engineering, 30 countably in¯nite sequence, 359 crisp number, 38, 398 crisp rule, 231 crisp set, 102 cumulative distribution function, 45 cytosine, 29 data mining, VIII, 3, 5, 86, 219, 247{249, 370, 385 decision making structure, 247 decision tree, 247{259 decoding error, 32 definite matrix positive, 310 symmetric, 309 defuzzification, 38{39, 389 degree of verb similarity, 238 Descartes, 1 deuterium, 28 deverbification, 245{246, 297, 298 datum, 301 diffeomorphic, 60 diffeomorphism, 60 differentiable dynamical system, 53 differential equation linear third-order, 53 digital camera, 347 digital video cam, 347 dilation, 65 direct-sequence code-division multiplex access, 123 discrete dynamical system, 53 discrete-time dynamical system, 53 disjoint, 359 disjunction, 21, 46 operator, 34 verb, 47, 222 displacement, 32 dissonance measure of, 201 Distance based on computational verb collapse, 130 distance of saturated verb, 166 distant function, 167 distant measure, 166 divide-and-conquer, 317 DNA, 29 domain, 42 domain name, 34 Dow, 371 Dow Jones Industrial Average Index, 371, 396 DriveQfy, 5 DS-CDMA, 124 dynamical experience, VIII, 4, 10, 239, 246, 281, 282, 316 dynamical system, VIII, 1, 7, 15, 52, 51{55, 58, 60, 73, 86, 101, 105, 107, 116, 130, 131, 137, 150, 221, 222, 241, 246, 261, 262, 317, 394, 395 -dimensional, 53{55 complex, 317 continuous-time, 138 finite-dimensional, 53 fuzzy, 56, 57, 182, 183, 403 in¯nite-dimensional, 53 linear, 53, 150, 261 low-dimensional, 394 nonlinear, VII, 63, 261 piecewise-linear, 61 qualitative theory, 290 stochastic, 57 edge, 248 edge detection, 340 edge-of-chaos, 391 elastic potential energy, 40 electron, 28 empirical science of natural languages, 4 endomorphism, 51 energy Cognition , 31, 32, 34, 35 configuration, 38 internal, 31 kinetic, 38 potential, 38 chemical, 13 conservation, 16 internal, 38 mechanical, 38 pattern, 16 physical, 15, 16 potential, 39 energy channel, 32 energy conservation, 16 energy of Becoming, 16 energy of Being, 16 energy-truth, 16 energy-Truth duality, 15 epimorphism, 51 equal almost everywhere, 129 equilibrium point, 53 equivalent verb number, 129 equivalent verb number classe, 129 erf, 206 error function, 206 Euclidean space, 217 event, 360 elementary, 360 modifier, 361 event copy, 366 event dilation, 365 event ensemble, 359 event erosion, 366 evolution function, 52 evolution parameter, 52 evolving function, 23, 24, 56{58, 58, 60, 64, 67, 68, 70, 71, 74{78, 86{100, 114, 116, 118, 121, 123{134, 140, 145, 151{153, 155, 157{159, 161, 165, 167, 169{171, 196, 199, 201, 205, 206, 211{217, 263, 265, 315, 319, 325, 326, 335, 337, 339, 344, 365, 366, 399, 402, 403, 405, 406 computational verb, 56 dynamical system, 56 linear, 206{207 monotonic, 78{84 monotonic and smooth, 77 nonlinear, 208{209 saturated, 165, 166, 169, 171{181, 367 expert system, 247 exponential distribution, 173 exponentially distributed, 211, 212 extension of verb number, 131 false alarm, 332 false-alarm rate, 332 FCM, 154 feature vector, 153 feeling, 388, 387{407 finitely additive probability space, 359 FireEye, 5 firing level, 163, 243, 261, 263, 264, 275, 279, 320 first-order Cognition , 17 ¯ssion, 26 flow, 52, 53 fmf, 46{48 force, 31, 36, 37 Cognition , 32{38 conservative, 38 gravitational, 37 nonconservative, 38 resultant, 36, 37 vector, 35 fractional, 38 gravitational, 37 macroscopic, 37 nonconservative, 38 resultant, 36 sensor, 13 formal grammar, 29 formal Language, 29 fraction, 39 function ¯tting, 145 functional analysis, 129 fusion, 26 fuzzification, 38{39 fuzziness, 196, 197, 211 computational verb, 196{199 measure, 198, 199 measure of, 197 fuzzy concentration, 48 fuzzy contrast intensi¯cation, 49 fuzzy dilation, 48 fuzzy gain scheduler, 281, 283{284 fuzzy linguistic hedge, 48 fuzzy logic, 2, 3, 14, 18, 21, 32, 39, 100, 223, 224, 226, 229, 236, 388{392, 407 ANDing, 91 medium, 39 operation, 73 ORing, 89 fuzzy medium, 39 fuzzy membership function, 79 fuzzy number, 38, 74, 75, 77, 97, 117, 118, 120, 121, 124{127, 130{133, 314, 342, 361, 397, 398 binary operation, 121 trapezoidal, 125{127, 361 triangular, 124, 125, 127, 130, 133, 134 fuzzy rule, 73, 97{99, 231, 232, 234, 261, 262, 279, 281, 284{287 fuzzy, 261 space, 285, 286 fuzzy set, 102 fuzzy singleton, 57, 58, 77, 110, 113, 114 fuzzy theory, 2, 4, 73, 100, 232, 261, 279, 287, 292, 394, 397, 398 GÄodel logic, 21 Gamma function, 173 Gaussian distribution, 57, 172 generalized synchronization, 16 generative grammar, 29 Georgian, 69{71 GMP, 237 verb, 231, 235, 238, 242{245, 282, 283 GMT verb, 244 Golden Gate Bridge, 353 gravitational constant, 37 gray-scale image, 332 greenhouse, 251{253 ground status, 117{119 grounded logic, 13, 14 grounded symbol, 406 group homomorphism, 51 guanosine, 29 HÄolder's inequality, 218 Henon map, 78, 323, 323, 324, 326, 327 half-normal distribution, 173 Harnad's model, 7 heat dissipation, 39 heat energy, 40 helium, 28 hidden Markov model, 69 high frequency component, 39 highway system, 331 homomorphism, 51 host verb, 117 human cognition, 8, 11, 16, 19 axiom, 16 computation, 8 human mind, 28, 41 irrational, 2 less than random, 369 mail, 391{392 measure, 1 rational, 2 human-machine co-evolution, 8, 11 hyperbolic, 138 hyperbolic saddle, 138 I Ching, 17 ID3 learning algorithm, 256 identity element, 51, 52, 56 IF/THEN rule, 231, 247 verb, 244 image processing, 331{357 image search engine, 332{333 image understanding, 331{333 e®ective and realtime, 333{334 image understanding engine, 356 indicator function, 43, 43, 102, 103 inertia, 37 information explosion, 22 information sciences, 6, 9, 10, 390 information sink, 32 information source, 32 initial condition, 313, 321, 326, 327 initial state, 53 intelligence intensive, 356 intelligent system, 4{5 artificial, 331 poor man's working de¯nition, 4 intelligent tra±c system, 331 internal node, 247 Internet, 7, 8, 14, 34, 38, 247, 333, 347 interval number, 108, 117, 120, 121, 124, 127, 130, 131 intraday data, 372 IP address, 34 isomorphism, 51 ITS, 331 Jacobian matrix, 60{62, 138 kinematic Truth , 39 kinematics, 23, 31 kinetic Truth , 32 knowledge, 36 knowledge discovery, 219{220 knowledge mining, 11 labor intensive, 356 Lao Tzu, 17 leading direction, 53, 54 leading plane, 54 leaf, 247, 258, 259 Lebesgue integrable, 129 Lebesgue integration, 129 Lebesgue-measurable, 274 liar's paradox non-self-referential, 224 life span, 17, 18, 24, 59, 59, 78, 84, 88, 90, 92, 93, 96, 97, 107, 124, 133, 153, 158, 163, 166, 171{173, 178, 181{183, 206, 215, 216, 265, 315, 325, 326, 335, 364, 396, 397, 405, 406 di®erent, 167{170, 183, 216{217 limited, 95, 170 light simulation problem, 391 light speed vacuum, 17, 44 linear predictive coding, 69 linearly separable, 158 linguistic hedge, 397 probability, 49 linguistic variable, 98 LMI, 320, 321 locally di®eomorphic, 53 logic AND, 34 logic OR, 34 logic system, 1{3, 21, 340, 387{389, 391, 392 context-free, 340 formal, 16 grounded, 407 machine, 406 of mind, 392 logical positivism, 29 Lyapunov equation, 309 Lyapunov function, 304, 310 machinality, 13 machinself, 2, 13, 14 machinself-sensor+actuator, 14 magma, 51 manifold, 53 map, 53 market sector basic material, 373 consumer discretionary, 373 consumer staples, 373 energy, 373 financial, 373 health care, 373 industrials, 373 technology, 373 telecom, 373 utilities, 373 mass, 36 Cognition , 31{35 mass channel, 32 mathematical operator, 59 mathematics most advanced and sophisticated, 3 Matlab, 145, 154, 161, 321 matter, 28 matter conservation, 16 mean-square distance, 130 meaning, 387 information, VIII, 36, 36, 387, 388, 392 word, 387 measurable cognition, 1 measure cognition, 1 measure of fuzziness, 199 measurement crisp, 43 fuzzy, 43 statistical, 43 mechanics, VIII, 1, 37 Cognition , VIII, 31, 32 Universe , 31 medium Cognition , 42, 387 fuzzy logic, 39 vacuum, 39 Truth, 387, 390{392, 407 cognitive, 18{26 membership function, 14, 17, 19, 43, 44, 44, 46, 48, 49, 57, 71, 73, 74, 77, 78, 84{87, 89{100, 102{104, 115, 118, 124{128, 132, 182, 184, 185, 197{199, 221{223, 232, 234, 261, 265, 266, 279, 284, 314, 318, 361{364, 394{398, 402{405 trapezoidal, 75, 111, 112, 233, 361 triangular, 109, 130, 131, 135, 198, 232, 234, 404 mental stress, 4 merge of two dynamical systems, 52 metric space, 129{130 metric system, 15, 18 Minkovski metric, 172 Minkowski inequality, 189{193 modifier, 47 computational verb, 59{71 molecule self-assembly technology, 13 molecule-electronics, 8 money energy, 372, 375 monoid, 51 monoid homomorphism, 52 monoid isomorphism, 52 monomorphism, 51 monotone increasing function, 184 most possible Cognition , 17 motion chaotic, 394 periodic, 394 multivalued logic, 18, 21, 22 mutual action, 37 n-tuples, 238 nano-electronics, 8 Nasdaq, 370, 371 natural language, VII, VIII, 1{4, 8{16, 18, 42, 61, 64, 65, 73, 117, 151, 227, 229, 240, 246, 281, 340, 348, 352, 387, 388, 392, 400 advanced, 14 adverb of degree, 85 average, 30 computability, 9 computable, 387 dynamics, 85 dynamics of verbs, 137 °exibility, 282 grammatical center, 41 in mind, 392 logic system, 388 mathematics, 392{399 meaning, 14 measurable, 9, 10 modeling ability, 247 natural science, 292 noun center, 262 number of verbs, 137 physical meaning, 9 program computer, 9 programming, 10 quantitative facet, 163 rule, 10 search engine, 388 simplest mathematical model, 2 statement, 246 uncertainty, 33 verb, 14 verb center, 262 natural language understanding, 4, 387 natural science, VIII, 8, 10, 11 negation fuzzy, 48 negative energy, 370, 372 curve, 369 neural network, 3, 6, 7 cellular, 7 neural pulse, 42 neuron, 7, 8, 161, 162 neutrino, 28 neutron, 28 Newton's law, 31 First, 36 of Gravitation, 37 Second, 36 Third, 37 NLU, 4 non-ambiguity, 332 non-leading direction, 54 non-leading plane, 53, 54 non-self-referential clause, 229 noncentral chi-square distribution, 173 nonlinear ¯lter, 339 normal distribution, 49, 85, 86, 205{207 normalized verb cross-correlation, 216 normally distributed, 45, 205, 209 number context, 117 OE day, 384 only constant, 17 OR verb, 222 orbit, 52, 55, 60, 64, 73 orbitally equivalent computational verb, 61 outcome space, 359 P-control, 284 P-controller, 293{300 pairs, 238 panda, 352 paradox verb, 228 weak, 229 paradox of liar, 223{229 paradoxical function, 229 particle, 23, 26, 31, 37 Cognition , 31, 34, 38 electrically charged, 37 information, 34, 37 matter, 37 PD-controller verb, 300{302 pdf, 43, 45, 45, 46, 47, 49, 57, 73, 74, 78, 84{86, 102, 103, 115, 142, 201, 202, 205, 210, 211, 221, 222, 233{235, 364, 367, 394, 395 joint, 46 perceptron, 158{161 phase of incidence, 24, 25 phase space, 52 photon, 28 physical linguistics, VII, VIII, 2{4, 7, 10, 14{16, 42, 51, 292, 340, 342{345, 350{352, 387, 391{393, 399, 406 physical symbol, 42 PicSeer, 350{356 PID controller, 149, 281 fuzzy, 281{286 verb, 281{303 piecewise composition of computational verb, 158 piecewise linear composition, 138{142 piecewise linear decomposition, 142{145 piecewise linear function, 127, 142, 143, 145{148, 335 three-segment, 140, 141 two-segment, 139 pixel, 339 PL, 341 PL image detector, 341 type-I, 341 type-II, 341 type-III, 342 PL image search engine, 350{356 PL network, 356 PL semantic decomposition, 343 plasma, 26, 33 pmf, 45 porn detection, 332 nudity, 332 removal, 332 porn-detection software, 332 PornSeer, 5 positive de¯nite matrix, 320 positive energy, 370 curve, 369, 372, 375 positive measure, 359 possibility, 46{48 with high probability, 394 with low probability, 394 potential Truth , 39 preverb, VIII, 69{71 probability, 45{47, 207, 249, 359, 360, 362{365, 367, 376, 389, 394 assignment, 359 measure, 45, de¯dx360, 364 operation, 73 rule, 231, 234 space, 360, de¯dx360, 366 discrete, 366 ¯nitely additive, 359 statement, 360 probability concentration, 49 probability contrast intensi¯cation, 49 probability density function, 45 probability dilation, 49 probability mass function, 45 probability rule, 231 probability theory, 3, 4, 43{45, 364, 369, 394 production rule, 248 proportional-integral-derivative controller, 281 proposition, 29{30, 223{225 Unicogse , 30 BE, 226 proton, 28 qualitatively invariant, 234{238 qualitatively similarity, 163 quantitative similarity, 163 quark, 28 quasi-periodic, 120 RC circuit, 57, 202 reaction, 37 real dynamical system, 53 real number, 117 reasoning, 71, 370, 389, 390 autonomous, 389 case, 395 chaotic attractor, 167 in natural language, 3 intuitive engine, 13 irrational, 390 process, 390 verb, VIII, 4, 158, 163, 164, 231{246 reference computational verb, 238 reference rule, 239 reference system, 66 re°ection of Truth, 22{26 refraction of Truth, 22{26 resultant, 36 risk management, 375 robot, 4, 6{9, 406, 407 autonomous, 6 industry, 11 intelligent, 12 next generation, IX nurse, 8 rough surface, 39 rule-wise linear system fuzzy, 318 verb, 317 Runge-Kutta method fourth, 67 Russell index, 376 rebalance, 376 rebalance pattern, 376 reconstitution, 376 reconstruction pattern, 377{385 Russell 1000, 376, 385 Russell 2000, 376, 385 Russell 3000, 376 Russell's annual index reconstitution, 376{385 saddle, 53 saddle (1,2), 53 saddle-focus (1,2), 55 saddle-focus (2,1), 55 saturate function, 152, 164, 164{166, 170, 325 saturated verb distance, 163 saturated verb similarity, 163 scale-up, 333, 346, 347, 356 scanner, 347 second-order plant, 293{300, 312{316 self-similarity, 163 semi-°ow, 53 sensor, 4, 6{8, 11, 43, 342, 406 arti¯cial, 13 audio, 13 e-nose, 13 force, 13 image, 13, 356, 403 machinself, 14 perceptual, 43 real number, 58 Shannon entropy, 201 signal processing, 66{69 similarity between triangular fuzzy membership function, 232 similarity between two pdf's, 233 similarity function, 240 similarity measure, 167 similarity of saturated verb, 167 simple verb set, 108 simplest Cognition , 17 singleton set, 52 skin tone, 343{344 African, 344 Asian, 344 Caucasian, 344 color, 332 skin-tone algorithm, 332 slope, 93, 146, 148, 149, 335 negative, 145 positive, 145 zero, 145 smooth, 79 smoothly conjugate, 60 smoothly equivalent, 60 smoothly orbitally equivalent computational verb, 61 SnP 500, 371 social science, VIII, 8, 10 spatial adverb, 65 spatial verb canonical, 335{339 spatial-temporal pattern, 393 speech sample, 67, 68, 70 speech signal, 356 speed of Truth , 1, 33 spring, 40 stable, 53 stable focus, 54 stable focus verb, 103 stable in large, 320 stable node, 53, 54 stable subspace, 53, 55 state, 52 state space, 52 static logic, 3 static verb, 398 sti®ness, 40 stochastic resonance, 69 stock, 372 stock model, 372, 375 stock price, 215, 258, 367, 370{372, 375, 377 daily, 107 stock trader, 399 strange attractor, 62, 324 structurally stable, 53 structuring element, 65 sub-image, 339 sun°ower, 354 switch, 57 symbol, 2, 4, 28, 29, 41 Unicogse , 41, 41, 42 cognitive, 41, 41 noun, 41 physical, 41, 41, 42 symbol grounding problem, 6, 7 symbol-grounding mechanism, 340 symbol-physics decoupling, 340 symbolic language, 229 symbolic linguistics, 42, 51 symbolic noun, 3, 42, 71 symbolic noun phrase, 42 symbolic string, 4 symbolic system, 6, 7 formal, 2 rational, 2 symbolized knowledge, 356 synchronization of H¶enon map, 323 system of metrics, 19 Takagi-Sugeno fuzzy system, 261, 262, 273, 317 tao, 19, 21 Taoism, 17 Taoist, 17 temperature of information, 31 template computational verb, 85{87, 103, 104, 130, 152, 155, 158, 181, 196, 197, 199, 213 template verb, 339 temporal adverb, 65 tense, 64, 359 terminal, 247, 252 texture, 340 texture enhancement, 339{340 texture segmentation, 339{340 the Fed, 370 Theory of the Unicogse , 13 Theory of the Unicogse, 387 thermal energy, 39 thymidine, 29 time series, 142, 143, 145, 149, 181, 199, 247, 253, 256, 257, 326, 366, 376, 381 ad hoc, 247 chaotic, 158 daily price, 107 stochastic, 369 symbolic, 158 token, 18 top-down, 340 topologically equivalent, 60, 150 topologically equivalent computational verb, 60{63 topologically equivalent computational verbs, 60 total internal re°ection, 22{23 total internal re°ection of Truth, 25 tra±c °ow, 404 TrafGo, 404 training sample, 256 training set, 258 trend of smooth waveform, 145{149 tritium, 28 trivial monoid, 52 truth conservation, 16 truth of Becoming, 16 truth of Being, 16 truth value, 2, 18, 20, 23, 28{30, 35, 101, 118, 223{227, 393, 394, 400, 405 Boolean, 226 fuzzy, 223 truth velocity, 17 truth-conserving system, 18 ugliest theory of everything, 1 unary operation, 121 uniform distribution, 201 uniform motion, 36 uniformly distributed, 117, 140, 142, 201, 202, 205, 206, 210, 367 United States Federal Reserve, 370 Universe-Cognition Duality, 391, 393, 407 unstable, 53 unstable subspace, 53, 55 vacuum, 391 Cognition , 17, 18, 29, 30, 32, 39 heat, 39 vacuum speed Truth , 33 Cognition , 33 vacuum velocity Truth, 17, 18, 22, 387 vagueness, 195, 196, 200, 333 computational verb, 195 intrinsic, 196 of similarity, 200 value of noun, 2 verb ,, 222 verb ), 222 verb algorithm, 236{238, 244, 244{246 verb AND, 221 verb arithmetic, 121{124 associativity, 121, 122 commutativity, 121, 122 distributivity, 122 verb bidirectional, 222 verb collapse, VIII, 5, 16, 71, 73{100, 103{105, 107{111, 114{116, 124, 127, 130, 132{135, 163, 167, 170, 171, 185{188, 196, 200{203, 205, 207, 209{212, 221, 225{229, 231, 366, 393 trapezoidal, 111, 114, 116 triangular, 109{111, 115, 116, 198 verb composition, 242 verb conjunction, 221 verb control of chaos, 317 verb control signal, 320 verb control system, 320 verb decision tree, 247{259 verb disjunction, 222 verb distance, 163 verb entropy, 5, 200, 247, 256{259 function, 200 verb equivalence, 222 verb event, IX, 359{364, 366, 367 benchmark, 359, 362, 363 dynamics of, 359 elementary, 359 modi¯er, 363{366 verb event close, 365 verb event dilation, 365 verb event erosion, 365 verb event open, 365 verb extension, VIII, 5, 16, 17, 73{100, 103, 110, 114, 130, 131, 134, 135 verb extension principle, 84{100 verb generalized modus ponens, 231 verb generalized modus tollen, 243 verb implication, 222, 238, 242, 244{246, 281 verb inference, 231 verb information gain, 256, 256 verb logic, VIII, 3, 4, 14, 16, 164, 221{229, 281, 388, 389, 407 verb negation, 221 verb NOT, 221 verb number, 117, 117{134, 246 addition, 121 bijective, 118 bipolar, 123 bottom, 119 division, 121 equality, 120 multi-valued, 118 multiplication, 121 negative, 119 reciprocal, 119 reverse, 118 singleton, 118 subtraction, 121 top, 119 type-I, 118 type-II, 118, 123 type-III, 118 unary operation, 121 verb OR, 222 verb paradox self-referential, 223 verb PID-controller, 302 verb relation, 238{246 verb relation operator, 238 verb rule, VIII, 16, 66, 68, 69, 73, 97{99, 151, 163, 219, 220, 231{246, 257{259, 261{264, 271, 275, 281, 285{291, 297, 300, 301, 303, 311, 314{317, 327 set, 261, 262 verb set, 4, 102, 101{116, 151, 158, 196{198, 201{206, 209{211, 238, 239, 242, 243, 245, 246 associative laws, 106 Cartesian product, 107 commutative laws, 106 complement, 105 cross product, 107 De Morgan's laws, 107 di®erence, 106 distributive laws, 106 idempotent law, 107 identity, 107 intersection, 106 law of double complementation, 107 law of excluded middle, 107 symmetric di®erence, 106 union, 106 verb similarity, 16, 152{154, 163, 199, 200, 218, 236{240, 242, 244, 245, 247, 253, 257, 258, 263{265, 267, 279, 282, 292, 293, 299{302, 312, 314, 319, 320, 322, 326, 327, 336{340, 374, 377{382, 405 balanced, 191{193 function, 245, 246, 294{297 verb similarity function convex, 246 normal, 246 verb-similarity-weighted sum, 238 verbi¯cation, 283 video stream, 347 visual path, 53 visual signal, 356 vitalism, 1 VLSI, 7 volume Cognition , 31, 32 Wall Street, 348, 369, 370, 375 wave-particle duality, 23 waveform chaotic, 107, 158 peak-valley, 158 valley-peak, 158 webcam, 347 webcam barcode scanner, 5 webpage, 349 White House, 333 winer-take-all, 311, 314 winner-take-all, 312, 316 work, VIII, 33, 387 Cognition , 34{35 work-Truth theorem, 34 zero measure, 129 zero-order Cognition , 17
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