# **SpringerBriefs in Computer Science**

**Jianfeng Xu · Shuliang Wang · Zhenyu Liu · Yashi Wang · Yingfei Wang · Yingxu Dang**

# **Objective Information Theory**

# SpringerBriefs in Computer Science

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Stan Zdonik, Brown University, Providence, RI, USA Shashi Shekhar, University of Minnesota, Minneapolis, MN, USA Xindong Wu, University of Vermont, Burlington, VT, USA Lakhmi C. Jain, University of South Australia, Adelaide, SA, Australia David Padua, University of Illinois Urbana-Champaign, Urbana, IL, USA Xuemin Sherman Shen, University of Waterloo, Waterloo, ON, Canada Borko Furht, Florida Atlantic University, Boca Raton, FL, USA V. S. Subrahmanian, University of Maryland, College Park, MD, USA Martial Hebert, Carnegie Mellon University, Pittsburgh, PA, USA Katsushi Ikeuchi, University of Tokyo, Tokyo, Japan Bruno Siciliano, Università di Napoli Federico II, Napoli, Italy Sushil Jajodia, George Mason University, Fairfax, VA, USA Newton Lee, Institute for Education, Research and Scholarships, Los Angeles, CA, USA

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Jianfeng Xu • Shuliang Wang • Zhenyu Liu • Yashi Wang • Yingfei Wang • Yingxu Dang

# Objective Information Theory

Jianfeng Xu Information Technology Service Center of People's Court Beijing, China

Zhenyu Liu School of Information Management for Law China University of Political Science and Law Beijing, China

Yingfei Wang Information Technology Service Center of People's Court Beijing, China

Shuliang Wang School of Computer Science and Technology Beijing Institute of Technology Beijing, China

Yashi Wang School of Information Management for Law China University of Political Science and Law Beijing, China

Yingxu Dang School of Computer Science and Technology Beijing Institute of Technology Beijing, China

This work was supported by Information Technology Service Center of People's Court

ISSN 2191-5768 ISSN 2191-5776 (electronic) SpringerBriefs in Computer Science ISBN 978-981-19-9928-4 ISBN 978-981-19-9929-1 (eBook) https://doi.org/10.1007/978-981-19-9929-1

© The Author(s) 2023. This book is an open access publication.

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# Foreword

Information is an indispensable component of our daily life and plays an ever more central role as we enter the information age. The rapid development of information technologies such as cloud computing, Internet of Things, mobile Internet, big data, and artificial intelligence has increased the scale and complexity of information systems. The numerous information systems in the real world have or will profoundly change our way of working and daily life. The widespread applications of information technology and systems have led to broadly significant progress in the Information Society.

However, our understanding of information itself is still very limited. Although great progress has been made in information theory since the pioneering work of Shannon in 1948, there is still no consensus on information in the wide sense concerning its model and measurement, let alone the nature of the information. Shannon's theory is hugely successful in the application to communication systems, but also limited to that. As Shannon himself put it as the title of his original paper "A mathematical theory of communication," his theory was developed mainly for the purpose of communication engineering, and it would be wishful thinking to expect that this theory also works for other systems involving information.

In his celebrated book Cybernetics or Control and Communication in the Animal and the Machine published in 1948, Wiener pointed out the pivotal importance of feedback information in the control of both biological and man-made systems. Decades of studies in control theory have found that the relation between feedback information and system controllability is much more complicated than explicable by Shannon's framework. Significant advances in biological sciences also uncovered many more complicated roles information plays in natural systems. For example, the information buried in the DNA sequence is far richer than the number of bits can describe.

Information systems abound in practice. The authors of this book have many years of experience in developing and researching information systems. In particular, Prof. Xu Jianfeng has been leading the nationwide construction, application, and promotion of the Smart Court system-of-systems engineering project of China, which guarantees the continuous upgrading of the level of informatization and reshaping of the judicial operation pattern of Chinese courts. These experiences have led the authors to a more profound view of the nature of information, and in particular, how information is measured in information systems.

In this book, Prof. Xu and his coauthors propose the concept of objective information, which is depicted from a philosophical view, a formal assumption, and a corresponding mathematical model. Furthermore, to demonstrate the feasibility of applying the objective information theory in the practice of large-scale complex information system constructions, two exemplificative cases of the air traffic control system and the Smart Court project of China are demonstrated. Therefore, this book provides a reference approach to quantitatively analyzing information operations in complex information system applications. Also, this book can serve as a textbook for senior students and graduate students in related majors.

I hope that Prof. Xu and his research team will continue to deepen the study of objective information theory and continue to make new progress in the development of information science.

Chinese Academy of Sciences, Beijing, China

Lei Guo

# Preface

Information is inseparable from an information society. The rapid development of information technologies such as cloud computing, Internet of Things, mobile Internet, big data, and artificial intelligence has increased the scale of information systems. The numerous information systems in the objective world have or will profoundly change human production and lifestyles. The popular applications of information technology and system bring a broad significance in the progress towards the information society.

People have made contributions to understanding and using information such as information theory. The information theory on change describes how things vary. It emphasizes the difference in size, amount, degree, or nature for telling an exact object from common objects. For example, Shannon defined information with the mathematical equation and made continuous contributions. He revealed the fundamentals of information transmission with probability and statistical methods for communication systems. However, there is no consensus on understanding information. It is difficult to distinguish the relationship between objectively existing information and subjective conscious activities. The measurement on information is inconsistent, leading to the problem of "difficult to model, difficult to measure, and difficult to evaluate" in the design, development, and application of complex information systems with big data. It has become a bottleneck to restrict the vigorous development and implementation of information systems in the national informatization strategy.

Actually, information is the link that connects everything in the world. Everything can express existence and movement through its objective reflection in the world. Information may reflect the existence, characteristics, and movement states of matter, and further helps matter interact with each other. In the implementation of information system engineering, information integration mirrors the static characteristics and dynamic changes of things. What information systems deal with is all kinds of objective information reflected in it, and does not first define whether it changes or not. The acquisition, transmission, processing, storage, and application of information may have not only individual function but also collective impact on the large-scale information systems. To understand the information from the view of reflection instead of change, we proposed objective information theory.

In this book, the objective information is depicted from the philosophical view, formal assumption, and mathematical model. Information is defined to reflect the world with the properties of objectivity, restorability, transitivity, combination, and relevance in the context of the sextuple model and mapping theory. By analyzing the similarities and dissimilarities between objective information and subjective knowledge, it uses mathematical language to give a sextuple model of information, discusses the basic properties of information, makes the definition of information more rigorous and clearer, and analyzes information systems. The metric system of objective information is presented in mainly 11 indicators: volume, delay, scope, granularity, variety, duration, sampling rate, aggregation, coverage, distortion, and mismatch. Each metric indicator is mathematically defined from the basic model with specific properties. To demonstrate the feasibility and practicality to investigate complex information systems at large scale with big data, the exemplification of the objective information theory takes Air Traffic Control System (ATCS) and Smart Court. The results show that objective information theory provides an alternative tool to fully master and take advantage of quantitative analysis of information operation in complex and uncertain scenarios, and precise control of information system applications. Especially, Xu Jianfeng put the objective information theory into the nationwide construction, application and promotion of the Smart Court system-of-systems engineering project of China for guaranteeing the upgrade and reshaping of the judicial operation pattern of Chinese courts. It has gained remarkable achievements and brought China to the leading position in judicial informatization in the world.

The work in this book is funded by the National Key R&D Program of China (2016YFC0800801, 2020YFC0832600) and National Natural Science Fund of China (62076027).

Many scholars have provided a lot of help for the writing of this book. Here, I would like to express our heartfelt thanks.

Beijing, China Jianfeng Xu November 2022

Shuliang Wang Zhenyu Liu Yashi Wang Yingfei Wang Yingxu Dang

# Contents

### Contents xi

# Chapter 1 Information Theory on Change to Reflection

Abstract Information reflects the existence, basic characteristics, and movement states of matter in the world. In the information society, it is difficult for human activities to leave information for the wide application of information technologies and systems bring a broad significance. Having achievements in both theory and application, information theory can be divided into the information theory on change and the information theory on reflection on the basis of cognitive perspective. In this chapter, the objective information theory (OIT) will be introduced to the background and motivation. It starts from the information theory on change. The bottleneck problems are discussed on information science, and information system. Then, the information theory on reflection is put forward. Two theories are further discriminated from the perspective of information. Finally, the monograph contents are briefed.

Keywords The information theory on change · The information theory on reflection · The state of the art

### 1.1 The Information Theory on Change

In the development of information theory, Harry Nyquist and Ralph Vinton Lyon Hartley have seminal role in the information theory, in the light of which Shannon defines information with the mathematical equation and made continuous contributions. They lay foundation for quantitative calculation and rapid development of the information theory on change.

### 1.1.1 Information Communication

In an early stage, the information describes how the things vary because of the initial knowledge and low calculations. To mathematically clarify the nature of communication, Harry Nyquist made fundamental theoretical and practical contributions to telecommunications. He [1] analyzed the relationship between the speed of a telegraph system and the number of signal values used by the communication system. His principles of sampling continuous signals to convert them to digital signals [2], showed that the sampling rate must be at least twice the highest frequency present in the sample in order to reconstruct the original signal.

To establish a quantitative measure to compare the capacities of various communication-systems to transmit information, Ralph Vinton Lyon Hartley [3] distinguished between meaning and information. Information was defined as the number of possible messages, independent of whether they are meaningful. He further thought that messages are concrete and diverse symbols, and information is the abstract parameter in the messages. Under the umbrella of this definition of information, he gave a logarithmic law for the transmission of information in discrete messages. With the logarithm of the probability on a message occurrence to measure information, he took sample from a set of symbols to create a word. If the probability of each symbol is the same, and the selection is random, you can get different words. In 1955, he further presented the information theory of the Fourier analysis and wave mechanics.

Claude Elwood Shannon [4] acknowledged his debt to works by Nyquist and Hartley in the first paragraph of his paper "The Mathematical Theory of Communication". He believed that the basic problem of communication was to accurately or approximately reproduce the message selected by another point at one point. Information is defined as a set of codes that encode the probability of occurrence of events, and the uncertainty of information is measured by using an entropy. A model of the communication system was given to solve the technical problems on accurately transmitting communication symbols, such as information capacity, statistical features of information sources, information source coding, channel coding, information measurement, and the relationship between channel capacity and noise.

As a result, for the first time in history, mankind clearly understood and grasped the essence of communication technology, which promoted the vigorous development of communication systems. Therefore, the information theory pays attention to the message transmission in information communication. The change of the symbol values transmits the message. The logarithm of the probability on a message occurrence models the information measurement.

### 1.1.2 Typical achievements of the information theory on change

Nowadays, there are many contributions to the information theory on change. The representative is Shannon information theory [4], including full information theory [5], general information theory [6, 7], representation information theory [8], unified information theory [9, 10], and information geometry theory [11]. These achievements strongly promoted human society from the industrial age to the information age.

### Shannon Information Theory

The idea is to extract the information from countless events. Information was described as "a group of codes with the probability of occurrence p1, p2, ..., pn" with the entropy [4].

$$H(\mathbf{X}) = H(p\_1, p\_2, \dots, p\_n) = -k \sum\_{i=1}^n p\_i \log p\_i \tag{1.1}$$

It is a measure that reflects the uncertainty of information and lays the theoretical foundation of Shannon information theory. Sequentially, Wiener [12], Ashby [13], De Luca, and Termini [14] used their entropy equations to measure information. Shannon entropy was further expanded, i.e. cumulative residual entropy [15–17], joint entropy [18, 19], conditional entropy [20–22], exponential entropy [23], mutual information entropy [24, 25], cross entropy [26–28], maximum entropy principle [29, 30] and so on. These results have prompted the development of information entropy.

### Expansion from Shannon Information Theory

Zhong Yixin [5] proposed full information theory by taking "self-representation/ self-revealing of the state of things and state changes" as the definition of information at the ontological level. He believes that full information includes the epistemological level information of the external form, internal meaning and utility value of the movement state of things and their changing modes. To truly master the epistemological information of the matter, you have to perceive its form, understand its meaning, and know its value. Simultaneously considering three factors, the formal is grammatical information, the meaning is semantic information, and the utility is pragmatic information.

General information theory [6, 7] treated information as the ability to cause changes under the principles of noumenon and value. In a information logic system IF(R), a mathematical model of triples was hC, I, Ri, where C is information carrier, I is information, and R is the system that information belongs to. A series of operators between C, I, R and IF(R) are used to transmit, receive, and store information. Taking information, carrier, and belonging system as a complete system, general information theory opens a door to study the nature and operation of information by using mathematical methods. Moreover, three types of structural metrics are given to measure the degree of information change: internal, intermediate, and external. They are namely abstract, reality, and experiment, defined under different target and requirement.

Representation information theory [8] is the proportion of structural complexity changes caused by a group of objects removed from their categories, by understandings the principles of human concept communication and learning. Compared with Shannon information theory, there are some expansions. The first is to replace the symbol sequence with the conceptual structure. The second is to replace the probability with the category invariance. And the third is replaces simple events with a collection of group objects.

Unified information theory [9, 10] uses a consistent concept to contain many different information connotations. It describes information as the structure, state, or state of the system, considering information from grammatical to semantic and pragmatic state. Its analysis of the product of behavior change is like full information theory [5], while the theoretical depth and systematisms are far inferior to the latter.

### Information Geometry

In contrast with Shannon information entropy, information geometry theory is known as the second generation of information theory [31]. It contaminates probability statistics and geometric methods. Rao [32] used the geodesic distance to measure the difference between the probability distribution functions by defining the Riemann metric on the manifold with Fisher information matrix [33]. Efron [34] gave the curvature on the statistical manifold. Chentsov [35] presented a family of affine connections on the statistical manifold. Amari [36] proposed the concept of dual affine connections.

### 1.1.3 The Limitations of the Information Theory on Change

The information theory on change emphasizes the difference in size, amount, degree, or nature for telling an exact object from common objects. However, many researchers lack an outline and consistent consensus on the objects, purposes, and methods of information. People's understanding of information is still divided, and the measurement on information is also inconsistent.

Regarding the object to study information, it is difficult to distinguish the relationship between objectively existing information and people's subjective conscious activities. The information in people's minds belongs to both objective and subjective categories, which is extremely complicated.

With regard to the purpose to study information, academic exploration and deduction are satisfied with. To a large extent, they ignore the numerous information systems in the objective world that have or will profoundly change human production and lifestyles. They also neglect the impact of acquisition, transmission, processing, storage and application in a information system, especially the actual guiding role of the research, design and development of large-scale integrated information systems.

For the method to study information, Shannon traditional probability and statistics methods for communication systems are inherited, which limits the proposal of more feasible and extensive method to comprehensively and systematically establish the theoretical foundation of information science. Furthermore, in the information theory represented by Shannon, the basic view is that only when the recipient of the information receives the information, the information can produce meaning.

### 1.2 Bottleneck Problems in Information Science

In an information system, information is integrated to reflect the static characteristics and dynamic changes of matter, which is a whole and difficult to separate in the real world. Nowadays, it is a trend to integrate, or make use of, different information systems to build a larger-scale system. However, there are various concepts on information with different connotations in information science. Lacking an outline on the objects, purposes, and methods, no consensus on the theory of information comes into being. The metrics is not systematic. It is difficult to guide the giant project.

### 1.2.1 Different Concept Descriptions

The pronouns of early information are with a certain meaning. The production, transmission and utilization of information is a human instinct. After the 1920s, with the large-scale production in the industrial age and the development of science and technology, the amount of information has increased rapidly. With the rapid increase in the amount of social information, people have to study how to obtain, process, transmit and use information in a timely and accurate manner. Some people go to explore information, and various definitions have been put forward for information.

In general terms, information is an objective phenomenon that can be described, and it is also knowledge with a certain physical meaning.

As Shannon said, "The basic results of information theory (in the narrow sense) are all aimed at some very special problems. They may not be suitable for psychology, economics, and some other social science fields." The limitations of the special information theory are mainly manifested in the three aspects:

### 1.2.2 Different Essential Connotation

There is a significant difference between objective information and subjective cognition. People can recognize, describe, and use information, but they cannot change information with their own subjective thoughts. Even if people have different interpretations of information under their respective subject areas and knowledge background, the objectivity of the source and content of information cannot be denied. As the material is not transferred by human will, the objectivity of information is also not shift with people's thoughts. Facing the same objective information, there various subjective cognition.

There are indeed fundamental differences between objective information and subjective cognition. People have to change the purpose, theories, methods and means of studying objective information and subjective cognition. Relatively, it is more intuitive, simpler and easier to study objective information than subjective cognition. So it is necessary to distinguish the objective and subjective attributes of information, and to define information in the category of objectivity.

### 1.2.3 Difficulty to Guide the Giant Project

It is difficult to adapt to the current needs of informatization when evaluating the effectiveness of complex systems such as operating status, fault discovery and handling. Good correspondence and quantification are the basis to build and evaluate a complex information system. A metric system with multiple indicators may lead to the improvement of overall construction effectiveness. The application effectiveness is the core to build a quality-efficiency indicator system, but there is still a lack of an indicator system to measure the quality and efficiency of complex system informatization. People have to pay attention to the operational quality and operational effectiveness of the complex system under the informatization. The traditional index system is often oriented to the quality of system equipment, such as resource allocation and failure occurrence. Nowadays, it imperative to develop the performance indicators for specific fields under the information metrics of information systems, so as to provide benchmark guidance for comprehensively analyzing, implementing and evaluating the complex system.

System-of-systems (SoS) is a system composed of a series of complex systems [37]. SoS engineering implementation combines many systems with independent operation, independent management, location distribution, emergent behavior and gradual development. The process not only maintains their respective independence, but also realizes more and stronger capabilities of the whole system. When various complex systems converge to form an SoS, it is necessary to further enrich and deepen related theories and methods to effectively solve main problems such as integration, sharing, coordination, evaluation and optimization.

To achieve a wider range and a higher level of sharing and collaboration, it is essential to optimize the overall performance of SoS under the basic requirements when integrating various systems. So it is necessary to effectively bridge the gap between the comprehensive integration of large-scale SoS and individual development of various systems. It may clearly expresses and maintains various information relationships among different systems without bogged down by tedious details, and reasonably balances the allocation of tasks. With the expansion of an SoS, it is important to reshape the original business model in an all-round way that the capacity and effectiveness of SoS as a whole can be improved from local to global, from short-term to lasting, and from relatively simple ideal settings to complex practical environments, through continuous quality-effectiveness analysis with key performance indicators and popularization, in order to achieve a good state of stability, universality, long-term operation, continuous evolution and continuous improvement.

According to the top-level design of the complex system, the system-level components of the complex system can be obtained, and each system-level component is an information system that manages objective information. The rapid development of information technologies such as cloud computing, Internet of Things, mobile Internet, big data, and artificial intelligence has increased the scale of information systems. Furthermore, the super-large-scale complex system needs to consider more factors, such as high-dimensional, multi-source, heterogeneous, nonlinear, uncertain, etc., making the information system more and more complex, and fewer and fewer problems can be solved correctly. Many problems need to be solved approximately by numerical calculation methods. The information system has encountered a major theoretical bottleneck of "difficult to model, difficult to measure, and difficult to calculate".

The operation of a complex system is a typical complex giant system problem. The factors that affect the results of the operation include basic equipment factors, user behavior factors, etc., and there are many cross-influences between each other. When the information system transitions from big data to strong intelligence, the scale of the information system becomes larger and more complex. As the scale and complexity of informatization in various industries continue to expand, such as air traffic control systems, Smart Court, etc. Focusing on the equipment of the complex system and taking fault handling as the goal, it has been difficult to adapt to the current needs of informatization. How to ensure the high-quality, efficient operation, and great value of informatization has become the focus of attention in the complex system.

The information theory on change said that it is impossible to effectively evaluate and guide the efficiency and capabilities of complex information systems and intelligent computing, which severely restricts the depth and breadth of the construction and application of big data intelligent information systems, and it has been difficult to fully satisfy big data. The increasingly complex and huge needs of intelligent systems have made it difficult to support the sustainable development of the complex world. It is urgent to break through the original shackles of information theory.

# 1.3 The Information Theory on Reflection

Information is the constituent element of the world, and it also reflects the nature of things in the world and the laws of their movement. In a information space, the real world is the essence and true nature of the information space, and provides the source of all information. In turn, the information space reflects the real world and feedbacks the real world by simulating and deducing objects and movements that are costly to conduct physically.

### 1.3.1 The Objective World and the Subjective World

The world is divided into two categories: the subjective world and the objective world [38]. Generally, the subjective world refers to the world of consciousness and ideas, and is the sum of the spiritual and psychological activities of understanding and grasping the entire world. It includes both the process of consciousness activities and the results of consciousness activities. Human concepts, will, desires, emotions, beliefs, etc., are different forms and manifestations of the subjective world. It relies on mankind's active thinking and awareness of the objective world, mainly cognition, understanding, control and utilization, and is the unity of knowledge, affection, and intention. The objective world refers to the material and perceivable world, which is the sum of all matter and its movement outside of conscious activities. It can be divided into two parts: natural existence and social existence. The former does not depend on anything but exists objectively, and the latter is formed in the process of human social practice but is not transferred by human will. The objective world does not depend on people's subjective consciousness. It is mainly the connotative nature of the world and the law of extensional motion, rather than a collection of consciousness and concepts.

The objective world and the subjective world are related and distinguished. The information in the objective world can act on the subjective world. We can treat the consciousness, concepts, thinking, and decision-making formed by humans and even other students from the information of the objective world as information. Owing to the objectivity of information, people can collect, transmit, process, aggregate, and apply information with various means. The rapid development of emerging technologies, such as artificial intelligence, brain-like systems, and brain– computer interfaces, is driven by advances that simulate the human mind and then transform humanity's subjective processes into objective information that can be processed by information systems. At the same time, there are still many essential differences between information in the objective world and consciousness or concepts in the subjective world.

The subjective world consciousness and the objective world information are heterogeneous. The objective world exists outside of human conscious activities, has direct reality, and moves according to its own inherent laws. The information it reflects is objectively existing, but it is different for different subjects. The material basis of external natural existence lies, and the material basis of human social existence is the material production method. Consciousness exists in the human brain, which is extensive and stretchable, and is mainly reflected in the spirit and thinking activities. In this sense, the subjective world is the range and boundary of the intelligence, wisdom, and thinking ability of the subject's conscious activities, as well as the range of thinking capacity that it can accept, understand, and process information. This also directly leads to individual differences, and different subjects may have different understandings when dealing with the same objective information.

On the other hand, the subjective world and the objective world are opposed and unified. Generally speaking, the objective world determines the subjective world, and the objective world is the external space of the subjective world. However, the subjective world is a kind of conscious existence after all. People can assemble and construct objects at will within consciousness, so that the subjective world can not only be the expression and reflection of the objective world, but also deviate from the objective world in some aspects, or even transcend the objective world. The world makes predictions for the future based on existing information. This has led to the incomplete synchronization of the development of the subjective world and the objective world. The subjective world and the objective world have a complicated contradictory relationship. On the one hand, the subjective world affirms and reflects the objective world; on the other hand, it deviates, denies, and transcends the objective world. The two are always intertwined.

Popper [39] further proposed three worlds. The first world is the world of physical objects or physical states composed of all matter and various phenomena in the objective world; the second world is the world of consciousness or mental state, or the world of behavioral intentions about activities; and the third world is the world of the objective content of thoughts. Actually, the third world is the world of human spiritual products, especially the world of scientific thoughts, poetic thoughts, and works of art, and especially emphasized the objective reality and independence of the third world.

Knowledge is the previously unknown patterns in the objective sense or in the subjective recognition. When people produce objective knowledge through subjective spirit, the third world becomes an objective reality, and its existence separates from human beings and exists alone. It can be seen that Popper's first world and third world are actually a further division of the objective world, that is, the first world contains a part of objective information, the second world is the subjective world in the usual sense, and the third world is completely composed of objective information. Composition, regardless of whether people are aware of the content of these thoughts, they all exist autonomously.

### 1.3.2 Trinity of the Objective World

In the objective world that does not depend on human subjective consciousness, matter, energy, information are its three major constituent elements, which is also called the ternary nature of the objective world. Wiener [12] pointed out that "information is information, not matter or energy." Steucke [40] deepened this view and said that "information is the third thing alongside matter and energy". Thus formed the world's material, energy, and information "trilogy". In fact, the world here refers to the objective world, and does not include the subjective world, because matter and energy obviously only belong to the category of the objective world.

Information is the third thing juxtaposed with matter and energy. It reflects the existence and movement of matter. The world is material, matter is in motion, and motion is regular. The true unity of the world lies in its materiality. From the perspective of nature, things are formed and developed under their own inherent laws, have their own origin and development history, and are all components of a unified material world. Human society is also the long-term development of the material world. Heraclitus said that "one cannot step into the same river twice." The whole world is the eternally moving material world. The movement in philosophy refers to the changes and processes of all things, which are the inherent attributes of matter and the form of matter existence. Everything is in motion. Some movement is obvious, and people can directly feel it, such as moving cars, flowing rivers, piercing meteors, etc. Some change slowly, and it is not easy to notice, for example, Mount Everest has risen 1600 m in 500,000 years. The laws of nature are mostly expressed as dynamic laws, revealing the one-to-one correspondence between certain things, and pointing out that the existence of one kind of thing must lead to the occurrence of another certain thing. The law of social development is mainly manifested as the law of statistics, which reveals not a simple one-to-one correspondence between things, but a regular relationship between a certain inevitability and a variety of random phenomena [41].

In the ternary nature of the objective world, matter is an objective existence that does not depend on human subjective consciousness but can be reflected by human consciousness. From the new perspective of the trinity of matter, energy, and information in the real world, energy is the material movement ability that does not depend on human subjective consciousness, and information uses matter or energy as a medium. Information reflects the material nature, internal connections and movement patterns of natural and human things in the objective and subjective worlds. In short, matter is the existence of origin, energy is the existence of movement ability, and information is the existence of the connection of things. All kinds of physical information systems can and can only acquire, transmit, process and apply various objectively existing information. Dynamics are essentially the mathematical expression of the regularities and mechanics of motion and change in space and time [42–45]. It is crucial to comprehensively study the principles of information movement and utilization from the overall perspective of the real world, society, and information systems. Therefore, the cognition of ternary nature satisfies the objective requirements of the operating objects of the information system. These views are completely based on the objective reality of information, which helps the birth of objective information theory [42].

### 1.3.3 Contributions from Objective Information Theory

Objective information theory (OIT) gives a philosophical view that information is the objective reflection of things and their motion states in the objective and subjective world. It presents the concept of information, mathematical definition,

Fig. 1.1 Definition, model, nature and metric system of information [43]

sextuple model, primary properties, and metric systems (see Fig. 1.1) for conducting profound and quantitative investigations.

The first is to clearly define the essence of information in the objective category. In response to the diverse conceptual connotations and lack of a comprehensive metric system of information, OIT studies the methods of information reflecting the nature of things in the world and the laws of their movements. It not only conforms to the fundamental positioning of information as one of the three major components of the objective world, but also adapts to the operational requirements that information systems can and can only process objective and real information. Since the information in people's minds belongs to both objective and subjective categories, many studies do not distinguish the relationship between objective information and subjective conscious activities, which makes the problem extremely complicated. OIT focuses on simplifying the complex, decoupling the objective and subjective attributes of information to a certain extent, and combining the operational requirements of the information system.

The second is to put forward the mathematical expression of information and the sextuple model, which provide a unified, clear, convenient and feasible mathematical theory basis for all subsequent studies. Based on the objective reality of information, the definition of information is proposed. Information is an objective reflection of things and their movement in the objective world and subjective world. In the light of the definition, the sextuple model of information includes noumenon, state occurrence time, state function carrier, reflection time, and reflection function. This model deconstructs the concept of information in three important ways: the dual deconstruction of information subjects, the temporal deconstruction of information duration, and the form deconstruction of information contents. Through these three important deconstructions, we can not only measure the capacity of information, but also other aspects of the information more profoundly, comprehensively and quantitatively.

The third is the information sextuple model combines the internal characteristics and external needs of information. It embodies the rationality of information and its model definition. Through the proof of the basic nature of the information, the model is further improved. It defines information as the reflection from a state set to a reflection set. Objectivity is the constraint of information perception. restorability refers to the fundamental premise of information application. Transitivity is the basic mode of information transmission. Relevance is the important meaning of information existence. Combinability proves that information has a combinatorial nature.

The fourth is that an information metric system is inferred from the sextuple model and the basic properties of information. The metric system of objective information mainly includes 11 indicators: volume, delay, scope, granularity, variety, duration, sampling rate, aggregation, coverage, distortion, and mismatch. The entire system is more comprehensive, and there is independence between measurements. Each metric is expressed in quantitative forms such as the measurement, potential and distance of a specific element. Each element is in several metrics. It provides a basis for quantitative research and analysis of information operation and application of information systems. The significance of such metric system is that it provides important measurement guidance for the evaluation of information systems, apart from providing the ground for information measurement. As the main efficacies of information systems depend on the input and output information, the benchmark for judging information systems should be based on the evaluation indicators of these input and output information.

### 1.4 Discrimination of Two Information Views

Information is the blood, food, and vitality of the world [46]. The achievements of information have promoted human society from the industrial age to the information age [47]. But people's understanding of information is various, and measuring information is also inconsistent consensus [38]. There is a lack of the widely adopted mathematical expressions to describe the concept of information.

The information theory on change believes that information can only produce meaning when the recipient of the information receives the information. The representative is Shannon information theory. It reveals that the capacity of information in communication systems can be defined as entropy, which has led many people to mathematically regard information as negative entropy. Probability theory and stochastic processes are taken as basic research tools to study the entire process of the generalized communication system, rather than the entire link. At present, there are still research directions such as the deepening of information concepts, the development of information distortion theory and its application in data compression, and the basic theories of computer-centric information processing systems. However, the concept of information is far beyond that in communication systems. Information entropy, as an information expression, cannot meet the general requirements of information science and technology. Shannon information theory cannot effectively evaluate and guide the efficiency and capabilities of information systems and intelligent computing increasing complexity. Scholars are far from reaching a consensus on the nature of information, which is considered as the key obstruction in developing a unified, convenient, and feasible mathematical information foundation. This severely restricts the depth and breadth of the construction and application of intelligent information systems in the era of big data.

The information theory on reflection think that information reflects the nature of things and the laws of their movement, regardless of whether things change or not. The representative is objective information theory. It defines the sextuple model and 11 metric indicators of information to depict the elements and characteristics in the complex information system. Its mathematical foundations are set theory, measurement theory, and topology. Although these theories are relatively abstract, they have a direct and clear correspondence with the statistics and calculation methods widely used by people every day. It is fully applicable to various specific tasks that are generally required in the process of informatization, such as big data, information stream, and information system design and evaluation.

The appearance of the two views of information theory has specific background. Before Shannon information theory, digital communication has made certain developments. With the mathematical principles of communication to create information theory, Shannon brought mankind into the information age and promoted the rapid development of information technology. In the twenty first century, with the development of communication hardware, the birth of big data has brought huge innovations to information systems. Big data has massive volume, a wide variety of categories, a relatively low value density, and a rapid growth rate. The infrastructures for big data have put forward new technical requirements. Shannon information theory on change cannot provide an ideal theoretical description of the complex information system on big data products. To sum up the previous theoretical results and current information changes, Xu Jianfeng et al. proposes objective information theory to revolve around the operating quality and results of complex information systems in the era of big data. From the new perspective of the trinity of matter, energy and information in the real world, it studies the methods of information reflecting the nature of things in the world and the laws of movement. It tries to solve the fundamental theoretical issues such as the nature of information, measurement methods, model algorithms, and computational controllability for the precise control of the applicable evolution, the quantitative calculation, and the inference of information patterns in information systems. Its completeness and practicality will also be tested and revised in many subsequent applications, because any kind of theories can reflect their value and be perfected only in practice.

### 1.5 Chapter Summary

Everything in the world can express existence and movement through its objective reflection, that is, information. Information helps things interact with each other. So information can be viewed at the same value as matter and energy, and it is one-third of the world. People have understood the importance of information, and achieved fruitful results on research and applications. In particular, Shannon revealed the fundamentals of information transmission, and made outstanding contributions to human beings' progress towards an information society; Zhong Yixin proposed the theory of total information, opening a door for the in-depth interaction of information with human cognition and intelligence. Under the axiomatization, Burgin established the model and operator system of general information theory, providing a demonstration for the use of mathematical tools to study the nature and operation of information. Based on previous results, this chapter explores the information connotation from the information theory on change to the information theory on reflection. There is no consensus on the conceptual description, essential connotation, and theoretical level, especially if the measurement system is not a system, it is difficult to guide the implementation of information system engineering. In complex information systems, information integration reflects the static characteristics and dynamic changes of things, and it is necessary to solve fundamental theoretical problems in information science, such as information essence, measurement methods, model algorithms, and controllability of calculations.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

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# Chapter 2 Recognizing Objective Information

Abstract To understand the importance and necessity of studying objective information, in this chapter, it will first analyzes the similarities and dissimilarities between objective information and subjective knowledge, then introduces the basic forms of information, and finally gives the role of objective information.

Keywords Classifying objective information · Characterizing objective information · Roles of objective information

### 2.1 Classifying Objective Information

Information could be classified in many ways. However, when it comes to studying basic information science problems, nothing is more suitable than classifying them into three basic types: nature information, behavior information, and media information.

### 2.1.1 Nature Information

Nature information is the first basic form of objective information, directly magnifying the motion states of matter and energy in the objective world. The sun, the moon and the stars, the rivers and mountains, the cities and streets, and the villages and fields display nature information at every moment. So do the tender buds on the trees in spring, the red leaves on the hill in autumn, the white hair on the head of the elderly, as well as the fit body of the young reflected in the mirror. From this, we can see that nature information acquaints us with nature's spectaculars in the most direct manner. Its main features are as follows:

• The subjects of nature information are objects in the objective world, such as geographical and astronomical phenomena, architecture and artifacts, and animals and plants. Though there also exists the complicated subjective world for human beings, nature information only reflects external images.

### 2.1.2 Behavior Information

Behavior information is the second basic form of objective information. It is the indirect reflection of the effects of consciousness and mentalities in the subjective world upon matter and energy in the objective world. Human beings display their state of mood through expressions or languages by laughing and raging. Birds express their longing for survival and courtship through body gestures and sound by flipping and tweeting. Elephants linger around their deceased mate for a long time to show their deep sorrow. All these indicate that behavior information indirectly reflects consciousness and mentalities in the subjective world of human beings and other living creatures through movements, expressions, language, and sound. Its main features are as follows:

In the view of the author of this thesis: though devices attempting to peep at the subjective world of human beings, such as lie detectors, have been in existence for over a hundred years, the subjective world of living creatures would never be directly exposed however developed the technology is because the subjective and objective worlds belong to two entirely different categories. It is this fact that makes the existence and development of the world more intriguing.

### 2.1.3 Media Information

Media information is the third basic form of objective information. It refers to the images stored in the form of matter and energy after collecting, transmission, and processing of nature and behavior information. A large number of social news, figure commentaries, and encyclopedic knowledge recorded in newspapers, journals, and

Table 2.1 Basic forms, connotations and characteristics of objective information

books is a kind of media information available for repeated reading. Audio and video broadcasts over the radio, film, and TV are media information available for repeated listening and watching. The huge amount of various data codes stored on Internet servers worldwide is also a form of media information capable of presenting or supporting the running of all kinds of information systems for their users. Media information records or copies all kinds of nature and behavior information, indirectly presenting the motion states of objects in the subjective and objective world in an enduring manner. It has the following features:

Table 2.1 summaries the basic types of objective information as well as their corresponding connotations and characteristics. It illustrates the typical characteristics of the three kinds of information, i.e., nature, behavior, and media information, in terms of their different noumenon, State Sets, Carriers, and Times.

Objective information is quite inclusive. In terms of space, we could see starlight from over 100 light-years away and feel it as near as our heartbeat. In terms of time, we could observe something as ancient as the microwave background radiation left after the Big Bang and see something as recent as the scenery that comes in sight upon opening our eyes. In terms of size, it includes the macroscopic as the picture of the whole universe and as microscopic as the movement traces of basic particles in the high energy accelerator. In terms of form, there are both natural scenes familiar to us and data coding stored in various forms of media. Therefore, objective information pervades all living creatures and the vast universe in different forms. We could conduct classified researches on it from multiple perspectives.

Firstly, objective information contains specific content. Bright sunshine and roaring waves belong to nature information. Happy smiles and angry reproaches indicate emotional information. Election debates on TV and newspaper news about leaders reflect political information. Stock prices on the screen and the financial statements of enterprises show economic information. Delicate pictures in a museum and beautiful melodies in the concert represent artistic information. All these kinds of information differ from each other in content, which is where the value and significance of objective information lie.

Secondly, objective information is time-related. Lofty mountains and ancient architectures display their magnificence in an enduring manner. The rising and setting of the sun and the ebb and flow of the tide reflect the periodically changing pattern of nature. Shooting stars flashing through the sky and athletes passing the finishing line leave only a transient impression on us. All these show that objective information can be enduring, intermittent and transient in terms of time. Besides, as objective information comes into existence at different times, their emergence sequence can also serve as an important criterion for information classification. Information of the past is different from that of the present.

Besides, objective information has various carriers. The disposition, shape, color, temperature, and volume of matter could all bear different information. Therefore, matter itself is an important direct carrier of objective information. Meanwhile, as light shows the color of matter, sound reflects the motion state, electric current indicates the movement of an electric charge, the magnetic field shows the geographical position, and force reflects the interaction between matter. All of them are indirect carriers of objective information.

### 2.2 Characterizing Objective Information

Objective information is quite inclusive. In terms of space, we could see starlight from over 100 light-years away and feel it as near as our heartbeat. In terms of time, we could observe something as ancient as the microwave background radiation left after the Big Bang and see something as recent as the scenery that comes in sight upon opening our eyes. In terms of size, it includes the macroscopic as the picture of the whole universe and as microscopic as the movement traces of basic particles in the high energy accelerator. In terms of form, there are both natural scenes familiar to us and data coding stored in various forms of media. Therefore, objective information pervades all living creatures and the vast universe in different forms. We could conduct classified researches on it from multiple perspectives.

### 2.2.1 Scene Features

Objective information contains specific content. Bright sunshine and roaring waves belong to nature information. Happy smiles and angry reproaches indicate emotional information. Election debates on TV and newspaper news about leaders reflect political information. Stock prices on the screen and the financial statements of enterprises show economic information. Delicate pictures in a museum and beautiful melodies in the concert represent artistic information. All these kinds of information differ from each other in content, which is where the value and significance of objective information lie.

### 2.2.2 Temporal Features

Objective information is time-related. Lofty mountains and ancient architectures display their magnificence in an enduring manner. The rising and setting of the sun and the ebb and flow of the tide reflect the periodically changing pattern of nature. Shooting stars flashing through the sky and athletes passing the finishing line leave only a transient impression on us. All these show that objective information can be enduring, intermittent and transient in terms of time. Besides, as objective information comes into existence at different times, their emergence sequence can also serve as an important criterion for information classification. Information of the past is different from that of the present.

### 2.2.3 Carrier Features

Objective information has various carriers. The disposition, shape, color, temperature, and volume of matter could all bear different information. Therefore, matter itself is an important direct carrier of objective information. Meanwhile, as light shows the color of matter, sound reflects the motion state, electric current indicates the movement of an electric charge, the magnetic field shows the geographical position, and force reflects the interaction between matter. All of them are indirect carriers of objective information.

### 2.2.4 Significance Analysis

Shannon Information Theory first studies the uncertainty metric of information in communication systems [1], implying that the significance of communication information lies in eliminating uncertainty, which is no doubt reasonable and feasible speculation for communication systems. Subsequent studies define information as the ability to cause changes [2] or the proportion of changes brought to the structural complexity [3], which expands the understanding of the significance of information but still follows the basic thought of Shannon Information Theory to a certain extent. We consider it neither appropriate nor adequate to simply apply the speculations suitable for communication information to objective information.

In terms of the cause of existence, nature information comes into being due to the law of motion in the objective world itself. As for behavior information, though part of it arises from the subjective wish to change or eliminate uncertainties in cognition, many contents, just like expressions of all kinds of emotion by animals, result from species instincts. Media information, which are projection images that record nature and behavior information mainly out of the need to store for replay, is closely related to eliminating and changing uncertainties. However, there are some parts like artistic products which come into existence for sensory and mental enjoyment. So the existence of lots of objective information is not determined by whether it can change or eliminate uncertainty or not.

Therefore, the significance of objective information is not confined to changing or eliminating uncertainties. It also lies in helping the receivers feel the actual existence of the objective and subjective world. In fact, according to studies on cosmic largescale structure formation and microwave background radiation, it is estimated that only 4% of the cosmos has been perceived, the remaining 23% being dark matter and 73% being dark energy. Dark matter and energy are called "dark" because they cannot produce information perceivable to us, namely objective information.

### 2.3 Roles of Objective Information

Based on the understanding of the objectivity of information, a large amount of nature, behavior, and media information could change or eliminate uncertainties in specific objects' cognition. However, the beautiful sceneries outside our window, though not constantly changing our understanding of the view, enable us to feel the beauty of nature all the time. Scenes of beasts chasing prey on the African plains, though not constantly altering the observers' perception of wild animals, strongly impresses them with the ruthless law-of-the-jungle. The music of the Blue Danube played on digital sounders, though not always strengthening the listeners' comprehension of the waltz, reminds them of the composers' briskness and serenity at heart every time it is played. From these examples, it is easy to see that the specific effects of objective information mainly lie in acquainting the receivers of the spectaculars of nature instead of changing or eliminating uncertainties.

The degree of uncertainty that could be changed or eliminated by nature information, behavior information, or media information varies with different receivers, displaying the distinct feature of relativity. However, through these kinds of objective information, it is beyond doubt that receivers feel the existence of the world around them. This indicates that the scope of change caused by objective information is relative, but the fact that objective information can inspire all receivers' feelings is absolute.

Objective information offers the only way for us to perceive the world. It has great significance and various specific effects, summarized as enabling the receivers to perceive the present, get to know the past, and foresee the future.

### 2.3.1 Summarizing the History

Getting to know the past is an indirect yet steady effect of objective information on receivers. The annual rings of trees reflect their past growth in the form of nature information. Books and media provide media information that reflects what happened in history. People get to know developments from all ages through these in an indirect yet accessible manner. Through this effect of objective information, the past gets closely and amply correlated to the present.

### 2.3.2 Perceiving the Present

Perceiving the present is the most transient and most direct effect of objective information on receivers. What people see when they look out is what nature information is in their current view, and what they hear when they talk is the behavior information they are communicating. As the "present" becomes the "past" in the twinkling of an eye, the effect of apperceiving the present is a transient one.

### 2.3.3 Foreseeing the Future

Foreseeing the future is an important effect of objective information that is easily ignored. We can foresee the success or failure of a hunt by watching the speed of the hunters' and their prey. An information system could figure out the target's future location with reasonable accuracy by observing the target's motion in the past and present. All these exhibit the critical role played by objective information in foreseeing the future. Because of this effect, a mutually traceable relationship gets established among the past, the present, and the future.

The above analysis shows that objective information is the link connecting everything in the world. Without which the world that we live in would be a broken, isolated, and dark one. Thus, objective information is considered one of the three key elements comprising the objective world, equivalent to matter and energy. For such an important subject as objective information, it deserves great effort to study its connotations, the mathematical expressions of its composition, and the mathematical models of its metrics. Only after solving these problems can we further understand the scientific mechanism whereby objective information connects the world and the operation processes and evolution laws of information systems acquiring, transmitting, processing, storing, and applying information.

### 2.4 Chapter Summary

Information is the link that connects everything in the world. It becomes one of the three major elements that constitute the objective world, and is equivalent to matter and energy. There are issues of objective information, i.e. defining connotation, expressing composition, and metric system. By solving these issues, we can further understand the mechanism of objective information linking everything in the world and the operation and evolution of acquisition, transmission, processing, storage, and application of information in a complex system.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Chapter 3 Modelling Objective Information: Sextuple

Abstract In this chapter, the objective information will be depicted from the philosophical view, formal assumption, and mathematical model. It uses mathematical language to give a sextuple model of information, discusses the basic properties of information, makes the definition of information more rigorous and clearer, and analyzes information systems.

Keywords Objective information theory (OIT) · Sextuple model · Information model · Information system model

# 3.1 Definition and Model

The objective information theory (OIT) has its roots in the philosophical view that information is the objective reflection of objects and their state of motion in the objective and subjective worlds, as inspired by Wiener's triadic theory of matter, energy, and information [1–3]. The attached information carrier reflects the motion state of the body at a specific time set. The specific time set is the time when the information state occurs, and the motion state is the information state set. Thus, we propose an accurate definition and mathematical model of information.

Assume that O, S, and T are the objective world set, subjective world set, and time set respectively. O includes the basic objects in Popper's first world and third world. S includes the basic objects in Popper's second world. T is the duration of information action. The elements in O, S, and T can be either continuous or discrete, subject to the specific requirement of the universe of discourse. The denotations of symbols or expressions are given in the following Table 3.1.

Let I denote objective information, o denote noumena (i.e., objects that originate information in the real world), c denote carriers (i.e., objects that transmit and maintain information in the real world), Th denote the occurrence time, and Tm denote the reflection time. The set of all information I is called an information space. o belongs to either an objective world or subjective world, mathematically o 2 2O [ S . c only belongs to an objective world, thus c 2 2O. Th and Tm are both in temporal domain, Th 2 2T and Tm 2 2T . When I is depicted in the context of o, Th, f, c, Tm, g simultaneously, a sextuple model comes into being.

Table 3.1 The denotations of symbols or expressions

$$I = \langle o, T\_h, f, c, T\_m, \mathbf{g} \rangle \tag{3.1}$$

Thus I is modeled mathematically as a full mapping from f(o, Th) to g(c, Tm).

$$I: f(o, T\_h) \to \mathcal{g}(c, T\_m) \tag{3.2}$$

where, a state set f(o, T ) of o on T , and a re ection set g(c, T ) of c on T are all h h fl m m non-empty. If I is the surjective mapping from f(o, Th) to g(c, Tm), Eq. (3.2) can be rewritten as

$$I(f(o, T\_{\hbar})) = \mathbf{g}(c, T\_{\hbar})\tag{3.3}$$

In the light of the sextuple model with mathematical information definition, we can realize the dual deconstruction of the subject, temporal domain, and form of the information, respectively. These deconstructions of information make it possible to conduct more profound and comprehensive investigations of information beyond Shannon information theory.

### 3.2 Model Properties

Based on the sextuple model, we can mathematically infer five primary properties of information: objectivity, restorability, transitivity, combinability, and relevance.

### 3.2.1 Objectivity

The world is material, and matter is in motion. The moving matter constitute an infinite objective world in space and time. Humans perceive the objective world through contact and observation. People are originally unknown to many things in the objective world, and the degree of unknown is uncertainty. They obtain information about a certain part of the objective world through perceptual means, which reduces the uncertainty and improves the degree of awareness.

In the sextuple model, the separation of o and c is the binary deconstruction of the subjects of information I. Based on the deconstruction, I can be reflected by mapping from f(o, Th) to g(c, Tm). Herein, c belongs to the objective world, and I can be perceived through the objective world, which is also why the OIT is named after the objectivity of information.

Owing to the objectivity of information, people can collect, transmit, process, aggregate, and apply information using various technical means. In fact, the rapid development of emerging technologies, such as AI, brain-like systems, and brain– computer interfaces, is driven by advances that simulate the human mind and then transform humanity's subjective processes into objective information that can be processed by information systems. Therefore, we consider the OIT to plays a fundamental role in the analysis and research of information systems and technologies.

### 3.2.2 Restorability

Defined as a type of mathematical mapping, I may have an inverse mapping I -1 . If g(c, Tm) is on of c, Tm can be reduced to the state f(o, Th) of o on Th by I -1 , then we call I restorable. This is the restorability of information. Here, f(o, Th) is also called the reduction state of I. The mathematical inference is as following.

I = ho, Th, f, c, Tm, gi is a surjective map of f(o, Th) onto g(c, Tm) and is called injective if for any oλ 2 o, Thλ 2 Th, fλ 2 f, oμ 2 o, Thμ 2 Th, fμ 2 f with fλ(oλ, Thλ) ≠ fμ(oμ, Thμ), there holds

$$\mathbf{I}(\mathbf{f}\_{\lambda}(\mathbf{o}\_{\lambda}, \mathbf{T}\_{\mathbb{H}^{\lambda}})) \neq \mathbf{I}(\mathbf{f}\_{\mu}(\mathbf{o}\_{\mu}, \mathbf{T}\_{\mathbb{H}\mu})) \dots$$

If I = ho, Th, f, c, Tm, gi is injective, then it is called invertible since it is surjective by definition. In this case, there exists an inverse map I-1 of I. That is, for any cλ 2 c, Tmλ 2 Tm, gλ 2 g, there exists a unique set of oλ 2 o, Thλ 2 Th, fλ 2 f, s.t.

$$\mathbf{I}^{-1}(\mathbf{g}\_{\lambda}(\mathbf{c}\_{\lambda}, \mathbf{T}\_{\mathbf{m}\lambda})) = \mathbf{f}\_{\lambda}(\mathbf{o}\_{\lambda}, \mathbf{T}\_{\mathbf{h}\lambda})\dots$$

This means that

$$\mathbf{I}^{-1}(\mathbf{g}(\mathbf{c}, \mathbf{T}\_{\mathfrak{m}})) = \mathbf{f}(\mathbf{o}, \mathbf{T}\_{\mathfrak{h}}) .$$

For the invertible information I = ho, Th, f, c, Tm, gi, g(c, Tm) of c on Tm can be reduced to the state f(o, Th) of o on Th by I-1 , and have I restorable and f(o, Th) the reduction state of I.

s Moreover, if there is a mapping J such that J g c, T ð Þ ð Þ m =~f ~o, T~h , where ~o i referred to as reflection noumena (~o 2 2O[SÞ, T~ h is referred to as reflection occurrence time (T~ h 2 2T), and ~f ~o, T~ h is a certain state set of ~o on T~ h, then J is called a reflection of I, and ~f ~o, T~ h is the reflection state of I based on J. When J = I -1 , ~f ~o, T~ h is the reduction state of I.

It should be noted that the isomorphism between the state set and the reflection set of restorable information is of great significance. Through isomorphism, the same mathematical method can be applied to two different sets of information, that is, noumenon states and carrier states. The objects in these sets have the same attributes and operations. The proposition established for one set can be established for another. This facilitates the use of abundant mathematical theories to support extensive research in the field of information science.

### 3.2.3 Transitivity

Information I can be transmitted from o to c and from c to other c ′ that is the reflection of c, from Th to Tm and from Tm to another reflection time Tm′ that is the reflection of Tm, and from f(o, Th) to g(c, Tm) and from g(c, Tm) to another reflection set g0 c0 , T0 m through the compound mapping I ′ (I( f(o, Th))); that is, via the transitivity of information.

Information I = ho, Th, f, c, Tm, gi is a surjective map from f(o, Th) onto g(c, Tm). If there exists the set c ′ in the objective world, the time set T0 m, the set g0 c0 , T0 m of all states of c ′ on T0 m, and a surjective map,

$$I': \mathcal{g}(c, T\_m) \to \mathcal{g}'(c', T\_m')$$

with

$$I'(g(c, T\_m)) = I'(I(f(o, T\_h))),$$

then

$$I \circ I' = \langle o, T\_h, f, c', T\_m', g' \rangle.$$

is also an information, where the set g0 c0 , T0 m is denoted by g′ . This is the transitivity of pieces of information.

It is reasonable to state that it is due to the transmissibility of information such that information movement in the collection, transmission, processing, convergence, and action links can be realized. In particular, serial information transmission is a common form of information movement in information systems, so it is significant to analyze the mechanism of a serial information transmission chain.

### 3.2.4 Compositionality

In I = ho, Th, f, c, Tm, gi, f(o, Th) and g(c, Tm) are mathematical sets. Thus I can naturally be decomposed or combined into different new sets, that is, information has compositionality. The mathematical inference is followed.

For two pieces of information I0 = o0 , T0 h, f0 , c0 , T0 m, g0 and I = ho, Th, f, c, Tm, gi with

$$\mathbf{o}' \subseteq \mathbf{o}, \,\mathbf{T}'\_{\mathbf{h}} \subseteq \mathbf{T}\_{\mathbf{h}}, \mathbf{f}' \subseteq \mathbf{f}, \mathbf{c}' \subseteq \mathbf{c}, \,\mathbf{T}'\_{\mathbf{m}} \subseteq \mathbf{T}\_{\mathbf{m}}, \mathbf{g}' \subseteq \mathbf{g}$$

if for any o0 1 2 o0 , T0 h1 2 T0 h, f0 1 2 f 0 , thereholds

$$\mathbf{I}'(\mathbf{f}\_1'(\mathbf{o}\_1', \mathbf{T}\_{\mathbf{h}1}')) = \mathbf{I}(\mathbf{f}\_1'(\mathbf{o}\_1', \mathbf{T}\_{\mathbf{h}1}'))$$

then I′ is called sub-information of I, denoted as I′ ⊆ I. We also say that I′ is in I.

If, moreover, there holds f′ ⊂ f, then I′ is called a proper sub-information of I, denoted as I′ ⊂ I. We also say that I′ is properly in I.

If I0 = o0 , T0 h, f0 , c0 , T0 m, g0 and I00 = o00, T00 h, f00, c00, T00 m, g00 are two pieces of proper sub-information of the information I = ho, Th, f, c, Tm, gi with o = o0 [ o00, Th = T0 h [ T00 h, f = f 0 [ f 00, c = c0 [ c00, Tm = T0 m [ T00 m, g = g0 [ g00 , then I is said to be a combination of I′ and I′′, denoted as I = I ′ [ I ′′ .

Naturally, information I can be decomposed or combined into several new sets. The compositionality of information determines that information can be flexibly split and arbitrarily combined, which creates sufficient conditions for people to determine the objects of information processing according to actual needs.

### 3.2.5 Relevance

The relevance of information manifests itself in at least three ways. Firstly, for information I = ho, Th, f, c, Tm, gi, o and c, Th and Tm, and f(o, Th) and g(c, Tm) all come in pairs. As a surjective map of f(o, Th) onto g(c, Tm), information I establishes a particular connection between o and c. In particular, the information transmission is an important embodiment of information relevance, in which things are bridged together. Thus, people usually state that information is a bridge that connects things. Secondly, there may be connections between different pieces of information or the containment of one piece of information in another. Because various mutual relationships can exist between different pieces of information, which is a form of information correlation, people can utilize various analytical approaches to uncover the values of information. Thirdly, the most important form of information relevance is the internal relationships in a reduction state. Here, reductional information can completely retain the internal structure of the original information, which is an important prerequisite for accessing, processing, and analyzing the internal structure of information.

### 3.3 The Sextuple Model of Information System

An information system is a human-computer integrated system for processing information flow, which is composed of computer hardware, network and communication equipment, computer software, information resources, information users, and rules and regulations. Processes such as control provide information that meets user requirements and has corresponding metrics.

Fig. 3.1 Information flows during the news gathering and release process

The information system to manage and use information has become an integral element of the social system. Numerous information systems have changed the way people live and work. In modern information systems, information integration reflects the static characteristics and dynamic changes of things, and it is necessary to solve fundamental theoretical problems in information science, such as the essence of information, measurement methods, model algorithms, and controllability of calculations.

Figure 3.1 visualizes the information flows in the news gathering and release process. This intuitive scenario can help us understand the sextuple model of information. In Fig. 3.1, the information collection link primarily collects the state information of the interviewees through video, audio, text, and other collection means. In addition, the information transmission link transmits the collected information to the corresponding processing system through the Internet and other wide area networks, and the information processing link performed video, audio, text, and mutual fusion processing to form various news materials. These news materials are gathered into a comprehensive news database to support extensive access and application. Then, in the information processing link, news information with richer content and forms is distributed and arranged to satisfy the publishing conditions. Through the information transmission link, all types of media news information are then transmitted to various information terminals over the Internet. Finally, in the information action link, all kinds of terminal devices display the corresponding news information to different audiences or readers in a variety of formats.

According to the analysis of information space framework, the entire news gathering and release process includes seven important links, where the information in each link has six elements, i.e., noumenon, occurrence time, state set, carrier, reflection time, and reflection set. Table 3.2 shows the specific content. Note that the information noumenon and carrier at each link in Table 3.2 differ. In particularly, the noumenon, occurrence time, and state set of the next link are the carrier, reflection time, and reflection set of the previous link, respectively, which reflects the concept of information flow (an important characteristic of information transmission). In addition, the news information itself reflects the subjective and objective state of the interviewee; thus, the noumenon and state of all links can be understood as the

Table 3.2 Information elements in main links of news gathering and release process

subjective and objective content of the interviewee's image, voice, and text during the interview period, which is also a basic characteristic of information transmission.

### 3.4 Chapter Summary

This chapter defines the essence of information in the objective category, which not only conforms to the fundamental positioning of information as one of the three major components of the objective world, but also adapts to the operational requirements that information systems can handle objective and real information. It puts forward the mathematical expression and six aspects of information, i.e. the sextuple model, providing a unified, clear, convenient and feasible theory for all subsequent studies. Moreover, it demonstrates the objectivity, restorability, transmission, combination and relevance of information.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Chapter 4 Measuring Objective Information

Abstract In the context of the sextuple model, this chapter presents a metrics system with 11 indicators to measure objective information. Under the measurement, potential, and features of the set, each indicator is mathematical explored from the methodological foundations, specific definitions, and related propositions for quantitative information analysis and information systems applications.

Keywords Objective information theory (OIT) · Information measurement · Metric system on on information · Metric system on on information system

### 4.1 Motivations to Measure Information

The philosophy of objective information and the related sextuple model provide a powerful and flexible tool to derive various information metrics, which is inspired by both theoretical research and practical experience in the light of purpose and principle to measure information.

### 4.1.1 Prerequisite to Well Manage Information

Peter Drucker said "If you can't measure it, you can't manage it." A set of reasonable and feasible metrics to measure information is the prerequisite for managing and using information well [1]. Measurement is a quantitative description to describe the relationship between objects. Information measurement is one of the basic tasks of information comprehensive effectiveness evaluation and comprehensive integration. It has a positive significance for the effective use of information resources and guiding the scientific development of information system construction.

### 4.1.2 Need to Comprehensively Master Information

There is a lack of a comprehensive of system for information metrics. To the best of our knowledge, Shannon information entropy is the only widely adopted information metric [2]. However, because information systems have evolved into different functionalities with complex structures far beyond those of communications, metrics based on entropy cannot comprehensively measure the complex dynamics of various information systems. Although other studies on the measurement of information exist, they lack for strict mathematical definitions and being systemic [3], making it difficult to establish an essential, basic reference framework for studying information mechanisms.

### 4.1.3 Requirement to Thoroughly Analyze Information Systems

There has been a lack of a thorough analysis of the efficacies of information systems. A dynamic mechanism describes how the target objects affect each other in a specific field. For example, in Newtonian mechanics, the efficiencies of speed, energy, and power were measured and analyzed to establish the theoretical system of mechanical dynamics. This methodology has also been applied in other fields. However, although information is pervasive in everyday life, there is a lack of a systematic approach to measuring and analyzing different efficacies of information systems. It is clear that only by establishing a complete analytical system of information efficacy can we accurately analyze the dynamic patterns in complex large-scale information systems.

### 4.1.4 Criteria to Measure Information

The information sextuple model depicts information with noumenon, state occurrence time, state set, carrier, reflection time, and reflection set, i.e. I = ho, Th, f, c, Tm, gi. To guide and regulate the derivation of information metrics, we should have come up with the following criteria.

To address these fundamental challenges, OIT takes information as the reflection of things and their movement states in the objective world and subjective world. It simplifies the concept of information from a general arbitrary idea to an objective concept in the real world, defining information as a mathematical mapping from objects in the real world to objects in information space. Based on the above principles, we will define 11 indicators for the metric system of information under the sextuple model. And the relevant basic propositions are also drawn from the set property such as metric, potential, and distance.

### 4.2 Metric System on Information

The metric system of objective information mainly includes 11 indicators: volume, delay, scope, granularity, variety, duration, sampling rate, aggregation, coverage, distortion, and mismatch (see Table 4.1). In Table 4.1, all of these indicators are defined and inferred with basic mathematical tools, such as set and measure, which allows them to accommodate classical information theories. For example, Shannon information entropy is a special case of the volume to measure the capacity of a communication system that transmits messages. In fact, for each metric indicator, a corresponding example can be found in classical or common information theories.

### 4.2.1 Volume

The value of information is closely related to its inherent containability, which is reflected by the required volume for the carrier. When other conditions are the same, the smaller the required volume is, the higher the information value is; and conversely, the lower the value is. So information may be measured by the volume.

Table 4.1 Classical theories of information science corresponding to different metrics

Let g(O × T) be the set of state sets, which contains g(c, Tm) on the objective world and temporal domain, and (g(O × T), 2g(O × T) , σ) be a measure space, where σ is a measure for the space. Then, the volume of information I relative to measure σ (viz. volumeσ(I)) is the measure σ(g(c, Tm)) of g(c, Tm), which is expressed as follows:

$$\text{volume}\_{\sigma}(I) = \sigma(\mathbf{g}(c, T\_{\mathfrak{m}})) \tag{4.1}$$

In information systems, the volume of information is usually measured in bits, which is the most understandable information metric. In practice, the measure σ of (g- (O × T), 2g(O × T) , σ) for a specific task is determined by the universe of discourse. Therefore, the volume metric defined here is in a general form, but can be defined differently according to the practical needs. The metrics defined in the reminder of this study follow the same principle when applied in practice.

Volume is one of the most familiar and applied information metrics. Almost all information resources in the form of files in information systems use volume as a typical metric. Many information technologies have been developed in response to the need to keep the substantive content unchanged and minimize the information volume of the files. Data compression is one of them. Many classical information principles are related to the volume metric, and it is not difficult to prove that the minimum restorable volume of information of random events is the Shannon information entropy.

### 4.2.2 Delay

The value of information I is related to its timeliness, embodied in the delay between reflection time and occurrence time. Generally, when other conditions are the same, the shorter the delay is, the higher the information value is; and conversely, the lower the value is. So I may be measured by the delay.

Delay reflects the speed of the carrier response to the state of noumena. Therefore, the delay of information I (viz. delay(I)) is the difference between the supremum of its reflection time (supTm) and the supremum of its occurrence time (supTh), which is expressed as follows:

$$\text{delay}(I) = \sup T\_{\text{m}} - \sup T\_{\text{h}} \tag{4.2}$$

It is should be noted that this definition of delay allows for both positive and negative values. In particular, when supTm < sup Th, delay(I) < 0. This represents the prediction of the future states by the carrier prior to the occurrence time Th of the state for noumena. For example, the motion of targeted objects and the occurrence of events of interest can be predicted in information systems.

Actually, delay is also one of the most concerned information metrics. In fact, it will be insignificant for most information if its delay is too large. For example, if the delay of a weather forecasting information is positive, indicating that the carrier's reflection time is later than the actual weather occurrence time in the relevant region, the information of weather forecasting loses the significance of forecast. Therefore, people develop high-speed communication and high-performance computing technologies often with the purpose of reducing the delay metric of information. It is not difficult to prove that the total delay of serial information transmission is equal to the sum of information delays of each link in an information system.

### 4.2.3 Scope

Information is the objective reflection of noumenon, and its value is related to the noumenon scope. When other conditions are the same, the wider the noumenon scope is, the higher the information value is; and conversely, the lower the value is. So information can be measured by scope (coverage, extensity).

Let (O [ S, 2O [ S , σ) be the measure space over O [ S. Let σ be some measure on (O [ S). According to the definitions of O and S, noumena o are elements of O and S, that is, o 2 2O [ S . Then, the scope of information I relative to measure σ (viz. scopeσ(I)) is the measure σ(o) of o, which is defined as follows:

$$\text{scope}\_{\sigma}(I) = \sigma(o) \tag{4.3}$$

The scope indicator reflects the extent of information noumena or information content, and is also an information metric that is highly valued by people. The Chinese idiom of "Climbing higher and looking farther" refers to the fact that a person can get a greater scope of visual field information when he/she climbs higher. Telescopes, radar and other detection tools are also designed to enable users to obtain a greater scope of information. It is not difficult to prove that the radar equation reveals the positive and negative proportional relationship between the scope of detection information and radar transmission power, antenna aperture, antenna gain and detection sensitivity.

### 4.2.4 Granularity

The value of information is directly related to the granularity of the noumenon it reflects. The granularity indicates the degree of coarseness of the particles into which the noumenon can be decomposed. When other conditions are the same, the finer the particles are, the higher the information value is; and conversely, the lower the value is. So information could be measured by granularity (detailedness).

For a pair of information I and I ′ , if I ′ is proper sub-information of I and there is no other proper sub-information I ′′ of I such that I ′′ ⊂ I ′ , then I ′ is called the atomic information of I. Here, let (O [ S, 2O [ S , σ) be a measure space, and let σ be some measure on the set (O [ S). The set of all atomic information in information I is denoted as A = {Iλ = hoλ, Thλ, fλ, cλ, Tmλ, gλi}λ 2 Λ, where Λ is an index set and λ is an index. In this case, let μ be the measure of the index set Λ and μ(Λ) ≠ 0. Then, the granularity of information I relative to measure σ (viz. granularityσ(I)) is the ratio of the integral of all atomic noumenon information measures in A to the measure μ of index set Λ, which is expressed as follows:

$$\text{(gramularity}\_{\sigma}(I) = \frac{\int\_{A} \sigma(o\_{\lambda}) \text{d}\mu}{\mu(\Lambda)}\tag{4.4}$$

where it is most appropriate to take μ as the counting measure.

The granularity indicator reflects the degree of detail of information noumena or information content, and is also an indicator that people attach great importance to in their daily lives. With the development of display technology, the resolution of color TV is getting higher and higher, meaning that the TV image information is getting increasingly smaller granularity, which can provide viewers with an immersive watching experience. It is not difficult to prove that the Rayleigh criterion reflects the positive and negative proportional relationship between the granularity of optical imaging information and the light wavelength and sampling pore width, respectively.

### 4.2.5 Variety

For information I, let R be an equivalence relation on the set of states f(o, Th), and the set of equivalence classes of the elements in f(o, Th) relative to R is [f(o, Th)]R. Then, the variety of information I relative to R (viz. varietyR(I)) is the cardinality of set [f(o, Th)]R, which is expressed as follows:

$$\text{variety}\_{\mathcal{R}}(I) = \overline{[f(o, T\_{\mathcal{h}})]\_{\mathcal{R}}} \tag{4.5}$$

For reducible information, the equivalence relation within state set can be transferred to reflection set. Therefore, the reflection set of the carrier can fully reflect the variety metric of information.

A rich variety of information provides a comprehensive understanding of what is of interest. Therefore, variety is an important indicator for measuring information. For example, when people have access to text, picture, audio and video information of the same object of interest, the variety of the whole information is increased, which thus enables them to know the object more comprehensively and deeply. Restorability is an extremely important property of information. It is not difficult to prove that the restorable information can maintain the invariance of the variety indicator.

### 4.2.6 Duration

The value of information is related to the duration of the noumenon it reflects. The duration indicates the density and span of occurrence time. When other conditions are the same, the higher the density is and the longer the span is, the higher the information value is. Conversely, the lower the value is. So information could be measured by the duration (sustainability).

The duration of information I (viz. duration(I)) is the difference between the supremum and infimum of Th, which is expressed as follows:

$$\text{duration}(I) = \sup T\_{\text{h}} - \inf T\_{\text{h}} \tag{4.6}$$

where infTh is the infimum of the occurrence time Th.

The duration indicator reflects the time span of information and is certainly an important indicator for measuring information. Manufacturers of monitoring equipment are always looking for ways to extend the stable working time of their products in order to meet the requirements of users to constantly increasing the information duration indicator. It is not difficult to prove that the average duration of information collected by a continuous monitoring system is equal to its mean time between failures (MTBF).

### 4.2.7 Sample Rate

For information I, if infTh ≠ sup Th, let {Uλ}λ 2 Λ be a family of pairwise disjoint connected sets that satisfy the following: for any λ2Λ, there are Uλ⊆ [infTh, sup Th], and Th \ Uλ = ∅. Then, the sampling rate of information I (viz. sampling \_ rate(I)) is simply the ratio of the cardinality of Λ to the Lebesgue measure jUj of U = S λ2Λ Uλ,

which is expressed as follows:

$$\text{sampling\\_rate}(I) = \frac{\overline{A}}{|\ U|}\tag{4.7}$$

Here, if infTh = sup Th or the Lebesgue measure of U is |U| = 0, then sampling \_ rate (I) = 1 is defined, which indicates that the state set of information I is completely continuous in time.

The sampling rate indicator reflects the sampling interval of information, which is closely related to the restorability of the information. It is not difficult to prove that Nyquist's sampling theorem reveals that the minimum restorable sampling rate of periodic function information is equal to half of its frequency, which is the essential requirement of information digitization.

### 4.2.8 Aggregation

The aggregation measures the similarity between different information, gathers similar information, separates dissimilar information, and divides the information into different clusters. The information similarity in the same cluster is the greatest, and the information dissimilarity between different clusters is the greatest. The aggregation starts from sample data, simplifies data through data modeling, and automatically clusters, which can be used as the basis for information classification.

For information I, if the cardinality of set f(o, Th) is f oð Þ , Th ≠ 0, let R be the set of relations between all elements on the state set f(o, Th). Then, the aggregation of I (viz. aggregation(I)) is the ratio of the cardinality of set R to that of set f(o, Th), which is expressed as follows:

$$\text{aggregation}(I) = \frac{\overline{\mathcal{R}}}{\overline{f(o, T\_h)}} \tag{4.8}$$

The aggregation metric characterizes the distance between the elements of the state set f(o, Th) in the information space. In general, the closer the distance is between the elements of the state set f(o, Th), the higher the degree of aggregation and the higher the value of the information.

The aggregation indicator reflects the degree of relevance of information content. Data fusion technology exactly uses the relevance of information content for in-depth processing to optimize other information metrics and meet the requirements of users in many aspects. It is not difficult to prove that the restorable information can maintain the invariance of the degree of aggregation.

### 4.2.9 Coverage

The value of information is often related to its distribution, which is reflected by the coverage of the carrier. For some information, the wider the coverage is (within a certain extent), the higher the Information value is. For other information, the narrower the coverage is, the higher the value is. So, in any case, information could be measured by the coverage (distribution).

 Here, let {Iλ = hoλ, Thλ, fλ, cλ, Tmλ, gλi}λ 2 Λ be a set containing information I and all of its copies. Then, the coverage of information I relative to some measure σ on a measurable set of c (viz. coverageσ(I)) is the integral of all measures cλ, which is expressed as follows:

$$\text{coverage}\_{\sigma}(I) = \int\_{\Lambda} \sigma(c\_{\lambda}) \text{d}\mu \tag{4.9}$$

The coverage indicator reflects the extent of distribution and knowledge of information content. Information encryption technology exactly adopts the approach to minimizing the coverage of information to the most extent in accordance with the right to know. Notwithstanding, the general online propaganda seeks to find ways to increase the coverage indicator of information. Therefore, the number of clicks and views has become a characteristic indicator to measure the value of a website in today's era, which is essentially the coverage of information. It is not difficult to prove that Metcalfe's law can also be expressed as the value of the network system is equal to the product of the maximum scope of information it can obtain and the maximum coverage of information it can distribute.

### 4.2.10 Distortion

As the mapping between noumenon state and carrier state, information should not be labeled true or false. According to the property of restorability, there always exists an inverse mapping of information in theory that could restore the actual state of information noumenon at the occurrence time. However, due to the complexity of the mapping process and the existence of subjective factors in the perception process, it is often impossible to acquire the exact inverse mapping of information. Therefore, only by deducing as far as possible can we get close to the actual state of noumenon at the occurrence time.

h i For information I and its reflection J, let the state set f(o, Th) and reflection state set ~f ~o, T~h - be elements in a distance space h i F , d , where F is the set of reflection sets and d is the distance on F . Then, the distortion of reflection J of information I (viz. distortionJ(I)) is the distance between ~f ~o, T~h - and f(o, Th) in the distance space F , d , which is expressed as follows:

$$\text{distortion}\_J(I) = d\left(f, \tilde{f}\right) \tag{4.10}$$

The distortion metric measures the degree of deviation between the reflection state and reduction state. The reflection state of information I is its reduction state if and only if the distortionJ(I) = 0.

Distortion is a negative indicator of the authenticity of information and is certainly one of the most concerned information metrics. It is hard to imagine that normal people need information that has lost its authenticity. Therefore, reducing errors by all means, that is, reducing information distortion, is one of the objectives for people to apply various technical means to process information since ancient times. It is not difficult to prove that Kalman filtering principle reveals the estimation method of the minimum information distortion for linear systems with known variance.

### 4.2.11 Mismatch

The value of information eventually depends on what degree it matches users' demand, embodied in the overall degree to which the various elements of information [11]. Generally speaking, the higher the overall degree is, the higher the information value is; conversely, the lower the value is. So information could be measured by mismatch (suitability).

 Let information I0 = ho0, Th0, f0, c0, Tm0, g0i be the target of information I = ho, Th, f, c, Tm, gi, let o0 and o, Th0 and Th, f0 and f, c0 and c, Tm0 and Tm, and g0 and g be elements in the sets Po, P Th , Pf , Pc, P Tm , and Pg, respectively, and let I0 and I be elements in the distance space Po, P Th , Pf , Pc, P Tm , Pg - , d . Then, the mismatch of information I to target information I0 (viz. mismatchI0 ð ÞI ) is the distance between I and I0 in the distance space Po, P Th , Pf , Pc, P Tm , Pg - , d , which can be expressed as follows:

$$\text{mismatch}\_{I\_0}(I) = d(I, I\_0) \tag{4.11}$$

For the distortion and mismatch of information, we have not got the relationship between the measurements of the part and the whole, showing that neither the distortion nor mismatch would change along with the amount of information. This conforms to both common sense and the operation rules of information systems.

Mismatch is a negative indicator reflecting the degree to which information meets the requirements of users. It is the eternal goal of information systems to deliver the right content to the right object at the right time and place, which actually requires the use of various information technologies to continuously reduce the mismatch of all output information of information systems. It is not difficult to prove that the average search length principle reveals the shortest search path for information with minimum mismatch in a finite set of information.

### 4.3 Metric System on Information System

The metrics system on information can be further used to evaluate the effectiveness of complex information system, and the quality of informatization.

### 4.3.1 System of Systems

An system-of-systems (SoS) is a collection of task-oriented systems that offer more functionality and a greater performance than simply the sum of the constituent systems. With the ever-growing scales and complexities of information systems, it is becoming increasingly difficult for people to understand and grasp information systems, particularly SoS. Owing to the influence of various internal and external uncertainties, the dynamic behavior of these complex SoS may deviate from their original purpose, and unstable phenomena may appear. Moreover, in the construction and application of such large-scale SoS, an emphasis on order alone, while ignoring vitality, will lead to system rigidity, whereas an emphasis on vitality alone, while ignoring order, will lead to system chaos. Therefore, there is an urgent need for measuring the information during the process of the design, development, application, and evaluation of large-scale SoS.

### 4.3.2 Metric Effects and Efficacies of Information Systems

Any information system can be simplified as a basic process of information input, process, and output. The significance of information systems lies in their various efficacies. That is, the abilities of information systems to act on the input information and the effects expressed through the output information. In a large scale SoS, different efficacies are usually intertwined owing to complex information movements. Without a comprehensive analysis, reasonable deconstruction, and quantitative expression of these efficacies, it is difficult to deeply understand the inherent rules of the operation mechanism of information systems. Consequently, it is impossible to develop theoretical information systems dynamics (ISD) to guide the construction and development of large-scale SoS. Therefore, accurately and comprehensively measuring various efficacies of information systems is of decisive significance for an in-depth study on ISD.

Efficacy cannot be expressed quantitatively without certain metrics; thus, there must be an effective metric for a specific efficacy. The aforementioned 11 metrics can be used to measure various aspects of the effect on the input and output of information systems, which we refer to as the metric effect. It is therefore natural to apply these metrics to comprehensively and quantitatively describe and analyze the main efficacies of information systems. Specifically, there are 11 information system efficacies that can be established through 11 types of metric effects—namely, the volume, delay, scope, granularity, class, duration, sampling rate, aggregation, coverage, distortion, and mismatch efficacies.

• The volume is the most fundamental one that information systems act on information amongst various efficacies. In practice, each link of information collection, transmission, processing, data space and action can affect the volume efficacy by changing the capacity of the system. For example, the insufficient storage of information collection, data space and information action will lead to the reduction of the volume efficacy of information systems. The insufficient channel bandwidth of information transmission system will lead to discard part of the information, thus reducing the volume efficacy. Information processing also needs enough storage space to support, thus also affects the volume of information. Note that information processing systems can affect volume efficacy by the data compression processing, that is, the total information volume of whole system is increased by the decompression of some information.

is possible to optimize the granularity of information. Data space can also affect the granularity effect of information by factors such as the model integrity, granularity, and database capacity. Similar to the analysis of scope efficacy, the information transmission does not directly have the granularity efficacy on information.

Fig. 4.1 Efficacy distribution for each link in information space [13]

Figure 4.1 illustrates the information efficiency distribution across the information space. In the figure, the star symbol indicates the existence of the corresponding efficiencies, represented by the sectors, of some link. The collection and action of information are positioned in the same ring at the periphery, and are distinguished by two different shades of blue: The dark blue represents the information collection link, and the light blue represents the information action link. Therefore, the functionality and performance of an SoS can be deconstructed through the efficacy distribution, which provides a sufficient and quantitative basis for the design, analysis, testing, and integration of an SoS.

Fig. 4.2 Complete dynamic configuration of information systems [13]

### 4.3.3 Dynamic Configurations of Information Systems

In Fig. 4.2, the operating mechanism and efficacy distribution of various information links—that is, information collection, transmission, processing, storage, and action are presented, along with the possible information flows between the links; these are referred to as the integral dynamic configuration of an SoS. Information flow is the form and carrier of information movement in an SoS. In general, as long as information flow maintains its continuity, we can use the local metric effects at various links to analyze the global functionalities and performance of an entire SoS, which is the starting point to investigate ISD and the original intent of applying ISD to guide the planning, design, research and development, and integration of information systems.

In Fig. 4.2, each link in an SoS can affect the functionality and performance of an entire system. In general, the effects of the same class at each link can have mutual superposition or mutual restraint. For example, the delay effect at each link can be superimposed to form the delay effect of the entire SoS. In addition, the volume metric of the previous link forms the volume requirement for the subsequent link. If the requirement is not satisfied at the subsequent link, the volume efficiency of the entire SoS is affected.

It should be noted that there are mutual effects among the different metrics. For example, the volume effect impacts the distortion effect of the system. With insufficient volume, the elements in the reflection set will be abandoned, which will result in an increase in the distortion metric. The degree of mismatch reflects the degree to which the information output of the system deviates from the needs of specific users. The volume, delay, scope, granularity, duration, class, sampling rate, and aggregation metrics are closely related to the needs of specific users, meaning that any of these metrics can affect the mismatch metric of the entire SoS. For example, to control the range of information acquisition, the coverage metric must be adjusted according to the desires of the users. Therefore, the coverage metric is not related to the mismatch metric in a single direction. In addition, the distortion metric is not positively correlated with the mismatch metric. For example, in an encrypted information system, higher distortion results in lower mismatch for a specific user.

In engineering practices, users do not always need to apply the integral dynamic configuration of information systems. In many cases, some links of the SoS may not determine or affect the key efficacies of the entire SoS. In such cases, it is possible to limit the consideration of system designers to relatively minor links and form the simplified dynamic configurations for the information systems. Studying the mechanism of the efficacies with various configurations to reveal the inherent operating regularities of information systems provides a powerful means of guiding the planning and development of the large-scale SoS in engineering practices.

### 4.4 Chapter Summary

This chapter systematically establishes 11 efficiencies of measurement systems for information, defines the mathematical expression of each measurement from the basic model, and discusses the basic properties of each measurement, providing a theory for quantitative research and analysis of information operation and information system applications, and method basis.

### References

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

# Chapter 5 Exemplifying Objective Information: Air Traffic Control System

Abstract Air Traffic Control System (ATCS) is a large-scale information system that spreads all over the world and is widely used (Zhang et al., Modern Air Traffic Management (in Chinese). Beijing University of Aeronautics and Astronautics Press, Beijing, 2005). Although it cannot represent the entire information system, its structure, functional composition and application methods can already reflect many basic characteristics of today's information systems, especially large-scale information systems. In this chapter, OIT is applied to explain the behavior of ATCS for demonstrating the feasibility and practicality to investigate information systems.

Keywords Objective information theory (OIT) · Air traffic control system (ATCS) · Exemplification

# 5.1 The Air Traffic Control System (ATCS)

ATCS is the core system for implementing airspace management, ensuring flight safety and maintaining the order of air transport, and also an information system of colossal volume, great complexity and typical significance [1]. The navigation intelligence, dynamic meteorological information, flight rules and regulations as well as static materials that are acquired, processed or generated by ATCS are the sources of operation for the whole system [2]. Thereinto, the flight status, meteorological and geographical information are the variations of the objective world, while flight regulations are subjectively stipulated by men. Once being imported and documented, all this information will become the objective reflection of the related objects and their motion states in ATCS, supporting the operation and management of ATCS [3].

### 5.2 The Sextuple of ATCS

For ATCS, its information set includes flight data, airport information, navigation and regional charts, national and international NOTAM, navigation chart information, route information, airport live weather reports, runway visual range, airport weather forecasts, significant weather alerts, weather images, air traffic control regulations, agreements with surrounding areas, procedures for handling special circumstances, airplane performance parameters and other data codes, etc. Table 5.1 characterizes the main elements of the aforesaid information in ATCS.

In Table 5.1, the noumenon of flight information include both objective objects, i.e. flights and other flying objects, and subjective consciousness of the pilots, since flight data reflects not only the motion parameters of flying objects, but also the pilots' operational intentions. Take the 9.11 attack for example, after the terrorists' hijacking of the plane, the most significant information in the air traffic control system is nothing but the subjective intention of the pilot, which is reflected through flight data. Thus, the state set of flight data contains not only the position, speed and direction of the flying object, but also subjective factors such as the pilot's operational intention. However, the reflection set only includes the motion parameters of the flying object, but not the subjective intention of the pilot.

The carriers of each information item are put into different subsystems of ATCS according to their features in retention and application. In fact, ATCS itself is the carrier of all these information items, while the subsystems in ATCS, such as communication gateway subsystem, information distribution subsystem and system data backup subsystem etc., may become the carriers of relevant information at the corresponding moments. For the sake of conciseness, many trivial details are omitted here.

The reflection time of all information items in Table 5.1 is "from acquisition & import to the outage of ATCS", because even when the aforesaid information items become outdated, they may still play a part in supporting and assisting the operation control in the future, and therefore, the advanced ATCS would save these information items in case of need.

In Table 5.1, the air traffic control regulations, the agreements with neighboring areas and the procedures for handling special circumstances etc. are all products of subjective will, but once issued, signed or stipulated, they will become important objective information stored in ATCS, supporting the air traffic control institutions and ATCS in the management and control of flying objects.

The sextuple model of OIT in ATCS indicates the specific components of an information item.

$$I = \langle o, T\_h, f, c, T\_m, g \rangle.$$

where, o, Th, f, c, Tm, g denote the noumenon, state occurrence time, state set, carrier, reflection time and reflection set of information I in ATCS respectively. Table 5.1 indicates 16 information items in ATCS. From the third row on Table 5.1,

Table 5.1 The main elements of the information set in ATCS (continued)

Table 5.1 (continued)

(continued)

$$o \supseteq \bigcup\_{i=1}^{16} o\_i, \quad T\_h \supseteq \bigcup\_{i=1}^{16} T\_{hi}, \quad f \supseteq \bigcup\_{i=1}^{16} f\_i, \quad c \supseteq \bigcup\_{i=1}^{16} c\_i, \quad T\_m \supseteq \bigcup\_{i=1}^{16} T\_{mi} \quad \text{and } g \supseteq \bigcup\_{i=1}^{16} g\_i,$$
 denote the nounmon, state occurrence time, state set, carrier, reflection time and reflection set of information  $I$  in ATCS respectively.

$$I\_i = \langle o\_i, T\_{hi}, f\_i, c\_i, T\_{mi}, \mathbf{g}\_i \rangle \quad (i = 1, 2, \cdots, 16)^t$$

Here, the relation between the former and the latter items in the equations is "⊇" but not "=" because Table 5.1 only lists the typical information of ATCS but not all.

### 5.3 The Information Metrics for ATCS

### 5.3.1 ATCS Volume