## 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.

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## 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 [1,2,3]. 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), *T*_{h} denote the occurrence time, and *T*_{m} 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^{O ∪ S}. *c* only belongs to an objective world, thus *c* ∈ 2^{O}. *T*_{h} and *T*_{m} are both in temporal domain, *T*_{h} ∈ 2^{T}and *T*_{m} ∈ 2^{T}. When *I* is depicted in the context of *o*, *T*_{h}, *f*, *c*, *T*_{m}, *g* simultaneously, a *sextuple model* comes into being.

Thus *I* is modeled mathematically as a full mapping from *f*(*o*, *T*_{h}) to *g*(*c*, *T*_{m}).

where, a state set *f*(*o*, *T*_{h}) of *o* on *T*_{h}, and a reflection set *g*(*c*, *T*_{m}) of *c* on *T*_{m} are all non-empty. If *I* is the surjective mapping from *f*(*o*, *T*_{h}) to *g*(*c*, *T*_{m}), Eq. (3.2) can be rewritten as

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*, *T*_{h}) to *g*(*c*, *T*_{m}). 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*, *T*_{m}) is on of *c*, *T*_{m} can be reduced to the state *f*(*o*, *T*_{h}) of *o* on *T*_{h} by *I*^{−1}, then we call *I* restorable. This is the restorability of information. Here, *f*(*o*, *T*_{h}) is also called the reduction state of *I*. The mathematical inference is as following.

I = 〈o, T_{h}, f, c, T_{m}, g〉 is a surjective map of f(o, T_{h}) onto g(c, T_{m}) and is called injective if for any *o*_{λ} ∈ *o*, *T*_{hλ} ∈ *T*_{h}, *f*_{λ} ∈ *f*, *o*_{μ} ∈ *o*, *T*_{hμ} ∈ *T*_{h}, *f*_{μ} ∈ *f* with *f*_{λ}(*o*_{λ}, *T*_{hλ}) ≠ *f*_{μ}(*o*_{μ}, *T*_{hμ}), there holds

If I = 〈o, T_{h}, f, c, T_{m}, g〉 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 c_{λ} ∈ c, T_{mλ} ∈ T_{m}, g_{λ} ∈ g, there exists a unique set of o_{λ} ∈ o, T_{hλ} ∈ T_{h}, f_{λ} ∈ f, s.t.

This means that

For the invertible information I = 〈o, T_{h}, f, c, T_{m}, g〉, g(c, T_{m}) of c on T_{m} can be reduced to the state f(o, T_{h}) of o on T_{h} by I^{−1}, and have I restorable and f(o, T_{h}) the reduction state of I.

Moreover, if there is a mapping *J* such that \( \mathrm{J}\left(\mathrm{g}\left(\mathrm{c},{\mathrm{T}}_{\mathrm{m}}\right)\right)=\overset{\sim }{\mathrm{f}}\left(\overset{\sim }{\mathrm{o}},\overset{\sim }{{\mathrm{T}}_{\mathrm{h}}}\right) \), where \( \overset{\sim }{\mathrm{o}} \) is referred to as reflection noumena (\( \overset{\sim }{\mathrm{o}}\in {2}^{\mathrm{O}\cup \mathrm{S}}\Big) \), \( \overset{\sim }{{\mathrm{T}}_{\mathrm{h}}} \) is referred to as reflection occurrence time (\( \overset{\sim }{{\mathrm{T}}_{\mathrm{h}}}\in {2}^{\mathrm{T}} \)), and \( \overset{\sim }{\mathrm{f}}\left(\overset{\sim }{\mathrm{o}},\overset{\sim }{{\mathrm{T}}_{\mathrm{h}}}\right) \) is a certain state set of \( \overset{\sim }{\mathrm{o}} \) on \( \overset{\sim }{{\mathrm{T}}_{\mathrm{h}}} \), then *J* is called a reflection of *I*, and \( \overset{\sim }{\mathrm{f}}\left(\overset{\sim }{\mathrm{o}},\overset{\sim }{{\mathrm{T}}_{\mathrm{h}}}\right) \) is the reflection state of I based on J. When J = I^{−1}, \( \overset{\sim }{\mathrm{f}}\left(\overset{\sim }{\mathrm{o}},\overset{\sim }{{\mathrm{T}}_{\mathrm{h}}}\right) \) 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 *T*_{h} to *T*_{m} and from *T*_{m} to another reflection time *T*_{m}′ that is the reflection of *T*_{m}, and from *f*(*o*, *T*_{h}) to *g*(*c*, *T*_{m}) and from *g*(*c*, *T*_{m}) to another reflection set \( {g}^{\prime}\left({c}^{\prime },{T}_m^{\prime}\right) \) through the compound mapping *I*^{′}(*I*(*f*(*o*, *T*_{h}))); that is, via the transitivity of information.

Information *I* = 〈*o*, *T*_{h}, *f*, *c*, *T*_{m}, *g*〉 is a surjective map from *f*(*o*, *T*_{h}) onto *g*(*c*, *T*_{m}). If there exists the set *c*^{′} in the objective world, the time set \( {T}_m^{\prime } \), the set \( {g}^{\prime}\left({c}^{\prime },{T}_m^{\prime}\right) \) of all states of *c*^{′} on \( {T}_m^{\prime } \), and a surjective map,

with

then

is also an information, where the set \( {\mathrm{g}}^{\prime}\left({\mathrm{c}}^{\prime },{\mathrm{T}}_{\mathrm{m}}^{\prime}\right) \) 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* = 〈o, T_{h}, f, c, T_{m}, g〉, f(o, T_{h}) and g(c, T_{m}) 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 \( {\mathrm{I}}^{\prime }=\left\langle {\mathrm{o}}^{\prime },{\mathrm{T}}_{\mathrm{h}}^{\prime },{\mathrm{f}}^{\prime },{\mathrm{c}}^{\prime },{\mathrm{T}}_{\mathrm{m}}^{\prime },{\mathrm{g}}^{\prime}\right\rangle \) and I = 〈o, T_{h}, f, c, T_{m}, g〉 with

if for any \( {\mathrm{o}}_1^{\prime}\in {\mathrm{o}}^{\prime },{\mathrm{T}}_{\mathrm{h}1}^{\prime}\in {\mathrm{T}}_{\mathrm{h}}^{\prime },{\mathrm{f}}_1^{\prime}\in {\mathrm{f}}^{\prime } \), thereholds

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 \( {\mathrm{I}}^{\prime }=\left\langle {\mathrm{o}}^{\prime },{\mathrm{T}}_{\mathrm{h}}^{\prime },{\mathrm{f}}^{\prime },{\mathrm{c}}^{\prime },{\mathrm{T}}_{\mathrm{m}}^{\prime },{\mathrm{g}}^{\prime}\right\rangle \) and \( {\mathrm{I}}^{\prime \prime }=\left\langle {\mathrm{o}}^{\prime \prime },{\mathrm{T}}_{\mathrm{h}}^{\prime \prime },{\mathrm{f}}^{\prime \prime },{\mathrm{c}}^{\prime \prime },{\mathrm{T}}_{\mathrm{m}}^{\prime \prime },{\mathrm{g}}^{\prime \prime}\right\rangle \) are two pieces of proper sub-information of the information I = 〈o, T_{h}, f, c, T_{m}, g〉 with \( \mathrm{o}={\mathrm{o}}^{\prime}\cup {\mathrm{o}}^{\prime \prime },{\mathrm{T}}_{\mathrm{h}}={\mathrm{T}}_{\mathrm{h}}^{\prime}\cup {\mathrm{T}}_{\mathrm{h}}^{\prime \prime },\mathrm{f}={\mathrm{f}}^{\prime}\cup {\mathrm{f}}^{\prime \prime },\mathrm{c}={\mathrm{c}}^{\prime}\cup {\mathrm{c}}^{\prime \prime },{\mathrm{T}}_{\mathrm{m}}={\mathrm{T}}_{\mathrm{m}}^{\prime}\cup {\mathrm{T}}_{\mathrm{m}}^{\prime \prime },\mathrm{g}={\mathrm{g}}^{\prime}\cup {\mathrm{g}}^{\prime \prime } \), 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* = 〈*o*, *T*_{h}, *f*, *c*, *T*_{m}, *g*〉, *o* and *c, T*_{h} and *T*_{m}, and *f*(*o*, *T*_{h}) and *g*(*c*, *T*_{m}) all come in pairs. As a surjective map of f(o, T_{h}) onto g(c, T_{m}), 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.

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 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

Xu, J., Wang, S., Liu, Z., Wang, Y.: Objective information theory exemplified in air traffic control system. Chin. J. Electron.

**30**(4), 743–751 (2021)Xu, J.F., Tang, T., Ma, X.F., Xu, B., Shen, Y.L., Qiao, Y.J.: Objective information theory: a sextuple model and 9 kinds of metrics. Arxiv:1308.3372v1.

Wiener, N.: Cybernetics: or the Control and Communication in the Animal and the Machine. The MIT Press, Cambridge (1961)

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Xu, J., Wang, S., Liu, Z., Wang, Y., Wang, Y., Dang, Y. (2023). Modelling Objective Information: Sextuple. In: Objective Information Theory. SpringerBriefs in Computer Science. Springer, Singapore. https://doi.org/10.1007/978-981-19-9929-1_3

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