The Mathematical Theory of Semantic Communication

　　自从1948年经 典信息论诞生以来，在其指导下，现代通信技术已经逼近了理论性能极限，例如信息熵 ，信道容量 以及率失真函数 。语义通信开辟了未来通信技术发展的新方向，但还缺乏数学指导理论。为了解决这一难题，本文构建了语义信息论的理论框架。通过研究语义通信的机制，我们发现同义性是其基本特征，由此定义了语义信息和语法信息之间的同义映射。基于同义映射 这一核心概念，我们引入了语义信息的度量体系，包括语义熵 ，上/下语义互信息 ，语义信道容量 以及语义率失真函数 。进一步，采用随机编码以及（联合）同义典 型序列编译码方法，证明了三个语义编码定理，即语义信源编码定理、语义信道编码定理以及语义率失真编码定理。我们发现同义映射扩展了通信系统的极限。进一步讨论了连续条件下的语义信息度量。特别地，对于带宽受限的高斯信道，我们得到了新的信道容量公式 ，其中平均同义长度 表征了信息的辨识能力，这一公式是香农信道容量公式的推广。 综上所述，本文提出的语义信息论，依据同义映射这一语义信息的本质特征，构建了语义信息的度量体系，引入新的数学工具，证明了语义编码的基本定理，论证了语义通信系统的性能极限，揭示了未来语义通信的巨大性能潜力。

　　Kai Niu received the B.S. degree in information engineering and the Ph.D. degree in signal and information processing from Beijing University of Posts and Telecommunications， Beijing， China， in 1998 and 2003， respectively. He is currently a Professor of the School of Artificial Intelligence， Beijing University of Posts and Communications. His research interests include information theory， coding theory， and semantic communication. He has published more than 200 academic papers， and proposed high performance construction algorithm of polar codes applied in 5G standard， and won the first prize of Natural Science of Science and Technology Award of the Chinese Institute of Electronics. Ping Zhang is currently a Professor of the School of Information and Communication Engineering at Beijing University of Posts and Telecommunications， the Director of State Key Laboratory of Networking and Switching Technology， a member of IMT-2020 （5G） Experts Panel， a member of Experts Panel for China’s 6G development. He served as Chief Scientist of National Basic Research Program （973 Program）， an expert in Information Technology Division of National High-tech Research and Development program （863 Program）， and a member of Consultant Committee on International Cooperation of National Natural Science Foundation of China. His research interests mainly focus on wireless communication. He is an Academician of the Chinese Academy of Engineering （CAE）.

Chapter 1 Introduction 001
Chapter 2 Semantic Communication System and Synonymous Mapping 013
2.1 Notation Conventions 013
2.2 Semantic Communication System 014
2.3 Synonymous Mapping 016
Chapter 3 Semantic Entropy 019
3.1 Semantic Information Measures 019
3.2 Semantic Joint Entropy and Semantic Conditional Entropy 022
Chapter 4 Semantic Relative Entropy and Mutual Information 031
4.1 Semantic Relative Entropy 031
4.2 Semantic Mutual Information 036
Chapter 5 Semantic Channel Capacity and Semantic Rate-distortion 041
5.1 Semantic Channel Capacity 041
5.2 Semantic Rate-Distortion 042
Chapter 6 Semantic Lossless Source Coding 045
6.1 Asymptotic Equipartition Property and Synonymous Typical Set 045
6.2 Semantic Source Coding Theorem 051
6.3 Semantic Source Coding Method 054
Chapter 7 Semantic Channel Coding 057
7.1 Jointly Asymptotic Equipartition Property and Jointly Synonymous
Typical Set 057
7.2 Semantic Channel Coding Theorem 065
7.3 Semantic Channel Coding Method 073
Chapter 8 Semantic Lossy Source Coding 081
8.1 Semantic Distortion and Jointly Typical Set 081
8.2 Semantic Rate-Distortion Coding Theorem 085
Chapter 9 Semantic Information Measure of Continuous Message 091
9.1 Semantic Entropy and Semantic Mutual Information for Continuous
Message 091
9.2 Semantic Channel Capacity of Gaussian Channel 098
9.3 Semantic Channel Capacity of Band-limited Gaussian Channel 101
9.4 Semantic Rate-Distortion of Gaussian Source 104
Chapter 10 Semantic Joint Source Channel Coding 107
Chapter 11 Conclusions 111
Appendix 115
References 117