Semantic communication initiates a new direction for future communication. In this paper, we aim to establish a systematic framework of semantic information theory (SIT). First, we propose a semantic communication model and define the synonymous mapping to indicate the critical relationship between semantic information and syntactic information. Based on this core concept, we introduce the measures of semantic information, such as semantic entropy $H_s(\tilde{U})$, up/down semantic mutual information $I^s(\tilde{X};\tilde{Y})$ $(I_s(\tilde{X};\tilde{Y}))$, semantic capacity $C_s=\max_{p(x)}I^s(\tilde{X};\tilde{Y})$, and semantic rate-distortion function $R_s(D)=\min_{p(\hat{x}|x):\mathbb{E}d_s(\tilde{x},\hat{\tilde{x}})\leq D}I_s(\tilde{X};\hat{\tilde{X}})$. Furthermore, we prove three coding theorems of SIT, that is, the semantic source coding theorem, semantic channel coding theorem, and semantic rate-distortion coding theorem. We find that the limits of information theory are extended by using synonymous mapping, that is, $H_s(\tilde{U})\leq H(U)$, $C_s\geq C$ and $R_s(D)\leq R(D)$. All these works composite the basis of semantic information theory. In summary, the theoretic framework proposed in this paper is a natural extension of classic information theory and may reveal great performance potential for future communication.

翻译：语义通信为未来通信开辟了新方向。本文旨在建立语义信息理论的系统性框架。首先，我们提出语义通信模型并定义同义映射，以揭示语义信息与语法信息之间的关键关系。基于这一核心概念，我们引入语义信息的度量，如语义熵$H_s(\tilde{U})$、上/下语义互信息$I^s(\tilde{X};\tilde{Y})$ $(I_s(\tilde{X};\tilde{Y}))$、语义容量$C_s=\max_{p(x)}I^s(\tilde{X};\tilde{Y})$和语义率失真函数$R_s(D)=\min_{p(\hat{x}|x):\mathbb{E}d_s(\tilde{x},\hat{\tilde{x}})\leq D}I_s(\tilde{X};\hat{\tilde{X}})$。进一步，我们证明了语义信息理论的三个编码定理，即语义信源编码定理、语义信道编码定理和语义率失真编码定理。我们发现，通过同义映射，信息论的极限得到了扩展，即$H_s(\tilde{U})\leq H(U)$，$C_s\geq C$，$R_s(D)\leq R(D)$。这些工作共同构成了语义信息理论的基础。总之，本文提出的理论框架是经典信息论的自然延伸，可能为未来通信揭示出巨大的性能潜力。