  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.
