  As conventional communication systems based on classic information theory
have closely approached the limits of Shannon channel capacity, semantic
communication has been recognized as a key enabling technology for the further
improvement of communication performance. However, it is still unsettled on how
to represent semantic information and characterise the theoretical limits. In
this paper, we consider a semantic source which consists of a set of correlated
random variables whose joint probabilistic distribution can be described by a
Bayesian network. Then we give the information-theoretic limit on the lossless
compression of the semantic source and introduce a low complexity encoding
method by exploiting the conditional independence. We further characterise the
limits on lossy compression of the semantic source and the corresponding upper
and lower bounds of the rate-distortion function. We also investigate the lossy
compression of the semantic source with side information at both the encoder
and decoder, and obtain the rate distortion function. We prove that the optimal
code of the semantic source is the combination of the optimal codes of each
conditional independent set given the side information.
