  Semantic communication has attracted significant interest recently due to its
capability to meet the fast growing demand on user-defined and human-oriented
communication services such as holographic communications, eXtended reality
(XR), and human-to-machine interactions. Unfortunately, recent study suggests
that the traditional Shannon information theory, focusing mainly on delivering
semantic-agnostic symbols, will not be sufficient to investigate the
semantic-level perceptual quality of the recovered messages at the receiver. In
this paper, we study the achievable data rate of semantic communication under
the symbol distortion and semantic perception constraints. Motivated by the
fact that the semantic information generally involves rich intrinsic knowledge
that cannot always be directly observed by the encoder, we consider a semantic
information source that can only be indirectly sensed by the encoder. Both
encoder and decoder can access to various types of side information that may be
closely related to the user's communication preference. We derive the
achievable region that characterizes the tradeoff among the data rate, symbol
distortion, and semantic perception, which is then theoretically proved to be
achievable by a stochastic coding scheme. We derive a closed-form achievable
rate for binary semantic information source under any given distortion and
perception constraints. We observe that there exists cases that the receiver
can directly infer the semantic information source satisfying certain
distortion and perception constraints without requiring any data communication
from the transmitter. Experimental results based on the image semantic source
signal have been presented to verify our theoretical observations.
