  This paper studies the fundamental limit of semantic communications over the
discrete memoryless channel. We consider the scenario to send a semantic source
consisting of an observation state and its corresponding semantic state, both
of which are recovered at the receiver. To derive the performance limitation,
we adopt the semantic rate-distortion function (SRDF) to study the relationship
among the minimum compression rate, observation distortion, semantic
distortion, and channel capacity. For the case with unknown semantic source
distribution, while only a set of the source samples is available, we propose a
neural-network-based method by leveraging the generative networks to learn the
semantic source distribution. Furthermore, for a special case where the
semantic state is a deterministic function of the observation, we design a
cascade neural network to estimate the SRDF. For the case with perfectly known
semantic source distribution, we propose a general Blahut-Arimoto algorithm to
effectively compute the SRDF. Finally, experimental results validate our
proposed algorithms for the scenarios with ideal Gaussian semantic source and
some practical datasets.
