  In this paper, we investigated semantic communication for multi-task
processing using an information-theoretic approach. We introduced the concept
of a "semantic source", allowing multiple semantic interpretations from a
single observation. We formulated an end-to-end optimization problem taking
into account the communication channel, maximizing mutual information (infomax)
to design the semantic encoding and decoding process exploiting the statistical
relations between semantic variables. To solve the problem we perform
data-driven deep learning employing variational approximation techniques. Our
semantic encoder is divided into a common unit and multiple specific units to
facilitate cooperative multi-task processing. Simulation results demonstrate
the effectiveness of our proposed semantic source and system design when
statistical relationships exist, comparing cooperative task processing with
independent task processing. However, our findings highlight that cooperative
multi-tasking is not always beneficial, emphasizing the importance of
statistical relationships between tasks and indicating the need for further
investigation into the semantically processing of multiple tasks.
