The paper considers stabilization under communication errors and limited data rate by means of realistic controllers with bounded (as time progresses) computational powers. Discrete-time partially observed noisy linear systems are studied for which the sensor signals are communicated to the controller over a finite capacity stochastic digital link. Addressed is stabilization in probability. It is shown that stability cannot typically be achieved by means of a finite memory decoder-controller so far as the boundary of the corresponding stabilizability domain is given by the zero-error capacity of the channel, which is typically zero. At the same time, stability can be achieved with keeping the expected values of the consumed computational resources bounded: the boundary of the domain where stabilization can be ensured by a coder and decoder with bounded expected computational powers is given by the ordinary Shannon capacity of the noisy channel.