  We investigate the problem of synthesizing optimal control policies for
Markov decision processes (MDPs) with both qualitative and quantitative
objectives. Specifically, our goal is to achieve a given linear temporal logic
(LTL) task with probability one, while maximizing the \emph{entropy rate} of
the system. The notion of entropy rate characterizes the long-run average
(un)predictability of a stochastic process. Such an optimal policy is of our
interest, in particular, from the security point of view, as it not only
ensures the completion of tasks, but also maximizes the unpredictability of the
system. However, existing works only focus on maximizing the total entropy
which may diverge to infinity for infinite horizon. In this paper, we provide a
complete solution to the entropy rate maximization problem under LTL
constraints. Specifically, we first present an algorithm for synthesizing
entropy rate maximizing policies for communicating MDPs. Then based on a new
state classification method, we show the entropy rate maximization problem
under LTL task can be effectively solved in polynomial-time. We illustrate the
proposed algorithm based on two case studies of robot task planning scenario.
