  Continuous-time Markov decision processes (CTMDPs) are canonical models to
express sequential decision-making under dense-time and stochastic
environments. When the stochastic evolution of the environment is only
available via sampling, model-free reinforcement learning (RL) is the
algorithm-of-choice to compute optimal decision sequence. RL, on the other
hand, requires the learning objective to be encoded as scalar reward signals.
Since doing such translations manually is both tedious and error-prone, a
number of techniques have been proposed to translate high-level objectives
(expressed in logic or automata formalism) to scalar rewards for discrete-time
Markov decision processes (MDPs). Unfortunately, no automatic translation
exists for CTMDPs.
  We consider CTMDP environments against the learning objectives expressed as
omega-regular languages. Omega-regular languages generalize regular languages
to infinite-horizon specifications and can express properties given in popular
linear-time logic LTL. To accommodate the dense-time nature of CTMDPs, we
consider two different semantics of omega-regular objectives: 1) satisfaction
semantics where the goal of the learner is to maximize the probability of
spending positive time in the good states, and 2) expectation semantics where
the goal of the learner is to optimize the long-run expected average time spent
in the ``good states" of the automaton. We present an approach enabling correct
translation to scalar reward signals that can be readily used by off-the-shelf
RL algorithms for CTMDPs. We demonstrate the effectiveness of the proposed
algorithms by evaluating it on some popular CTMDP benchmarks with omega-regular
objectives.
