  In this paper, we consider the optimal stopping problem on semi-Markov
processes (SMPs) with finite horizon, and aim to establish the existence and
computation of optimal stopping times. To achieve the goal, we first develop
the main results of finite horizon semi-Markov decision processes (SMDPs) to
the case with additional terminal costs, introduce an explicit construction of
SMDPs, and prove the equivalence between the optimal stopping problems on SMPs
and SMDPs. Then, using the equivalence and the results on SMDPs developed here,
we not only show the existence of optimal stopping time of SMPs, but also
provide an algorithm for computing optimal stopping time on SMPs. Moreover, we
show that the optimal and "-optimal stopping time can be characterized by the
hitting time of some special sets, respectively.
