A run-time monitor shares computational resources, such as memory and CPU time, with the target program. Furthermore, heavy computation performed by a monitor for checking target program's execution with respect to requirement properties can be a bottleneck to the target program's execution. Therefore, computational characteristics of run-time monitoring cause a significant impact on the target program's execution. We investigate computational issues on run-time monitoring. The first issue is the power of run-time monitoring. In other words, we study the class of properties run-time monitoring can evaluate. The second issue is computational complexity of evaluating properties written in process algebraic language. Third, we discuss sound abstraction of the target program's execution, which does not change the result of property evaluation. This abstraction can be used as a technique to reduce monitoring overhead. Theoretical understanding obtained from these issues affects the implementation of Java-MaC, a toolset for the run-time monitoring and checking of Java programs. Finally, we demonstrate the abstraction-based overhead reduction technique implemented in Java-MaC through a case study.