Approximate Bisimulation Equivalence and Variable Refinement

In this paper, we consider an operator for refinement of variables to be used in the design of hybrid system. Variables on a given level of abstraction are replaced by more complicated processes on a lower level just like the function are called in the program. Then we established the equivalence, bisimulation equivalence and approximate bisimulation equivalence which are by polynomial flow event structures. These equivalence, bisimulation equivalence and approximate bisimulation equivalence are based on the common forms of their zeros. The example show that the equivalence, bisimulation equivalence and approximate bisimulation equivalence are preserved or not under the variables refinement, if the equivalence is not preserved precisely, then we can use the approximate methods to make them approximate equivalence. Lastly we show that our refinement has some nice properties.