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Given two states i), u G Vp, we say that u is reachable from v if there is an execution fragment with V as the first state and u as the last state. The state u is reachable in P if there is an 28 Yan Zhang, Xiaofeng Yu, Tian Zhang, Xuandong Li, Giioliang Zheng Let Fp denote the set of all execution fragments in lA P . For every r] e Fp, write the first state of r] as first{ri), the last state of rj as last{ri) and the set of all states of rj as V{rj). Definition 2 (interface automata product). Two lAs P and Q are composable if Aff^AQ = 0, A^nAp = 0, A^pnA'q = $ and A$nA'^ = 0.

However, /2^ and O^ are finite sets and the lengths of all simple runs and simple loops are finite. Furthermore, cpc {^R) and tpc {€>R) are finite sets. Additionally, we also notice that it is impossible to eliminate the undesiied behavior represented by £> G £~ from R® E hy not constructing the corresponding step in E for any step in R "after" the proper occurrence of g. A step T "after" a proper occurrence rj means that head{T) = last{rj) or headir) is reachable from last{r}). Suppose that C is a proper occurrence of £> e £~ in a simple run a of lA i?.

Accordingly, 4>£,^ (SR) , (pc+i^R) ^^^ the lengths of some runs in them may be infinite. Thus, it is unfeasible to traverse all runs in ^ £ - {SR) directly. For getting a feasible approach, we introduce the concepts of the simple run and simple loop. 32 Yan Zhang, Xiaofeng Yu, Tian Zhang, Xuandong Li, Giioliang Zheng Given an lA i? and a set C of action sequences, a run a = woaoflai • • • an-iVn of i? is a simple run when it satisfies the following conditions: 1. if a e