Definition 8: Given an EFSM M = (S, T) over X, Y, R, V, D Rx, D V and a parameterized input x, select a set P x = {p x1, p x2 … p xl } such that for each state s, each configuration (s, v) and each outgoing transition from state s under the parameterized input x with the predicate P, there exists p x ∈ P x such that (v, p x ) satisfies P, written (v, p x ) | = P.
The (parameterized) input projection α of a trace σ is α = σ↓X = (x 1, p x1 )….
In addition, in fact, there is a need to determine the parameterized input sequence (with input parameters values) that can be used to take the distinguishing machine into a fail state.
For EFSM M, we call the parameterized inputs (x, p x ), p x ∈ P x, selected (parameterized) inputs and other (parameterized) inputs (x, p x ), p x ∈ D Rx P x, are non-selected (parameterized) inputs.
In fact, in this case, in the abstraction function (Definition 9), for a given configuration and parameterized input, there could be many enabled transitions at a state with the same parameterized input and the abstraction function takes into consideration all such transitions.
Another transition has the guard odd z) which equals 1 iff z is odd; otherwise, the predicate equals 0. Assume that (1, z = 0) is a current configuration of the EFSM and the machine receives a parameterized input a(i), then the machine checks the predicates of outgoing transitions from state 1 that are satisfied for the current configuration under the input a with parameter value a.i.
Correspondingly, from each subset, we select one value of a parameterized input and add the value to the set P a.
Definition 4: Given input x and the possibly empty set D Rx of input parameter valuations, a parameterized input (or an input) is a tuple (x, p x ), where p x ∈ D Rx. Recall in case R x = ∅, D Rx = , and the input (x, ⊥) is denoted as x.
A parameterized input (x, p x ) is selected in such a way that the average number of outgoing transitions under (x, p x ) at the m abstract states is minimized.
Hereafter, for a parameterized input say a, we let a(0) or (a, 0) denote the fact that the machine receives the input a with the parameter value a.i = 0.
An FSM abstraction derived at Step 2 over the non-parameterized input b does not distinguish the sets C′ and C′′.
