Algorithm 2. Sat-set of a quantified LTL formula Eφ.

1: procedure Sat∃LTL(M, φ)

2:   $\mathcal{L}(\mathcal{M}\otimes \mathcal{A})$ ← GBA of φ

3:   $\mathcal{A}$ ← BA of Kripke model M(RG), with S0 = RS

4:   $\mathcal{M}$ ← synchronized product of $\mathcal{A}$ with $\mathcal{M}$.

5:   $\mathcal{A}$ is the set of the acceptance sets of $\mathcal{F}$

6:   $Z_0\leftarrow succ(Q_0^{\mathcal{M}\otimes \mathcal{A}},{\delta }^{\mathcal{M}\otimes \mathcal{A}})$

7:  M( $\mathcal{M}$) is the Kripke model of $\mathcal{A}$

8:  switch type( $\mathcal{A}$) do

9:   case weak:

10:    AS ← SATCTL(M( $\mathcal{A}$), EF EG F1)

11:   case terminal:

12:    AS ← SATCTL(M( $\mathcal{A}$), EF F1)

13:   case otherwise

14:    AS ← SATEFAIRG(M( $\mathcal{A}$), true, fair = $\mathcal{A}$):

15:  return map (AS ∩ Z0) over RS

16: end procedure