At section 8 of the paper On the Synthesis of a Reactive Module, the authors state the following theorem:
Proposition 2 (Non-Emptiness Condition)
Let $A = (Q, \delta, Q_0, \Omega)$ be a tree automaton with $\Omega = \{(L_i, U_i) | 1 \leq i \leq m\}$ and $|Q|=n$. Then, $T_\omega(A) \neq \emptyset$ if and only if for some finite tree $E$ and a run $r=(\nu, E')$ of $A$ on $E$, for every path $\pi$ in $E$, there exist an integer $i$, $1 \leq i \leq m$, and strings $x_0, x_1, \dots, x_{n+1}$, such that the following hold:
- $x_0 \leq x_1 < \dots < x_n < x_{n+1} \in \pi \cap Ft(E)$.
- For all $l$, $0 < l \leq n+1$, $\nu(x_l) \in U_i$.
- For all $j$, $1 \leq j \leq m$, $\nu[x_1,x_{n+1}] \cap L_j \neq \emptyset \Rightarrow \nu[x_0, x_1] \cap L_j \neq \emptyset$.
- $\nu[x_0, x_{n+1}] \cap L_i = \emptyset$.
They do not provide a proof, but add that:
The following proposition and its proof essentially consist of delicate modifications of the results reported in [HR72] as theorem 1 and its proof.
Since I could not grasp the intuition beyond the third condition, I tried to prove the theorem by myself, with the proof of the mentioned theorem in mind. Proving the 'only if' direction was easy, but I can not come up with a proof for the 'if'.
The idea in [HR72] follows these steps:
- build a mapping $\eta$ from an infinite tree $T$ to the finite tree $E$
- define a run $\overline{r}$ of $A$ on $T$ by means of $\eta$ and the run $r = (\nu, E')$
- show that $\overline{r}$ is an accepting run for $T$, thus $T \in T_\omega(A)$
In [HR72] the conditions are different from those stated by Pnueli and Rosner and I am not able to adapt that proof to this alternative Non-Emptiness condition.
How can I adapt the proof reported in [HR72], in order to prove Proposition 2?
