Question 2
Notation:
• For an LTL formula $\varphi$ over AP, we denote the infinite words that satisfy $\varphi$ by $L(\varphi) = \{\pi \in (2^{AP})\omega : \pi \models \varphi\}$.
• Let AP' such that AP $\subseteq$ AP'. For $\pi = \pi_1, \pi_2, ... \in (2^{AP'})\omega$ denote the projection of $\pi$ on the subset AP of AP' by
$\pi|_{AP} = \pi_1|_{AP}, \pi_2|_{AP}, ... \in (2^{AP})\omega$.
Recall that we saw in class that LTL $\subset$ NBW; that is, there is an NBW A over an alphabet $2^{AP}$ such that there is no LTL
formula $\varphi$ with $L(A) = L(\varphi)$. In this question, you will suggest a remedy for this limitation of LTL by extending AP.
Given an NBW A = ($2^{AP}$, Q, $\delta$, $Q_0$, $\alpha$), construct an LTL formula $\varphi_A$ over AP' = AP $\cup$ Q such that
• For $\pi' \in L(\varphi_A)$, we have $\pi'|_{AP} \in L(A)$.
• For $\pi \in L(A)$, there is $\pi' \in L(\varphi_A)$ such that $\pi = \pi'|_{AP}$.
(Hint: Construct $\varphi_A$ so that for $\pi \in L(\varphi_A)$ the word $\pi|_{Q}$ is an accepting run of A on $\pi|_{AP}$)
