Problem $3.[10$ points$]$

An alternative to the language acceptance is to require the stack to be empty when the end of the input string is reached. Formaly, an NPDA $M$ is said to accept the language $N(M)$ by empty stack if

where $p$ is any element in $Q.$ Show that this notation is effectively equivalent to Definition in the sense that for any NPDA $M,$ there exists an NPDA hat$(M)$ such that $L(M)=N(hat(M)),$ and vice versa. $q,$

$q,$