$($a$)(10$ points$)$ Consider the language over the alphabet $\{0,1\}$ :

$A=\{1^i(01{)}^j{0}^i|i,j0\}$

i$.$ Draw the state diagram of a PDA that recognizes the language $A.$

ii$.$ Provide the formal definition of a CFG that describes the language $A.$

$($b$)(12$ points$)$ Consider the PDA:

$Q=\{q0,q1,q2,q3\}$

$=\{0,1\}$

$=\{$$$,0\}$

$(q0,1,)=\{(q1,$$$)\}$

$(q1,0,)=\{(q1,0)\}$

$(q1,1,)=\{(q2,)\}$

$(q2,0,0)=\{(q2,)\}$

$(q2,1,$$$)=\{(q3,)\}$

$(r,a,b)=\frac{O}{?}$

$q=q0$

$F=\{q3\}$

otherwise

i$.$ Draw the state diagram for the PDA.

ii$.$ Describe the language recognized by this PDA using set builder notation.

iii. Provide the formal definition of a CFG that describes this language.