1. Top-Down PDA Sequence of steps to which string X= [][[][]] has to be accepted by NPDA NT(G) Grammar has productions s  [S]S|^

2. We compare the moves made by NT(G) in accepting this string be leftmost derivation of this string. S=> [S]S =>[ ] S => [ ][S]S => [ ][ [S] S ] S =>[ ] [[ ] S] S => [ ] [ [ ] [S] S] S  [ ] [ [ ] [ ] S] S  [ ] [ [ ] [ ] ] S => [ ] [ [ ] [ ] ]

3. Transition table for Top Down PDA NT(G) Move#StateInputStack symbolMove 1q0^Z0(q1, SZ0) 2q1^S(q1, [S]S), (q1, ^) 3q1[[(q1, ^) 4q1]](q1, ^) 5q2^Z0(q2, Z0)

4. To the right of the each move that replace a variable on stack, we show the corresponding step in the left most derivation. (q0, [ ] [ [ ] [ ] ], Z0 ) Steps are derived by using the moves in transition table |- (q1, [ ] [ [ ][ ] ], SZ0) S |- (q1, [ ] [ [ ] [ ] ], [S]SZ0) => [S]S |- (q1, ] [ [ ] [ ] ], S]SZ0) |- (q1, ] [ [ ] [ ] ], ]SZ0) |- (q1, [ [ ] [ ] ], SZ0) |- (q1, [ [ ] [ ] ], [S]SZ0) =>[ ] [S]S |- (q1, [ ] [ ] ], S]SZ0) |- (q1,, [ ] [ ] ], [S] S]SZ0) =>[ ] [ [S]S]S |- (q1, ] [ ] ], S] S]SZ0) |- (q1, ] [ ] ], ] S]SZ0) |- (q1, [ ] ], S]SZ0) |- (q1, [ ] ], [S]S]SZ0) => [ ] [ [ ] [S]S]S |- (q1, ] ], S]S]SZ0) |- (q1, ] ], ]S]SZ0)

5. |- (q1, ], S]SZ0) |- (q1, ], ]SZ0) |- (q1, ^, SZ0) |- (q1, ^, Z0) = > [ ] [ [ ] [ ] ] |- (q2, ^, Z0) These are the sequence of steps that string x to be accepted

6. Thank you