1. Balanced Parentheses G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\}

2. One Derivation Tree for ()(())() G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S

3. First Expansion Phase G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS

4. Second Expansion Phase G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS SS()S

5. Third Expansion Phase G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S

6. Fourth Expansion Phase G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S

7. Final Expansion Phase G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S

8. Leftmost/Rightmost Derivations G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost:S ==> SS Leftmost: S ==> SS

9. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS Leftmost: S ==> SS

10. Second Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) Leftmost: S ==> SS ==> SSS

11. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) Leftmost: S ==> SS ==> SSS

12. Third Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S(/\) Leftmost: S ==> SS ==> SSS ==> (S)SS

13. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() Leftmost: S ==> SS ==> SSS ==> (S)SS

14. Fourth Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> (/\)SS

15. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS

16. Fifth Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S

17. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S

18. Sixth Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S

19. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S

20. Seventh Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() ==> S((/\))() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S ==> ()((/\))S

21. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() ==> S(())() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S ==> ()(())S

22. Eighth Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() ==> S(())() ==> (S)(())() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S ==> ()(())S ==> ()(())(S)

23. Leftmost/Rightmost Variables G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() ==> S(())() ==> (S)(())() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S ==> ()(())S ==> ()(())(S)

24. Final Derivation Steps G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S Rightmost: S ==> SS ==> S(S) ==> S() ==> SS() ==> S(S)() ==> S((S))() ==> S(())() ==> (S)(())() ==> (/\)(())() Leftmost: S ==> SS ==> SSS ==> (S)SS ==> ()SS ==> ()(S)S ==> ()((S))S ==> ()(())S ==> ()(())(S) ==> ()(())(/\)

25. Ambiguous Grammar G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S SS ()S /\ S ()S S ()S ()S S S ()S S S ()S S ()S ()S We’ll show a second derivation tree for the string ()(())(). This proves the grammar G is ambiguous.

26. Unambiguous Grammar G = (V, , S, P) V = {S}  = {(,)} Start variable is S P = { S --> (S) | SS | /\} S’ T () /\() G’ = (V’, , S’, P’) V = {S’,T}  = {(,)} Start variable is S’ P = { S’ --> S’T | /\ T --> (S’)} S’ /\ T ()S’ /\ S’ /\ T ()S’ /\ T