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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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)

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) ==> ()(())(/\)

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.

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