CS606 Assignment 3 Solution and Discussion

zareen

The FIRST of a sentential form is the set of terminal symbols that lead any sentential from derived from the very first sentential form. In this particular case X and Y only derive the empty string and as a result the empty string is the FIRST set of both non-terminal symbols X and Y. The FIRST of S, however, includes “a” as in the first production once can derive a sentential form that starts with an “a” given that X can be replaced by the empty string. X similar reasoning allows you to include “b” in the FIRST(S).
summary: FIRST(X) = {e}, FIRST(Y) = {e}, FIRST(S) = {a, b}

The FOLLOW set of a non-terminal symbol is the set of terminals that can appear after that non-terminal symbol in any sentential form derived from the grammar’s start symbol. By definition the follow of the start symbol will automatically include the $ sign which represents the end of the input string. In this particular case by looking at the productions of S one can determine right away that the follow of X includes the terminal “a” and “b” and that the FOLLOW of Y includes the terminal “b”. Given that the non-terminal S does not appear in any productions, not even in its own productions, the FOLLOW of S is only $.
summary: FOLLOW(S) = {$}, FOLLOW(X) = {a, b}, FOLLOW(Y) = {b}.

b) YES, because the intersection of the FIRST for every non-terminal symbol in empty. This leads to the parsing table for this LL method as indicated below. As there is no conflict in entry then grammar is clearly LL (1).
[23/01, 21:33] Waqas Ahmad: a b $
S S→XaXb S → Yb
A X→ε X→ε
B Y→ε