LL(1) Parsing Table Error

Question

I start with having this grammar

E -> E + T | T
T -> id | id[] | id[X]
X -> E, E | E

It's obvious that there's left recursion in the first rule, so I remove it to get the following

E  -> TE'
E' -> +TE' | epsilon
T  -> id | id[] | id[X]
X  -> E, E | E

From here I produce the FIRST and FOLLOW sets (id is a single terminal).

FIRST(TE') = FIRST(E, E) = FIRST(E) = {id}
FIRST(+TE') = {+}
FIRST(epsilon = {epsilon}
FIRST(id) = FIRST(id[]) = FIRST(id[X]) = {id}

FOLLOW(E) = {$}
FOLLOW(E') = {$}
FOLLOW(T) = {$}
FOLLOW(X) = {]}

But when I try to construct the parse table, I end up with multiple values for a couple of cells.

    |  id               |  +     | [  |  ]  |  ,  |  $  |
----+-------------------+--------+----------+-----+-----+
E   |  TE'              |        |    |     |     |     |
E'  |                   |  +TE'  |    |     |     |  ε  |
T   |id or id[] or id[X]|        |    |     |     |     |
X   |E, E or E          |        |    |     |     |     |

Am I missing something? How do rectify this so that I can parse id + id[id + id, id[]] using this grammar?

Answer 1

Your grammar, even after removing left recursion, is not LL(1). These productions are problematic:

T -> id | id[] | id[X]
X -> E , E | E

Try to apply the following transformation:

A -> ab_1 | ... | ab_n  ~~~~~~> A  -> aA'
                                A' -> b_1 | ... | b_n

In your case it is:

T  -> id T'
T' -> [] | [X] | epsilon

X  -> EX'
X' -> , E | epsilon

But T' still needs to apply the transformation one more time:

T  -> id T'
T' -> [ T" | epsilon
T" -> ] | X ]

Full grammar:

E  -> TE'
E' -> +TE' | epsilon
T  -> id T'
T' -> [ T" | epsilon
T" -> ] | X ]
X  -> EX'
X' -> , E | epsilon

Moreover, you should do the steps in the following way:

1) Compute FIRST for all nonterminals.

2) Compute FOLLOW for all nonterminals.

3) Compute SELECT for all productions. Keep in mind that SELECT sets for one nonterminal should be disjoint.