Computing the FOLLOW sets

Question

My task is to calculate FIRST and FOLLOW sets for the following grammar:

P ::= S CS .
S ::= ( int , int )
CS ::= C CS | epsilon
C ::= left int | right int | low C

I got the following first sets:

FIRST(S) = {'('}
FIRST(C) = {left,right,low}
FIRST(CS) = {left,right,low,epsilon}
FIRST(P) = FIRST(S) = {'('}

For the following sets I calculated:

FOLLOW(P) = $ (or empty)
FOLLOW(C) = {left,right,low,'.'}
FOLLOW(CS) = {'.'}
FOLLOW(S) = {left,right,low}

I tried out my solution using first and follow sets generator and what I got with FOLLOW(S) was: FOLLOW(S) = {'.', left, right, low}. Is generator's solution correct and why? I calculated my solution using formula: FOLLOW(S) = FIRST({left,right,low} concat FOLLOW(P)) = {left, right, low}. Can someone please explain me why my/ generator's solution is not correct and check whether I got everything else right? I also want to know why I don't have int in any first or follow set and if this will be okay with building parser later anyway. Thank you

Answer 1

When you compute FOLLOW sets you have to be careful with​ empty productions.

In this case, CS has an empty production, which means that S might be followed by a . in P → S CS .. Similarly, the C in C CS might be at the end of the production, so C could also be followed by a .

int can only appear after a left or right token. It can never appear at the beginning of a nom-terminal nor immediately following a non-terminal. So it is entirely expected that it not be in any FIRST or FOLLOW set.