How to handle epsilon production in CLR(1) parsing?

Question

Given the following grammar:

1. S -> A
2. A -> AB
3. A -> ϵ
4. B -> aB
5. B -> b

Check whether it is CLR(1) or not.

I have drawn the canonical itemsets as well as the parse table. But I'm not sure whether it is actually correct or not. I have tried searching for some similar examples that have epsilon productions but could not find any. Could someone please help me out?

Answer 1

The initial state of your parser is I0, and $ is the only lookahead-terminal for which I0 has an action. So if the input is anything other than the empty string, your parser will raise an error. But the grammar has non-empty strings in its language. So this can't be a correct parser for the grammar.

I think your main mistake is in propagating lookaheads. For example, when you close A -> .AB, $, you need to add A -> .whatever items with lookaheads equal to each terminal in FIRST(B$), and you haven't done that.