Consider the Grammar:
S→A
A→$B$|id
B→B, A|A
If I0 = CLOSURE ({[S → .A]}) then, how many items be in the set for GOTO (I0,$)

Correct option is C

The GOTO function describes the transitions between different items in an LR(1) parser. For the given grammar, applying CLOSURE and using the GOTO function will lead to five items in the set.
· Production Rules:
· S → A: This means that the start symbol S can be derived from the non-terminal A.
· A → $B$ | id: A can either produce a symbol $B$ or the terminal id.
· B → B, A | A: B can either produce a comma-separated list of non-terminals B and A, or just A.
We will now look at the CLOSURE and GOTO functions as they pertain to this grammar.
Step-by-Step Solution:
1. Initial Closure: We start with the initial item, I₀ = CLOSURE ({[S → .A]}). This means we are considering the start symbol's production but haven’t consumed any of the right-hand side yet (that’s why the dot . appears before A). We apply the closure function to this item.
Closure expands any non-terminal after the dot, which in this case is A. So, we include the items:
· [A → .$B$] (We haven't consumed $B$ yet)
· [A → .id] (We haven't consumed id yet)
Now, we look at $B$, since it's after the dot in the first production of A → $B$. So, we add productions for $B$:
· [B → .B, A] (Haven’t consumed B, A yet)
· [B → .A] (Haven’t consumed A yet)
2. Goto (I₀, $): To compute GOTO (I₀, $), we look for items that have $ after the dot. From the closure we computed above, we have:
· [A → .$B$] is one such item, since $ is directly after the dot in this production.
Moving the dot past $, we get:
· [A → $B$ .]
So, the GOTO function leads us to this item, and there are 5 items in total in the closure and the GOTO.
Detailed Items in the Set:
1. [S → .A]
2. [A → .id]
3. [A → .$B$]
4. [B → .B, A]
5. [B → .A]
Information Booster:
1. LR(1) Parsing: A type of bottom-up parsing that uses lookahead tokens to decide the next action.
2. CLOSURE: A set of items that includes the initial item and all items derived from it by applying grammar rules. The CLOSURE function computes all possible production rules that can be applied starting from a particular item, adding new items whenever a non-terminal symbol appears after a dot.
3. GOTO Function: The GOTO function moves the dot past a particular symbol (in this case, the terminal $) and computes the next set of items based on that move.
Additional Knowledge:
· Items in GOTO depend on the productions derived from a non-terminal symbol.
· Parsing actions are based on item sets and how grammar productions are closed.
· The GOTO function leads to transitions between states in LR parsing.
· In this case, we use LR(1) parsing, which considers one token of lookahead to guide the parsing decisions.