Question

Consider the following augmented grammar, which is to be parsed with an SLR parser. The set of terminals is \(\{a, b, c, d, \#, @\}\). \[ S' \rightarrow S \quad S \rightarrow SS \, | \, Aa \, | \, bAc \, | \, BC \, | \, bBa \quad A \rightarrow d\# \quad B \rightarrow @ \] Let \(I_0 = \text{CLOSURE}(\{S' \rightarrow \cdot S\}\). The number of items in the set \(GOTO(I_0, S)\) is ............

Hint:

To compute the number of items in a GOTO set, always analyze the closure operation and include all applicable productions for the given grammar symbol.

Answer 1

Step 1: Analyze the given DFA diagram. 
The given DFA represents the items in the closure of \(I_0\) and the GOTO function for the symbol \(S\). 

Step 2: Identify the items in the closure of \(I_0\). 
The closure of \(I_0\) contains the following items: \(S \to \cdot S\) \(S \to \cdot SS\) \(S \to \cdot Aa\) \(S \to \cdot bAc\) \(S \to \cdot Bc\) \(S \to \cdot bAb\) \(A \to \cdot d\#\) \(B \to \cdot @\) 

Step 3: Apply the GOTO function on the symbol \(S\). 
When the GOTO function is applied to \(S\), the following transitions occur: \(S \to S\cdot\) \(S \to S\cdot S\) \(S \to SS\cdot\) \(S \to Aa\cdot\) \(S \to bAc\cdot\) \(S \to Bc\cdot\) \(S \to bAb\cdot\) \(A \to d\#\cdot\) \(B \to @\cdot\) 

Step 4: Count the total number of items. 
From the diagram, the GOTO function \(GOTO(\text{closure}(I_0), S)\) results in 9 items: \[ S \to S\cdot, \, S \to S\cdot S, \, S \to SS\cdot, \, S \to Aa\cdot, \, S \to bAc\cdot, \, S \to Bc\cdot, \, S \to bAb\cdot, \, A \to d\#\cdot, \, B \to @\cdot. \] 

Final Answer: From the above DFA, we can conclude that: \[ \boxed{\text{GOTO}(\text{closure}(I_0), S) \text{ contains 9 items.}} \]