Consider the following augmented grammar, which is to be parsed with a SLR parser. The set of terminals is $\{a,b,c,d,\#,@\}$ $$\begin{aligned} S^{\prime} &\rightarrow S S &\rightarrow SS \mid Aa \mid bAc \mid Bc \mid bBa A &\rightarrow d\# B &\rightarrow @ \end{aligned}$$ Let $I_0 = CLOSURE(\{S^{\prime} \rightarrow \bullet S\})$. The number of items in the set $GOTO(I_0, S)$ is _______

### Step 1: Find the initial state $I_0 = \text{CLOSURE}(\{S^{\prime} \rightarrow \bullet S\})$ Starting with $S^{\prime} \rightarrow \bullet S$:

• Add all $S$-productions with the dot at the beginning:

  • $S \rightarrow \bullet SS$

  • $S \rightarrow \bullet Aa$

  • $S \rightarrow \bullet bAc$

  • $S \rightarrow \bullet Bc$

  • $S \rightarrow \bullet bBa$

• From $S \rightarrow \bullet Aa$, add $A$-productions:

  • $A \rightarrow \bullet d\#$

• From $S \rightarrow \bullet Bc$, add $B$-productions:

  • $B \rightarrow \bullet @$

### Step 2: Compute $\text{GOTO}(I_0, S)$ Find all items in $I_0$ where the dot is immediately before $S$, and advance the dot past $S$:

1. From $S^{\prime} \rightarrow \bullet S$, we get the kernel item: $S^{\prime} \rightarrow S \bullet$

2. From $S \rightarrow \bullet SS$, we get the kernel item: $S \rightarrow S \bullet S$

### Step 3: Compute the closure of this kernel set Kernel items:

1. $S^{\prime} \rightarrow S \bullet$

2. $S \rightarrow S \bullet S$

Since item 2 has the dot before non-terminal $S$, add all $S$-productions with the dot at the beginning:

3. $S \rightarrow \bullet SS$

4. $S \rightarrow \bullet Aa$

5. $S \rightarrow \bullet bAc$

6. $S \rightarrow \bullet Bc$

7. $S \rightarrow \bullet bBa$

From item 4 ($S \rightarrow \bullet Aa$), the dot is before $A$, so add $A$-productions:

8. $A \rightarrow \bullet d\#$

From item 6 ($S \rightarrow \bullet Bc$), the dot is before $B$, so add $B$-productions:

9. $B \rightarrow \bullet @$

No other non-terminals have a dot immediately before them in newly added items.

### Step 4: Count the items The total number of items in the set $\text{GOTO}(I_0, S)$ is 9.