Consider the following grammar. $$S \rightarrow aSB \mid d$$ $$B \rightarrow b$$ The number of reduction steps taken by a bottom-up parser while accepting the string $aaadbbb$ is _______.

A bottom-up parser traces the rightmost derivation in reverse to reduce the input string to the start symbol $S$.

Let us construct the rightmost derivation of the string $aaadbbb$ from $S$:

1. $S \Rightarrow aSB$

2. $\Rightarrow aSb$ (using $B \rightarrow b$)

3. $\Rightarrow a(aSB)b = aaSBb$ (using $S \rightarrow aSB$)

4. $\Rightarrow aaSbb$ (using $B \rightarrow b$)

5. $\Rightarrow aa(aSB)bb = aaaS Bbb$ (using $S \rightarrow aSB$)

6. $\Rightarrow aaaSbbb$ (using $B \rightarrow b$)

7. $\Rightarrow aaa(d)bbb = aaadbbb$ (using $S \rightarrow d$)

The bottom-up parser performs these 7 derivation steps in reverse order as reductions:

• Step 1: $d \rightarrow S$, reducing $aaadbbb$ to $aaaSbbb$

• Step 2: $b \rightarrow B$, reducing $aaaSbbb$ to $aaaSBbb$

• Step 3: $aSB \rightarrow S$, reducing $aaaSBbb$ to $aaSbb$

• Step 4: $b \rightarrow B$, reducing $aaSbb$ to $aaSBb$

• Step 5: $aSB \rightarrow S$, reducing $aaSBb$ to $aSb$

• Step 6: $b \rightarrow B$, reducing $aSb$ to $aSB$

• Step 7: $aSB \rightarrow S$, reducing $aSB$ to $S$

Total number of reduction steps is 7.