Given the grammar G = (\{S\}, \{a, b\}, P, S) P: S \rightarrow aSb | ab Derive the string aabb using leftmost derivation, rightmost derivation and als

Question

Given the grammar
$G = (\{S\}, \{a, b\}, P, S)$
$P: S \rightarrow aSb | ab$

Derive the string **aabb** using leftmost derivation, rightmost derivation and also draw the parse tree.

Accepted Answer

## Answer

**Grammar:** $G = (\{S\}, \{a,b\}, P, S)$ with $P: S ightarrow aSb \mid ab$

### 1. Leftmost Derivation (LMD)
At each step, replace the **leftmost non-terminal**.

$$
\begin{aligned}
S &\Rightarrow_{	ext{lm}} aSb & 	ext{(using $S ightarrow aSb$)} \
  &\Rightarrow_{	ext{lm}} aaSbb & 	ext{(using $S ightarrow aSb$ on leftmost $S$)} \
  &\Rightarrow_{	ext{lm}} aabbb & 	ext{(using $S ightarrow ab$ on leftmost $S$)} \
  &= \mathbf{aabb}
\end{aligned}
$$

**LMD Steps:** $S \xRightarrow{lm} aSb \xRightarrow{lm} aaSbb \xRightarrow{lm} aabbb = aabb$

### 2. Rightmost Derivation (RMD)
At each step, replace the **rightmost non-terminal**.

$$
\begin{aligned}
S &\Rightarrow_{	ext{rm}} aSb & 	ext{(using $S ightarrow aSb$)} \
  &\Rightarrow_{	ext{rm}} aaSbb & 	ext{(using $S ightarrow aSb$ on rightmost $S$)} \
  &\Rightarrow_{	ext{rm}} aabbb & 	ext{(using $S ightarrow ab$ on rightmost $S$)} \
  &= \mathbf{aabb}
\end{aligned}
$$

**RMD Steps:** $S \xRightarrow{rm} aSb \xRightarrow{rm} aaSbb \xRightarrow{rm} aabbb = aabb$

*Note: Since this grammar is unambiguous and has only one non-terminal, LMD and RMD coincide in the sequence of sentential forms.*

### 3. Parse Tree (Derivation Tree)

```
            S
           /|\
          a S b
            |
            S
           /|\
          a S b
            |
            ab
```

**Text/Mermaid representation:**

```mermaid
graph TD
    S1[S]
    S1 --> a1[a]
    S1 --> S2[S]
    S1 --> b1[b]
    S2 --> a2[a]
    S2 --> S3[S]
    S2 --> b2[b]
    S3 --> a3[a]
    S3 --> b3[b]
```

**Tree structure:**
- **Root:** $S$
- **Internal nodes:** $S$ (non-terminals)
- **Leaves (yield):** $a, a, b, b$ (reading left to right = `aabb`)
- **Height:** 3 (excluding root)

### Verification
- String generated: `aabb` ✓
- Language of grammar: $L(G) = \{a^n b^n \mid n \ge 1\}$
- `aabb` corresponds to $n=2$ ✓

### Final Answer
**LMD:** $S \xRightarrow{lm} aSb \xRightarrow{lm} aaSbb \xRightarrow{lm} aabbb$  
**RMD:** $S \xRightarrow{rm} aSb \xRightarrow{rm} aaSbb \xRightarrow{rm} aabbb$  
**Parse Tree:** As shown above (yield = `aabb`).

Answer

1. Leftmost Derivation (LMD)

2. Rightmost Derivation (RMD)

3. Parse Tree (Derivation Tree)

Verification

Final Answer

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