Write derivations for four strings of various lengths and describe what is
the language generated, L(G) by the following grammar
V = {S, B} T = {a, b}, S is the start nonterminal
S → aSaa | B
B → bB | lamda

### Grammar Components

Given grammar:
- **Variables (Non-terminals)**: \( V = \{S, B\} \)
- **Terminals**: \( T = \{a, b\} \)
- **Start Symbol**: \( S \)
- **Production Rules**:
  1. \( S \rightarrow aSaa \) 
  2. \( S \rightarrow B \)
  3. \( B \rightarrow bB \) 
  4. \( B \rightarrow \lambda \) (where \( \lambda \) denotes the empty string)

### Derivations for Four Strings

1. **String: "baa"**
   - \( S \rightarrow B \)
   - \( B \rightarrow bB \)
   - \( B \rightarrow \lambda \)
   - Resulting string: "baa"

2. **String: "bbaaa"**
   - \( S \rightarrow B \)
   - \( B \rightarrow bB \)
   - \( B \rightarrow bB \)
   - \( B \rightarrow \lambda \)
   - Resulting string: "bbaaa"

3. **String: "abbbaaabbaa"**
   - \( S \rightarrow aSaa \)
   - \( S \rightarrow aBaaa \)
   - \( B \rightarrow bB \)
   - \( B \rightarrow bbB \)
   - \( B \rightarrow \lambda \)
   - Resulting string: "abbbaaabbaa"

4. **String: "aaabaa"**
   - \( S \rightarrow aSaa \)
   - \( S \rightarrow aBaaa \)
   - \( B \rightarrow bB \)
   - \( B \rightarrow \lambda \)
   - Resulting string: "aaabaa"

### Language Generated by the Grammar \( L(G) \)

The language \( L(G) \) generated by this grammar consists of strings that:
- Begin with zero or more 'b's.
- Contain exactly one segment of 'a' sequences which are always of even length (since they are in pairs due to the production \( S \rightarrow aSaa \)).
- These 'a' sequences can be preceded by any number of 'b's (due to the rules involving \( B \)).
- The string can also just be a sequence of 'b's, or a sequence of 'b's followed by an even number of 'a's.

### Summary of \( L(G) \)
The language \( L(G) \) consists of all strings that can be expressed as \( b^n a^m b^k \) where \( n, k \geq 0 \) and \( m \) is even.

---

Would you like more details or have any questions?

Here are some related questions:

1. How does the structure of production rules affect the language generated?
2. What is the significance of the empty string in a grammar?
3. Can this grammar generate all possible strings of 'a's and 'b's?
4. How would the language change if the production rule \( S \rightarrow aSaa \) was replaced with \( S \rightarrow aSa \)?
5. What would the derivation look like for a string like "baaa"?

**Tip:** Understanding how production rules build up strings step-by-step is key to mastering context-free grammars.

Explore derivations and language generation by a context-free grammar with detailed examples. Understand how production rules shape strings of 'a's and 'b's, and delve into the language generated by this grammar.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation