When using the pumping lemma with length to prove that the language L = {anban, n>0} is nonregular, the following approach is taken.

Assume L is regular.
Then there exists an FA with k states which accepts L.
We choose any word w which is a word in L.
Which one of the following would be an appropriate word w?

1.
ambam for m>0

2.
akbak

3.
a100ba100

4.
akbkak

To prove that the language \( L = \{ a^n b a^n \mid n > 0 \} \) is nonregular using the pumping lemma, we need to choose an appropriate word \( w \) that is a member of \( L \) and has a length of at least \( p \) (the pumping length). This word should be carefully selected so that it can be shown that no matter how we divide it into three parts \( xyz \) (with conditions from the pumping lemma), the word \( x y^i z \) for \( i \neq 1 \) will not always be in \( L \).

The pumping lemma for regular languages states:

1. For any regular language \( L \), there exists an integer \( p \) (the pumping length) such that any string \( w \in L \) with \( |w| \geq p \) can be written as \( w = xyz \), where:
   - \( |xy| \leq p \)
   - \( |y| > 0 \)
   - For any \( i \geq 0 \), the string \( xy^i z \) is in \( L \)

Given the options:

1. \( a^m b a^m \) for \( m > 0 \)
2. \( a^k b a^k \)
3. \( a^{100} b a^{100} \)
4. \( a^k b k^a \)

We need to choose a word that is clearly in \( L \) and can be used to show that \( L \) cannot satisfy the pumping lemma conditions.

Let's examine each option:

1. \( a^m b a^m \) for \( m > 0 \):
   - This word is in the form \( a^n b a^n \) with \( n = m \). It is suitable because it fits the structure required by \( L \).

2. \( a^k b a^k \):
   - This word is also in the form \( a^n b a^n \) with \( n = k \). It fits the required structure.

3. \( a^{100} b a^{100} \):
   - This word is in the form \( a^n b a^n \) with \( n = 100 \). It fits the required structure and is appropriate since it has a definite length greater than any possible pumping length \( p \).

4. \( a^k b k^a \):
   - This word does not fit the form \( a^n b a^n \) because the last part is \( k^a \) instead of \( a^k \). This word is not in \( L \).

Given these analyses, the appropriate word \( w \) to use is:

**3. \( a^{100} b a^{100} \)**

### Explanation for Choice

The word \( w = a^{100} b a^{100} \) is in \( L \) and has a length that is certainly greater than any typical pumping length \( p \). Using this word, we can apply the pumping lemma to show that no matter how \( w \) is divided into \( xyz \) with the conditions \( |xy| \leq p \) and \( |y| > 0 \), pumping \( y \) (repeating it zero or more times) will result in a string that does not remain in the form \( a^n b a^n \).

For example, if \( y \) consists solely of \( a \)s from the first block of \( a \)s, then pumping \( y \) would result in more or fewer \( a \)s before the \( b \), disrupting the balance required by the structure \( a^n b a^n \).

### Do you want details or have any questions?

### Related Questions

1. What is the pumping lemma for regular languages?
2. How does the pumping lemma help in proving that a language is nonregular?
3. Why is the structure \( a^n b a^n \) problematic for regular languages?
4. What is the significance of choosing a word longer than the pumping length \( p \)?
5. How can we divide the word \( w = a^{100} b a^{100} \) into \( xyz \)?
6. Can the pumping lemma be used to prove that a language is regular?
7. Are there other methods besides the pumping lemma to prove nonregularity of a language?
8. How would you approach proving that the language \( \{a^n b^n \mid n > 0\} \) is nonregular?

### Tip

When using the pumping lemma, it's crucial to carefully choose a word \( w \) from the language and demonstrate that no division into \( xyz \) satisfies the lemma's conditions, which proves that the language is nonregular.

Learn how to apply the pumping lemma to prove that the language L = {anban | n > 0} is nonregular. Explore the conditions and selection of the appropriate word w = a^{100} b a^{100} for demonstrating nonregularity. Suitable for advanced undergraduate students and anyone interested in formal languages and automata theory.

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