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Fill in the blank in terms of p, where p is the maximum string length in L.
Statement: Finite languages trivially satisfy the pumping lemma by having n = ______
Statement: Finite languages trivially satisfy the pumping lemma by having n = ______
- a)p*1
- b)p+1
- c)p-1
- d)None of the mentioned
Correct answer is option 'B'. Can you explain this answer?
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Fill in the blank in terms of p, where p is the maximum string length ...
Finite languages trivially satisfy the pumping lemma by having n equal to the maximum string length in l plus 1.
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Fill in the blank in terms of p, where p is the maximum string length ...
Understanding the Pumping Lemma
The pumping lemma is a fundamental concept in formal language theory, particularly for regular languages. It states that for any infinite regular language, there exists a pumping length n such that any string s in the language of length at least n can be divided into three parts, xyz, fulfilling specific conditions.
Finite Languages and the Pumping Lemma
- Finite languages contain a finite number of strings.
- The maximum string length in a finite language L is denoted as p.
- For any string of length greater than p, the pumping lemma requires that the string can be "pumped" (i.e., repeated) while still remaining in the language.
Why n = p + 1?
- The pumping length n must be greater than any possible string length in the finite language.
- If we take n = p + 1, it guarantees that any string s chosen from L, which has a maximum length of p, is eligible for division into parts xyz.
- Since all strings in L are of length p or less, n = p + 1 effectively covers all strings, ensuring compliance with the pumping lemma conditions.
Conclusion
- Therefore, the correct answer is option b) p + 1.
- This choice ensures all strings can be "pumped" as per the lemma’s requirements and highlights the trivial nature of finite languages concerning the pumping lemma.
In summary, finite languages satisfy the pumping lemma by having n set to p + 1, allowing for the necessary conditions to be met without exception.
The pumping lemma is a fundamental concept in formal language theory, particularly for regular languages. It states that for any infinite regular language, there exists a pumping length n such that any string s in the language of length at least n can be divided into three parts, xyz, fulfilling specific conditions.
Finite Languages and the Pumping Lemma
- Finite languages contain a finite number of strings.
- The maximum string length in a finite language L is denoted as p.
- For any string of length greater than p, the pumping lemma requires that the string can be "pumped" (i.e., repeated) while still remaining in the language.
Why n = p + 1?
- The pumping length n must be greater than any possible string length in the finite language.
- If we take n = p + 1, it guarantees that any string s chosen from L, which has a maximum length of p, is eligible for division into parts xyz.
- Since all strings in L are of length p or less, n = p + 1 effectively covers all strings, ensuring compliance with the pumping lemma conditions.
Conclusion
- Therefore, the correct answer is option b) p + 1.
- This choice ensures all strings can be "pumped" as per the lemma’s requirements and highlights the trivial nature of finite languages concerning the pumping lemma.
In summary, finite languages satisfy the pumping lemma by having n set to p + 1, allowing for the necessary conditions to be met without exception.
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