What are the two cases to consider when dividing a string to apply the pumping lemma?
In the study of computational complexity theory, specifically within the context of context-sensitive languages, the Pumping Lemma is a powerful tool used to prove that a language is not context-sensitive. When applying the Pumping Lemma, there are two cases to consider when dividing a string: the pumping up case and the pumping down case.
1. Pumping up case:
In this case, we assume that the language in question is context-sensitive and proceed to divide a string into five parts: uvwxy. The Pumping Lemma states that for any context-sensitive language, there exists a constant p, such that for any string s in the language with a length of at least p, we can write s as uvwxy, satisfying the following conditions:
– |vwx| ≤ p: The length of vwx, the substring formed by concatenating v, w, and x, must be less than or equal to p.
– |vx| ≥ 1: The length of vx, the substring formed by concatenating v and x, must be greater than or equal to 1.
– For all natural numbers i, the string uv^iwx^iy is also in the language.
To prove that the language is not context-sensitive, we choose a string from the language and demonstrate that it cannot be pumped, i.e., there exists at least one i for which the pumped string is not in the language. By selecting a suitable string and showing that it fails to satisfy the conditions of the Pumping Lemma, we can conclude that the language is not context-sensitive.
2. Pumping down case:
In this case, we assume that the language in question is context-sensitive and proceed to divide a string into five parts: uvwxy. Similar to the pumping up case, the Pumping Lemma states that for any context-sensitive language, there exists a constant p, such that for any string s in the language with a length of at least p, we can write s as uvwxy, satisfying the conditions mentioned earlier.
To prove that the language is not context-sensitive, we again choose a string from the language and demonstrate that it cannot be pumped, i.e., there exists at least one i for which the pumped string is not in the language. By selecting a suitable string and showing that it fails to satisfy the conditions of the Pumping Lemma, we can conclude that the language is not context-sensitive.
It is important to note that the pumping up case and the pumping down case are not mutually exclusive. In some cases, a language may fail to satisfy the conditions of the Pumping Lemma in both cases, providing strong evidence that the language is not context-sensitive.
When applying the Pumping Lemma to prove that a language is not context-sensitive, we consider two cases: the pumping up case and the pumping down case. By selecting a suitable string and demonstrating that it cannot be pumped, we can conclude that the language is not context-sensitive.
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