The pumping property, also known as the pumping lemma, is a fundamental tool in the field of computational complexity theory for analyzing context-sensitive languages. It helps determine whether a language is context-sensitive by providing a necessary condition that must hold for all strings in the language. However, in the case of language B and the string a^Pb^Pc^P, the pumping property does not hold.
To understand why the pumping property does not hold for this particular string, let's first review the pumping lemma for context-sensitive languages. According to the lemma, if a language L is context-sensitive, then there exists a constant n (the pumping length) such that any string w in L with |w| ≥ n can be divided into five parts: w = uvxyz, satisfying the following conditions:
1. |vxy| ≤ n
2. |vy| ≥ 1
3. For all i ≥ 0, the string uv^ixy^iz is also in L.
Now, let's consider the string a^Pb^Pc^P, where P is a prime number. This string consists of a sequence of 'a's followed by the same number of 'b's and 'c's. Suppose we divide this string into five parts as w = uvxyz, where |vxy| ≤ n and |vy| ≥ 1.
In this case, since the pumping length n is a constant, it is not possible to choose a partition that satisfies the conditions of the pumping lemma. This is because the number of 'a's, 'b's, and 'c's in the string is fixed and equal to P, which is a prime number. Therefore, it is not possible to divide the string into five parts such that the number of 'a's, 'b's, and 'c's in the pumped strings remains the same.
For example, let's consider a possible partition where vxy consists of only 'a's. In this case, pumping up by increasing the exponent of 'a' would result in a string that has a different number of 'a's compared to 'b's and 'c's, violating the condition that all pumped strings must be in the language. Similarly, any other possible partition would lead to a violation of the pumping lemma conditions.
Therefore, we can conclude that the pumping property does not hold for the string a^Pb^Pc^P in the example of language B. This means that the language B, which includes strings of the form a^Pb^Pc^P, is not a context-sensitive language.
The string a^Pb^Pc^P does not satisfy the conditions of the pumping lemma for context-sensitive languages due to its fixed and prime number of 'a's, 'b's, and 'c's. This violation of the pumping property indicates that the language B, which includes this string, is not a context-sensitive language.
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