Which of the following is true for the language: { ap| p is a prime}
2025
Which of the following is true for the language: { ap| p is a prime}
- A.
It is regular but not context-free
- B.
It is neither regular nor context-free, but accepted by a Turing machine
- C.
It is not accepted by a Turing Machine
- D.
It is context-free but not regular
Attempted by 32 students.
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Correct answer: C
Analysis of the Language { a^p | p is a prime }
To determine the properties of the language L = { a^p | p is a prime }, we analyze its classification within the Chomsky hierarchy.
1. Is it Regular?
No. The set of prime numbers is not a regular set. A finite automaton cannot count to infinity to verify primality. By the Pumping Lemma for regular languages, any sufficiently long string in a regular language can be pumped, but pumping a string of prime length often results in a composite length, violating the language definition.
2. Is it Context-Free?
No. The Pumping Lemma for Context-Free Languages also fails for this language. If we pump a string of prime length, the resulting length is unlikely to remain prime. Thus, the language is not context-free.
3. Is it Accepted by a Turing Machine?
Yes. A Turing machine can compute whether a given number is prime. Since primality testing is a decidable problem, a Turing machine can accept strings of prime length and reject others. Therefore, the language is recursive (and thus recursively enumerable).
Conclusion: The language is neither regular nor context-free, but it is accepted by a Turing machine.