Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider the following decision problems:(P1)...
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Consider the following decision problems:
(P1) Does a given finite state machine accept a given string
(P2) Does a given context free grammar generate an infinite number of stings
(P2) Does a given context free grammar generate an infinite number of stings
Q. Which of the following statements is true?
- a)Both (P1) and (P2) are decidable
- b)Neither (P1) nor (P2) are decidable
- c)Only (P1) is decidable
- d)Only (P2) is decidable
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the following decision problems:(P1) Does a given finite stat...
- A finite state machine always halts in final or non-final state.Therefore, problem P1 is decidable.
- We check if the context free language generates any string of length between n and (2n – 1). If so, context free language is infinite else it is finite.Therefore, problem P2 is decidable. Thus, option (A) is correct. Please comment below if you find anything wrong in the above post.
Most Upvoted Answer
Consider the following decision problems:(P1) Does a given finite stat...
Introduction:
The given problem includes two decision problems: (P1) determining whether a given finite state machine accepts a given string, and (P2) determining whether a given context-free grammar generates an infinite number of strings. We need to determine which of these problems are decidable.
Decidability of (P1):
(P1) refers to the problem of determining whether a given finite state machine accepts a given string. This problem is decidable. We can construct an algorithm to solve this problem as follows:
1. Start at the initial state of the finite state machine.
2. For each character in the input string, follow the transition corresponding to that character.
3. If the machine reaches an accepting state after processing the entire string, then the answer is "yes". If not, the answer is "no".
This algorithm will always halt and produce a correct answer for any given input, making (P1) decidable.
Decidability of (P2):
(P2) refers to the problem of determining whether a given context-free grammar generates an infinite number of strings. This problem is also decidable. We can construct an algorithm to solve this problem as follows:
1. Convert the given context-free grammar into Chomsky normal form.
2. Check if the grammar contains a production rule of the form A → ε, where A is a non-terminal symbol.
3. If such a production rule exists, then the grammar can generate an infinite number of strings by repeatedly applying the rule. The answer is "yes".
4. If no such production rule exists, then the grammar can only generate a finite number of strings. The answer is "no".
This algorithm will always halt and produce a correct answer for any given input, making (P2) decidable.
Conclusion:
Both (P1) and (P2) are decidable problems. (P1) can be solved by simulating the given finite state machine on the input string, while (P2) can be solved by checking for a particular production rule in the Chomsky normal form of the given context-free grammar. Therefore, the correct answer is option 'A' – Both (P1) and (P2) are decidable.
The given problem includes two decision problems: (P1) determining whether a given finite state machine accepts a given string, and (P2) determining whether a given context-free grammar generates an infinite number of strings. We need to determine which of these problems are decidable.
Decidability of (P1):
(P1) refers to the problem of determining whether a given finite state machine accepts a given string. This problem is decidable. We can construct an algorithm to solve this problem as follows:
1. Start at the initial state of the finite state machine.
2. For each character in the input string, follow the transition corresponding to that character.
3. If the machine reaches an accepting state after processing the entire string, then the answer is "yes". If not, the answer is "no".
This algorithm will always halt and produce a correct answer for any given input, making (P1) decidable.
Decidability of (P2):
(P2) refers to the problem of determining whether a given context-free grammar generates an infinite number of strings. This problem is also decidable. We can construct an algorithm to solve this problem as follows:
1. Convert the given context-free grammar into Chomsky normal form.
2. Check if the grammar contains a production rule of the form A → ε, where A is a non-terminal symbol.
3. If such a production rule exists, then the grammar can generate an infinite number of strings by repeatedly applying the rule. The answer is "yes".
4. If no such production rule exists, then the grammar can only generate a finite number of strings. The answer is "no".
This algorithm will always halt and produce a correct answer for any given input, making (P2) decidable.
Conclusion:
Both (P1) and (P2) are decidable problems. (P1) can be solved by simulating the given finite state machine on the input string, while (P2) can be solved by checking for a particular production rule in the Chomsky normal form of the given context-free grammar. Therefore, the correct answer is option 'A' – Both (P1) and (P2) are decidable.
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Consider the following decision problems:(P1) Does a given finite state machine accept a given string(P2) Does a given context free grammar generate an infinite number of stingsQ. Which of the following statements is true?a)Both (P1) and (P2) are decidableb)Neither (P1) nor (P2) are decidablec)Only (P1) is decidabled)Only (P2) is decidableCorrect answer is option 'A'. Can you explain this answer?
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