Computer Science Engineering (CSE) Exam > Computer Science Engineering (CSE) Questions > Consider three decision problems P1, P2 and P...
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Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?
- a)P3 is decidable if P1 is reducible to P3
- b)P3 is undecidable if P3 is reducible to P2
- c)P3 is undecidable if P2 is reducible to P3
- d)P3 is decidable if P3 is reducible to P2's complement
Correct answer is option 'C'. Can you explain this answer?
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Consider three decision problems P1, P2 and P3. It is known that P1 is...
Background: In computational complexity theory, a decision problem has only two possible outputs yes or no. A decision problem is said to be decidable if there exists an effective method or algorithm that returns a correct yes/no answer to that problem. A decision problem is said to be undecidable if there does not exist a single algorithm that always lead to a correct yes/no solution. In terms of reducibility: A ≤p B denotes A is a decision problem that is reducible to B in polynomial time p. This simply means that A’s instance can be transformed into B’s instance and following the solution of B we can get a solution for the problem A. So here we can draw some conclusions:
1. If A ≤p B and B is decidable then A is also decidable. This is because if there exists a specific algorithm for solving B and we can also reduce A to B then we can have a solution of A as well. Hence A is decidable.
However the reverse is not true i.e. if A ≤p B and A is decidable then B is also decidable because A can have an algorithm existing for its correct solution but might be the case that B does not.
2. If A ≤p B and A is undecidable then B is also undecidable. This is because if A is undecidable even when it can be reduced to B that simply reflects even B cannot provide an algorithm by which we can solve B and hence A. So decision problem B is also undecidable.
However the reverse is not true i.e. if A ≤p B and A is decidable then B is also decidable because A can have an algorithm existing for its correct solution but might be the case that B does not.
2. If A ≤p B and A is undecidable then B is also undecidable. This is because if A is undecidable even when it can be reduced to B that simply reflects even B cannot provide an algorithm by which we can solve B and hence A. So decision problem B is also undecidable.
However the reverse is not true here as well i.e. if A ≤p B and B is undecidable then A is also undecidable because there might exist an algorithm for A that can provide a solution to A. Using the above stated conclusions we can say that option 1, 2 and 4 are false and option 3 is true.
Option 1: P1 ≤p P3 and given P1 is decidable gives no conclusion for P3.
Option 2: P3 ≤p P2 and given P2 is undecidable gives no conclusion for P3.
Option 3: P2 ≤p P3 and given P2 is undecidable gives conclusion for P3 to be undecidable.
Option 4: P3 ≤p P2’s complement and given P2 is undecidable therefore P2’s complement is also undecidable gives no conclusion for P3.
Option 2: P3 ≤p P2 and given P2 is undecidable gives no conclusion for P3.
Option 3: P2 ≤p P3 and given P2 is undecidable gives conclusion for P3 to be undecidable.
Option 4: P3 ≤p P2’s complement and given P2 is undecidable therefore P2’s complement is also undecidable gives no conclusion for P3.
Most Upvoted Answer
Consider three decision problems P1, P2 and P3. It is known that P1 is...
Decidability and Reducibility
To understand why option 'C' is the correct answer, we need to understand the concepts of decidability and reducibility.
Decidability:
A decision problem is said to be decidable if there exists an algorithm that can solve it for all possible inputs and always terminate with a correct answer.
Undecidability:
A decision problem is said to be undecidable if there is no algorithm that can solve it for all possible inputs. In other words, there are inputs for which the algorithm may run forever or produce incorrect results.
Reduction:
In the context of decision problems, reduction is a way to transform one problem into another in such a way that the solution to the second problem can be used to solve the first problem. If problem A is reducible to problem B, it means that an algorithm for solving problem B can be used to solve problem A.
Explanation of the Options:
a) P3 is decidable if P1 is reducible to P3:
This option states that if P1 is reducible to P3, then P3 must be decidable. However, the reducibility of P1 to P3 does not guarantee the decidability of P3. It is possible that P1 is decidable and reducible to P3, but P3 itself is still undecidable.
b) P3 is undecidable if P3 is reducible to P2:
This option states that if P3 is reducible to P2, then P3 must be undecidable. This is not necessarily true. The reducibility of P3 to P2 does not imply the undecidability of P3. It is possible that P3 is reducible to P2, but P3 itself is still decidable.
c) P3 is undecidable if P2 is reducible to P3:
This option states that if P2 is reducible to P3, then P3 must be undecidable. This is the correct answer. If P2 is undecidable and reducible to P3, it means that an algorithm for solving P3 can be used to solve P2, which contradicts the undecidability of P2. Therefore, P3 must also be undecidable.
d) P3 is decidable if P3 is reducible to P2's complement:
This option is unrelated to the given information. It introduces the concept of the complement of P2, which is not mentioned in the question. Therefore, this option is not relevant to determining the decidability of P3 based on the given information.
Conclusion:
The correct answer is option 'C'. If P2 is undecidable and reducible to P3, then P3 must also be undecidable.
To understand why option 'C' is the correct answer, we need to understand the concepts of decidability and reducibility.
Decidability:
A decision problem is said to be decidable if there exists an algorithm that can solve it for all possible inputs and always terminate with a correct answer.
Undecidability:
A decision problem is said to be undecidable if there is no algorithm that can solve it for all possible inputs. In other words, there are inputs for which the algorithm may run forever or produce incorrect results.
Reduction:
In the context of decision problems, reduction is a way to transform one problem into another in such a way that the solution to the second problem can be used to solve the first problem. If problem A is reducible to problem B, it means that an algorithm for solving problem B can be used to solve problem A.
Explanation of the Options:
a) P3 is decidable if P1 is reducible to P3:
This option states that if P1 is reducible to P3, then P3 must be decidable. However, the reducibility of P1 to P3 does not guarantee the decidability of P3. It is possible that P1 is decidable and reducible to P3, but P3 itself is still undecidable.
b) P3 is undecidable if P3 is reducible to P2:
This option states that if P3 is reducible to P2, then P3 must be undecidable. This is not necessarily true. The reducibility of P3 to P2 does not imply the undecidability of P3. It is possible that P3 is reducible to P2, but P3 itself is still decidable.
c) P3 is undecidable if P2 is reducible to P3:
This option states that if P2 is reducible to P3, then P3 must be undecidable. This is the correct answer. If P2 is undecidable and reducible to P3, it means that an algorithm for solving P3 can be used to solve P2, which contradicts the undecidability of P2. Therefore, P3 must also be undecidable.
d) P3 is decidable if P3 is reducible to P2's complement:
This option is unrelated to the given information. It introduces the concept of the complement of P2, which is not mentioned in the question. Therefore, this option is not relevant to determining the decidability of P3 based on the given information.
Conclusion:
The correct answer is option 'C'. If P2 is undecidable and reducible to P3, then P3 must also be undecidable.
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Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer?
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Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer?.
Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer? for Computer Science Engineering (CSE) 2024 is part of Computer Science Engineering (CSE) preparation. The Question and answers have been prepared according to the Computer Science Engineering (CSE) exam syllabus. Information about Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Computer Science Engineering (CSE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider three decision problems P1, P2 and P3. It is known that P1 is decidable and P2 is undecidable. Which one of the following is TRUE?a)P3 is decidable if P1 is reducible to P3b)P3 is undecidable if P3 is reducible to P2c)P3 is undecidable if P2 is reducible to P3d)P3 is decidable if P3 is reducible to P2's complementCorrect answer is option 'C'. Can you explain this answer?.
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