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Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f-1 be also polynomial time computable. Which of the following CANNOT be true?
  • a)
    L1 ∈ P and L2 is finite
  • b)
    L1 ∈ NP and L2 ∈ P
  • c)
    L1 is undecidable and L2 is decidable
  • d)
    L1 is recursively enumerable and L2 is recursive
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Consider two languages L1 and L2 each on the alphabet ∑. Let f :...
We have one to one mapping for all instances of L1 to L2. L1 is given to be undecidable. Further L1 is polynomial time reducible to L2. (By given mapping). Now if L2 is decidable then there is algorithm to solve L2 in polytime. But then we can solve every instance of L1 in polytime, making L1 also decidable. Contradiction
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Most Upvoted Answer
Consider two languages L1 and L2 each on the alphabet ∑. Let f :...
Explanation:

Given Information:
- We have two languages L1 and L2 on the same alphabet.
- There exists a polynomial-time computable bijection f such that (x) [x L1 iff f(x) L2].
- The inverse of f, denoted as f-1, is also polynomial time computable.

Analysis of Options:

a) L1 P and L2 is finite:
This option can be true. If L1 is in P, it means it can be solved in polynomial time. If f is a polynomial-time bijection between L1 and L2, then L2 is also efficiently computable. L2 being finite does not contradict this scenario.

b) L1 NP and L2 P:
This option can be true as well. If L1 is in NP, it means it can be verified in polynomial time. If f is a polynomial-time bijection, then L2 can also be verified in polynomial time. L2 being in P is consistent with this setup.

c) L1 is undecidable and L2 is decidable:
This option cannot be true. If L1 is undecidable, it means there is no algorithm that can decide membership in L1. However, the bijection f allows us to decide membership in L2 based on membership in L1. This contradicts the fact that L2 is decidable.

d) L1 is recursively enumerable and L2 is recursive:
This option can be true. If L1 is recursively enumerable, it means there exists a Turing machine that can enumerate the strings in L1. The bijection f allows us to enumerate the strings in L2 based on the enumeration of L1. L2 being recursive (decidable) is consistent with this setup.
Therefore, option c) is the one that cannot be true based on the given information and the properties of the languages L1 and L2.
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Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f-1be also polynomial time computable. Which of the following CANNOT be true?a)L1 ∈ P and L2 is finiteb)L1 ∈ NP and L2 ∈ Pc)L1 is undecidable and L2 is decidabled)L1 is recursively enumerable and L2 is recursiveCorrect answer is option 'C'. Can you explain this answer?
Question Description
Consider two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f-1be also polynomial time computable. Which of the following CANNOT be true?a)L1 ∈ P and L2 is finiteb)L1 ∈ NP and L2 ∈ Pc)L1 is undecidable and L2 is decidabled)L1 is recursively enumerable and L2 is recursiveCorrect 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 two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f-1be also polynomial time computable. Which of the following CANNOT be true?a)L1 ∈ P and L2 is finiteb)L1 ∈ NP and L2 ∈ Pc)L1 is undecidable and L2 is decidabled)L1 is recursively enumerable and L2 is recursiveCorrect 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 two languages L1 and L2 each on the alphabet ∑. Let f : ∑ → ∑ be a polynomial time computable bijection such that (∀ x) [x ∈ L1 iff f(x) ∈ L2]. Further, let f-1be also polynomial time computable. Which of the following CANNOT be true?a)L1 ∈ P and L2 is finiteb)L1 ∈ NP and L2 ∈ Pc)L1 is undecidable and L2 is decidabled)L1 is recursively enumerable and L2 is recursiveCorrect answer is option 'C'. Can you explain this answer?.
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