(Solution)
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(D) <math>L_1</math> is recursively enumerable and <math>L_2</math> is recursive
 
(D) <math>L_1</math> is recursively enumerable and <math>L_2</math> is recursive
  
===Solution===
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==={{Template:Author|Arjun Suresh|{{arjunweb}} }}===
  
 
  Since, <math>f</math> is a polynomial time computable bijection and <span class="nocode" style="color:#48484c"> $f^{-1}$ </span> is also polynomial time computable, $L_1$ and $L_2$ should have the same complexity (isomorphic). This is because, given a problem for $L_1$, we can always do a polynomial time reduction to $L_2$ and vice verse. Hence, the answer is 'C', as in 'A', $L_1$ and $L_2$ can be finite, in 'B', $L_1$ and $L_2$ can be in $P$ and in 'D', $L_1$ and $L_2$ can be recursive. Only, in 'C' there is no intersection for $L_1$ and $L_2$, and hence it canʼt be true.
 
  Since, <math>f</math> is a polynomial time computable bijection and <span class="nocode" style="color:#48484c"> $f^{-1}$ </span> is also polynomial time computable, $L_1$ and $L_2$ should have the same complexity (isomorphic). This is because, given a problem for $L_1$, we can always do a polynomial time reduction to $L_2$ and vice verse. Hence, the answer is 'C', as in 'A', $L_1$ and $L_2$ can be finite, in 'B', $L_1$ and $L_2$ can be in $P$ and in 'D', $L_1$ and $L_2$ can be recursive. Only, in 'C' there is no intersection for $L_1$ and $L_2$, and hence it canʼt be true.

Revision as of 14:04, 14 April 2014

Consider two languages $L_1$ and $L_2$ each on the alphabet $\Sigma$. Let $f : \Sigma → \Sigma$ be a polynomial time computable bijection such that $(\forall x) [ x\in L_1$ iff $f(x) \in L_2]$. Further, let $f^{-1}$ be also polynomial time computable.

Which of the following CANNOT be true ?

(A) <math>L_1</math> $\in P$ and <math>L_2</math> is finite

(B) <math>L_1</math> $\in NP$ and <math>L_2</math> $\in P$

(C) <math>L_1</math> is undecidable and <math>L_2</math> is decidable

(D) <math>L_1</math> is recursively enumerable and <math>L_2</math> is recursive

Solution by Arjun Suresh

Since, <math>f</math> is a polynomial time computable bijection and  $f^{-1}$  is also polynomial time computable, $L_1$ and $L_2$ should have the same complexity (isomorphic). This is because, given a problem for $L_1$, we can always do a polynomial time reduction to $L_2$ and vice verse. Hence, the answer is 'C', as in 'A', $L_1$ and $L_2$ can be finite, in 'B', $L_1$ and $L_2$ can be in $P$ and in 'D', $L_1$ and $L_2$ can be recursive. Only, in 'C' there is no intersection for $L_1$ and $L_2$, and hence it canʼt be true.

Alternatively, we can prove 'C' to be false as follows:

Given $L_2$ is decidable. Now, for a problem in $L_1$, we can have a $TM$, which takes an input x, calculates $f(x)$ in polynomial time, check $f(x)$  is in $L_2$ (this is decidable as $L_2$ is decidable), and if it is, then output yes and otherwise no. Thus $L_1$ must also be decidable. 




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Consider two languages $L_1$ and $L_2$ each on the alphabet $\Sigma$. Let $f : \Sigma → \Sigma$ be a polynomial time computable bijection such that $(\forall x) [ x\in L_1$ iff $f(x) \in L_2]$. Further, let $f^{-1}$ be also polynomial time computable.

Which of the following CANNOT be true ?

(A) <math>L_1</math> $\in P$ and <math>L_2</math> is finite

(B) <math>L_1</math> $\in NP$ and <math>L_2</math> $\in P$

(C) <math>L_1</math> is undecidable and <math>L_2</math> is decidable

(D) <math>L_1</math> is recursively enumerable and <math>L_2</math> is recursive

Solution by Arjun Suresh[edit]

Since, <math>f</math> is a polynomial time computable bijection and  $f^{-1}$  is also polynomial time computable, $L_1$ and $L_2$ should have the same complexity (isomorphic). This is because, given a problem for $L_1$, we can always do a polynomial time reduction to $L_2$ and vice verse. Hence, the answer is 'C', as in 'A', $L_1$ and $L_2$ can be finite, in 'B', $L_1$ and $L_2$ can be in $P$ and in 'D', $L_1$ and $L_2$ can be recursive. Only, in 'C' there is no intersection for $L_1$ and $L_2$, and hence it canʼt be true.

Alternatively, we can prove 'C' to be false as follows:

Given $L_2$ is decidable. Now, for a problem in $L_1$, we can have a $TM$, which takes an input x, calculates $f(x)$ in polynomial time, check $f(x)$  is in $L_2$ (this is decidable as $L_2$ is decidable), and if it is, then output yes and otherwise no. Thus $L_1$ must also be decidable. 




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