View source for GATE2003 q52 ← GATE2003 q52

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[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 can't 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.