View source for GATE2003 q54 ← GATE2003 q54

Define languages L0 and L1 as follows :

$L_0 = \{< M, w, 0 > |$ $M$ halts on $w\} $

$L_1 = \{< M, w, 1 > |$ $M$ does not halts on$ w\}$

Here $< M, w, i >$ is a triplet, whose first component. $M$ is an encoding of a Turing Machine, second component,$ w$, is a string, and third component, $i$, is a bit. Let $L = L_0 ∪ L_1$. Which of the following is true ?

(A) $L$ is recursively enumerable, but is not

(B) $L$ is recursively enumerable, but$ L'$ is not

(C) Both $L$ and $L'$ are recursive

(D) Neither $L$ nor $L'$ is recursively enumerable

Solution[edit]

Both $L$ and $L'$ are undecidable. Because halting problem can be solved with both $L$ and $L'$.

Halting problem can be stated as follows: A machine $M$ and a word $w$ are given. You have to tell, if $M$ halts on $w$.

So, to solve halting problem $<M,w>$ using $L$, just give $<M,w,0>$ and $<M,w,1>$ to two instances of $T$ which is the Turing machine for $L$. If $T$ accepts the triplet $<M,w,0>$, it means $M$ halts on $w$ => we have solved halting problem. If $T$ accepts the triplet $<M,w,1>$, it means $M$ doesn't halt on $w$ => we have solved halting problem. We know that either $<M,w,0>$ or $<M,w,1>$ is in $L$. So, if $L$ is recursively enumerable, $T$ is bound to stop on at least one of these inputs. Hence, using $L$ we can solve halting problem => $L$ is not recursively enumerable.

Similarly, we can also show that halting problem can be solved with $L'$.

Hence, neither $L$ nor $L'$ is recursively enumerable.

Alternate Solution[edit]

$L_0$ is recursively enumerable. (Given $<M,w,0>$, we can just give $w$ to $M$. If $M$ halts on $w$, $<M,w,0>$ is element of $L_0$.

$L_1$ is not recursively enumerable. Because, halting problem can be solved with it. To decide if a turing machine $M$ accepts a word $w$, just give $<M,w,1>$ to the Turing machine for $L_1$ and also give $w$ to $M$. Either $M$ accepts $w$, or the Turing machine for $L_1$ accepts $<M,w,1>$. In either case we have solved halting problem. Hence, $L_1$ is not recursively enumerable.

$L_1$ can be reduced to $L_0'$, and hence $L_0'$ also is not recursively enumerable.

$L_1'$ can be reduced to $L_0$, and hence $L_1'$ is recursively enumerable.

Now, $L$ = $L_0 U L_1$

= re U not re

= not re

$L' = (L_0 \cup L_1)'$

$= L_0' ∩ L_1'$

$=$ not re $∩$ re

$=$ not re