Confusion of halting problem

Question

Show that the following problem is solvable.Given two programs with their inputs and the knowledge that exactly one of them halts, determine which halts.

lets P be program that determine one of the program will be halted.

P(Program x,Program y){

    if(x will be halted)

          then return 1;

    else

           then return 2;
}

since we know that exactly one of them will be halted,if 1 then program x will be halted.Otherwiae program y will be halted.

then we construct a new program call D

D(X,Y){

     if(P(X,Y) == 2)

         D will halt;

      else

         while(1)//D will not halt;

  }

lets S be aritbrary program.

So if we have D(D,S)

if D will halt then D will not halt

if D will not halt then D will halt

It impiles a contradiction same as halting problem.

But the question stated that it is solvable.

Answer 1

To start with an important side note: what are the inputs for $X$ and $Y$ that they will halt on? you need to specify a specific input for the machines in order for the question to be well-defined (for example, the question could be "given $X,Y$ where exactly one terminates on $\epsilon$, find who halted." Or maybe, "given ... where exactly one always halts on itself as an input...")

I see what confused you there. The problem is indeed solvable: in $P$ emulate both $X$ and $Y$ and answer whoever halted first.

The problem in your solution is that an input to $P$ must be $X,Y$ where exactly one of them halts, and the other doesn't.

So, a proper input to $D$ must be a proper input to $P$. Now notice you called $D(D,S)$. To make sure that $D$ will return a correct output, we must have proper inputs: only one of $D$ or $S$ halts! But... If $S$ halts then your $D$ also halts and this is not a proper input to $P$, and therefore also not a proper input to $D$. Thus you cannot rely on output from $D(D,S)$ as it may not be correct