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Is the halting problem for the Collatz Program computable?

In related questions, we asked whether the Collatz Conjecture is true and when it will be resolved one way or another. Here we ask more specifically whether we can predict the behavior of the corresponding program.

Let's define the Collatz Program in pseudocode as

collatz(n) = 
  if (n is 1) return 1
  else if (n is even) return collatz(n/2)
  else return collatz(3n + 1)
where input n is a positive integer.

The Collatz Conjecture is that this program halts (and returns 1) for all integer inputs.

Let's imagine a companion program called collatz_halts(), which takes an integer input n, always halts, and returns 1 if collatz() halts, and 0 otherwise.

Does collatz_halts() exist? If collatz() always halts, then collatz_halts() definitely exists, because the answer is 1 for all inputs. If collatz_program() only halts for some n, then collatz_halts() might or might not exist.

Note that if the Collatz Conjecture is false for only a finite number of inputs, then collatz_halts() exists, since the program could test against an enumeration of the the inputs for which collatz() does not halt. Also note that if collatz() always either halts or encounters a cycle, then collatz_halts() exists by modifying collatz() to check for cycles.

Resolution:

  • This question will resolve positively if it is demonstrated that a program must exist that always halts and tests whether the Collatz program halts with a given input.

  • It will resolve negatively if the Conjecture is proven to be false and such a halting-test program is proven not to exist.

  • Both of these resolutions will be via publication in a major mathematics journal.

If no such proof is published before June 21, 2520, then the question will resolve as ambiguous.


Other questions on the Collatz Conjecture:

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