Termination analysis

In computer science, termination analysis is program analysis which attempts to determine whether the evaluation of a given program halts for each input. This means to determine whether the input program computes a total function.

void f(int n) {
   while (n > 1)
      if (n % 2 == 0)
          n = n / 2;
      else
          n = 3 * n + 1;
}

It is closely related to the halting problem, which is to determine whether a given program halts for a given input and which is undecidable. The termination analysis is even more difficult than the Halting problem: the termination analysis in the model of Turing machines as the model of programs implementing computable functions would have the goal of deciding whether a given Turing machine is a total Turing machine, and this problem is at level ${\displaystyle \Pi _{2}^{0}}$ of the arithmetical hierarchy and thus is strictly more difficult than the Halting problem.

Now as the question whether a computable function is total is not semi-decidable, each sound termination analyzer (i.e. an affirmative answer is never given for a non-terminating program) is incomplete, i.e. must fail in determining termination for infinitely many terminating programs, either by running forever or halting with an indefinite answer.