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The halting problem relies on the fluidity of Turing machines. That is, a string can represent a machine.

Can you do the same for C++ on a modern computer?

Let's see my first attempt. Let bool h(string x, string y) be the purported function that decides haltingness and always halts. You can easily turn it into a full program.

Now define a self-contradictory function f:

bool f(string x) {
    if (h(x, x))
        for (;;);
    return true;
}

int main() {
    cout << h(f, f) << endl;
}

The problem is that I can't feed the code of f into h in main().

My second attempt: Prepare 4 files: h.cpp, h.exe, f.cpp, f.exe. And suppose the call formats are

h.exe filename1 filename2
f.exe filename

My problem is that in f.cpp I need to call h.exe, which in turn has to be part of f.cpp.

What's your working scheme?

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  • $\begingroup$ I'm not sure what you are trying to achieve here. Proving the halting problem is undecidable by C++ machines, I assume? $\endgroup$ – dkaeae Jul 12 at 9:16
  • $\begingroup$ @dkaeae Is turning a C++ program into a Turing machine the only way to prove that this program can't solve the halting problem? I want to do the argument in C++ and hence bypassing the idea of Turing machines altogether. I don't like the idea of Turing machines. Or is it that in C++ this argument is not viable and hence Turing machines are necessary? I want to shoot down the belief that there is no halting problem in C++ programs, i.e., you can decide haltingness for the set of C++ programs. If you believe the set of Turing machines and the set of C++ programs are indeed equal, do it in C++. $\endgroup$ – Zirui Wang Jul 12 at 9:52
  • $\begingroup$ Surely you can feed the code of f into h. Just write it again as a literal string. $\endgroup$ – Dmitri Urbanowicz Jul 12 at 10:11
  • $\begingroup$ But also keep in mind, that then you either need to include the source of h into that string, or you need to be sure that h knows that "h(x,x)" means call of h itself. $\endgroup$ – Dmitri Urbanowicz Jul 12 at 10:12
  • $\begingroup$ The key problem is to solve the fluidity problem. It's easy to convert strings to Turing machines and back. But how to do that in C++? That's the main difficulty. I think it's too easy with TMs and conjecture that it's impossible with C++, under my current understanding. $\endgroup$ – Zirui Wang Jul 12 at 11:23
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If you want to prove that the halting problem for C++ programs can't be decided by a C++ program, just copy the proof of the Halting problem but replace every use of a universal Turing machine with a C++ interpreter/compiler that's written in C++.

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  • $\begingroup$ Yes, the translation is not as trivial as advertised in textbooks. $\endgroup$ – Zirui Wang Jul 12 at 11:15

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