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Let's assume there exists hardware that is able to compute the halting function H(n). That is, if you give it the number of a Turing Machine program/input combination, it will output a 1 if the TM halts and a 0 otherwise. “Super Turing Machine” (STM) based on this are constructed that can run “super algorithms.” STMs are ordinary TMs that can call the magic hardware as though it were a subroutine. Super algorithms are therefore ordinary TM programs that can make a call to the function H whenever necessary, and actually get a result. Clearly these are more powerful than ordinary computers, since they can solve the halting problem for ordinary TMs. Can they also solve the halting problem for themselves?

Thanks much in advance!!!

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  • $\begingroup$ Hint: try the very same reasoning that applies to ordinary Turing machines. $\endgroup$
    – dtldarek
    Jan 31 '13 at 23:08
  • $\begingroup$ Turing Machines are not constructible in hardware, since they are partially defined by infinite tape - whereas real computers are finite state machines with finite memory. Try shopping online for an infinite terabyte solid state drive. $\endgroup$ Feb 1 '13 at 4:15
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Congratulations! You've invented an oracle machine. It's fairly easy to show that a no Halt-testing oracle machine is capable of deciding the halting problem for Halt-testing oracle machines. The proof is similar to that for ordinary TMs.

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  • $\begingroup$ The OP said "there exists hardware" whereas TMs and oracles are mathematical concepts. $\endgroup$ Feb 1 '13 at 4:13

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