# Is something more than Turing complete Turing complete?

In complexity theory, we do not call a decision problem that is not in NP "NP-complete".

But in computability, do we call a machine model "Turing complete" if it can compute functions which Turing machines can not?

The definition from Wikipedia didn't address this problem explicitly, and likely assumed yes:

A computational system that can compute every Turing-computable function is called Turing complete (or Turing powerful). Alternatively, such a system is one that can simulate a universal Turing machine.

For example, if there is a system that can:

• Compute every Turing-computable function.
• Compute the halting problem for a Turing machine.

And nothing else. Is it Turing complete?