Natural problem arising from diagonalization?

Question

Diagonalization is a frequently used technique in complexity theory. However, the problems (sets) that are created by diagonalization rarely correspond to anything natural. It would be interesting to collect examples, when a problem was originally defined via diagonalization, but later it was found to coincide with a natural problem.

it is in a way hard to argue that any TM complete problem is any more "natural" than any other, they are equivalent and there seems to be no "natural" way of discriminating them except perhaps by # of states in the TMs etc.... — vzn, Feb 08 '14 at 01:45
fairly close to easy to understand undecidable problem, [math.se] — vzn, Feb 08 '14 at 01:56
@vzn: The question didn't say anything about completeness... — Joshua Grochow, Feb 08 '14 at 02:17
Real numbers are fairly "natural", and when you use diagonalization to prove that naturals are not countable you "define" (in a loose sense) the unit interval of reals (that can be 1:1 mapped to the whole real line). — Marzio De Biasi, Feb 08 '14 at 11:13

score 7 · Accepted Answer · answered Feb 08 '14 at 01:34

In some sense the halting problem satisfies this request (I was almost ready to say "There are no such examples!" when I thought of this one.) By doing diagonalization against computable sets, you can of course get sets of very high complexity, but the seemingly-most-natural way of diagonalizing against computable sets gives you a problem that is equivalent to the halting problem. As we now know, the halting problem is equivalent to many other computational problems, such as deciding many properties of TMs, telling whether a Diophantine equation has solutions, the word problem in finitely presented groups, telling whether a high-dimensional knot is the unknot, etc.

I am still ready to say that I would be surprised to see other examples (that aren't just, say, relativized versions of this).

Natural problem arising from diagonalization?

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