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As it says in the title:

Are there problem instances which we know to be unsolvable?

Or equivalently

Are there any promise problems with a finite number of possible inputs which are undecidable?

Please note: I realize that many computational problems are known to be unsolvable, but to the best of my knowledge (limited it may be) all must take an infinite number of inputs. Therefore, their existence does not imply that there cannot be an infinite number of algorithms each solving a subset of these problems.

Remark: I only wish to consider well-defined problems, i.e. every input has a correct output.

On a related note, is there a complexity hierarchy for promise problems with only a finite number of possible inputs?

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Every problem with a finite number of inputs is computable.

Proof: Suppose that the inputs are $i_1,\ldots,i_n$ and that the desired outputs are $o_1,\ldots,o_n$. Then the following program solves the problem:

  • If the input is $i_1$, output $o_1$.
  • If the input is $i_2$, output $o_2$.
  • ...
  • If the input is $i_{n-1}$, output $o_{n-1}$.
  • Output $o_n$.
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  • 2
    $\begingroup$ As an aside, the analogous approach to "demonstrate" computability with an infinite number of inputs doesn't work: it would require a program with an infinite number of steps, which does not meet the definition of an algorithm. (This may be very obvious to most readers, but I felt like pointing it out anyway.) $\endgroup$ – Jeroen Mostert Dec 21 '17 at 22:47

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