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Only language i cant think of, that does not belong in RE class is diagonal language, but unfortunately its complementary language is recursively enumerable. Does anyone have any ideas?

Eric
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  • I wonder if this better belongs on.... http://programmers.stackexchange.com/ ? – scunliffe Nov 23 '13 at 15:02
  • Did you search google? See for example http://objectmix.com/theory/713406-example-non-rec-enum-language-whose-complement-isntrec-enum-too.html and http://math.stackexchange.com/q/388380 – l4mpi Nov 23 '13 at 15:11
  • Actually, [cs.stackexchange.com](http://cs.stackexchange.com/) would be the appropriate place to ask this. – mabi Dec 16 '13 at 16:33

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Consider the following set: all machines which fail to halt on some input. In other words, all machines M for which there exists an input X such that M does not halt on input X. This is not recursively enumerable, since there's no algorithmic way to determine that a given machine fails to halt on a given input. I mean, how would you do it? Simulate the machine running on the input? For how long?

The complement of this set is the following: all machines which halt on all inputs. In other words, all machines M which always halt on any input X. This is not recursively enumerable either, since there's no algorithmic way to determine that a given machine halts on all possible inputs. How would you do it? Check them all?

Patrick87
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