What are some examples of non-enumerable languages whose complement isn't either? I.e., a language L such that L is not Turning-recognizable and L’ is not Turing-recognizable either.
Update: Found some examples:
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$L_{0,1}={\langle M\rangle\mid M\text{ halts on } 0\text{ but not on } 1 }$ – John L. May 05 '22 at 14:59
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A random language is neither recognizable nor co-recognizable almost surely.
Every language which is complete for $\Sigma_k$ or $\Pi_k$, for any $k \geq 2$ (these are classes in the arithmetical hierarchy), is neither recognizable nor co-recognizable. Some examples:
- The language of all oracle TMs $M$ such that $M$ halts on the empty input when given access to an oracle of the halting problem ($\Sigma_2$-complete).
- The language of all TMs $M$ such that $M$ halts on finitely many inputs ($\Sigma_2$-complete).
- The language of all TMs $M$ such that $M$ halts on all inputs ($\Pi_2$-complete).
- The language of all TMs $M$ such that "$M$ halts on $x$" is decidable ($\Sigma_3$-complete).
- The language of all TMs $M$ such that $M$ halts on all but finitely many inputs ($\Sigma_3$-complete).
Yuval Filmus
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