I'm reading about a Turing Machine $M$ and it says the problem of deciding whether M accepts a string is "$\Sigma^0_2$-hard and $\Pi^0_2$-hard".

I haven't seen this kind of notation before and haven't found a good answer from searching. Is this a more specific form of saying NP-hard?


No, it's unrelated to NP-hardness. $\Sigma_n^0$ and $\Pi_n^0$ are the levels of the arithmetical hierarchy. $\Sigma_2^0$ is the class of problems that can be decide by Turing machines that have an oracle for the halting problem and $\Pi_2^0$ is the class of problems whose complement is in $\Sigma_2^0$.

There is the corresponding notion of the polynomial hierarchy, in which $\Sigma_1^\mathrm{P}$ is NP and $\Pi_1^\mathrm{P}$ is co-NP.

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  • $\begingroup$ Thank you! Is there any information related to decidability that I can assume then for the Σ02 class of problems and Π02 class of problems? It seems like given Σ02 is the class of problems that can be decided by a TM that has an oracle for the halting problem, that it cannot be semi-decidable (computably enumerable) and must be fully undecidable. And then given Π02 is its complement, Π02 problems must be fully undecidable as well. Is this the right thought process? $\endgroup$ – Sarina Jan 4 '15 at 22:18
  • $\begingroup$ Also, if I wanted to prove that a problem is in the Σ02 class, should I show that there is a TM with an oracle for the halting problem that decides it? Thanks so much! $\endgroup$ – Sarina Jan 4 '15 at 22:19
  • $\begingroup$ All decidable and recursively enumerable problems are in $\Sigma_0^2$, too: you can decide them with an oracle machine that just happens to never use the oracle, for example. This is just like every problem in P being in NP, too, except that we know that $\Sigma^0_2\neq \Sigma^0_1=\mathrm{RE}$ so a $\Sigma^0_2$-complete problem is definitely not RE. And, yes, to show that something is in $\Sigma^0_2$, you just need to show you can decide it with a Turing machine that has an oracle for the halting problem. $\endgroup$ – David Richerby Jan 4 '15 at 22:31

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