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My professor gave an introduction to the arithmetic hierarchy via Turing reductions, stating that, for instance, $\Sigma_2 = \text{r.e.}^\text{r.e.}$ (namely an $\text{r.e.}$ pseudocode with access to an $\text{r.e.}$ oracle) or $\Pi_3 = \text{co-r.e.}^{\text{r.e.}^\text{r.e.}}$. Later, the equivalent formulation via alternating quantifier descriptions was discussed, but I found it difficult to link the two definitions. I have not been able to find any references describing the connection between the above definitions, with everything I've seen having solely to do with the quantifier-based description. Are there any texts/notes that might cover what I'm looking for?

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  • $\begingroup$ Isnt the definition of the hirarchy using oracles to $NP$ and $coNP$? $\endgroup$
    – nir shahar
    Commented Oct 28, 2021 at 6:15
  • $\begingroup$ @nirshahar Not sure what you mean. I'm referring to the arithmetic hierarchy, not the polynomial hierarchy. $\endgroup$
    – gf.c
    Commented Oct 28, 2021 at 6:18
  • $\begingroup$ Oh, ok. The arithmetic hirarchy indeed uses oracle calls to $RE$ and $coRE$. For some reason I was sure you meant to talk about the polynomial hirarchy... $\endgroup$
    – nir shahar
    Commented Oct 28, 2021 at 6:21

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I found Dexter Kozen's book "Theory of Computation" to contain a very understandable introduction to these topics.

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