At the Wikipedia page, the polynomial hierarchy also defines the following:

$\Delta_0^\text{P} = P$, $\Delta_i^\text{P} = \text{P}^{\Sigma_{i-1}^\text{P}}$

However, the only usage of this anywhere else on the page is the following set of inclusions:

$\Pi_i^P \subseteq \Delta_{i+1}^P \subseteq \Pi_{i+1}^P$.

What usage does this $\Delta_i^P$ class have?


The polynomial hierarchy is an analog of the arithmetical hierarchy in recursion theory, which also has $\Sigma,\Pi,\Delta$, and several other hierarchies also have $\Delta$ terms.

The usage is exactly the same as the usage of $\Sigma_i^P$ and $\Pi_i^P$. For a given problem, you try to find its exact location in the polynomial hierarchy. If a problem is in $\Delta_i^P$ then it can't be $\Sigma_{i+1}^P$-hard (unless the polynomial hierarchy collapses), but perhaps it is $\Delta_i^P$-hard. So you might need these classes for some problems.

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  • $\begingroup$ If I recall correctly, in recursion theory the $\Delta$ is the intersection of the $\Pi$ and $\Sigma$ of the same level? Here this seems not to be the case (or at least it is not known)? $\endgroup$ – Hendrik Jan May 22 '15 at 17:16
  • $\begingroup$ @HendrikJan At least it is contained in the intersection of $\Pi$ and $\Sigma$. I haven't heard before of equality. $\endgroup$ – Ryan May 22 '15 at 17:49
  • $\begingroup$ @HendrikJan That also confused me, but apparently it's conjecture to be different from the intersection. $\endgroup$ – Yuval Filmus May 22 '15 at 21:52
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    $\begingroup$ @Ryan not that I know of. $\endgroup$ – Luke Mathieson May 24 '15 at 1:45
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    $\begingroup$ @Ryan and having real trouble finding one! The $P \subset NP \cap coNP$ is obviously well know, but I think most people prefer to deal with the $PH$ using the alternating quantifier definition, in which case they ignore $\Delta$ classes, so it doesn't get talked about a lot. $\endgroup$ – Luke Mathieson May 25 '15 at 1:17

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