# polynomial hierarchy

The polynomial hierarchy is a hierarchy of complexity classes  generalizing the relationship between $\mathcal{P}$ and $\mathcal{NP}$.

We let $\Sigma^{p}_{0}=\Pi^{p}_{0}=\Delta^{p}_{0}=\mathcal{P}$, then $\Delta^{p}_{i+1}=\mathcal{P}^{\Sigma^{p}_{i}}$ and $\Sigma^{p}_{i+1}=\mathcal{NP}^{\Sigma^{p}_{i}}$. Define $\Pi^{p}_{i}$ to be $co\Sigma^{p}_{i}$.

For instance $\Sigma^{p}_{1}=\mathcal{NP}^{\mathcal{P}}=\mathcal{NP}$.

The complexity class $\mathcal{PH}=\bigcup_{i\in\mathbb{N}}\Sigma^{p}_{i}$.

The polynomial hierarchy is closely related to the arithmetical hierarchy; indeed, an alternate definition is almost identical to the definition of the arithmetical hierarchy but stricter rules on what quantifiers  can be used.

When there is no risk of confusion with the arithmetical hierarchy, the superscript $p$ can be dropped.

Title polynomial hierarchy PolynomialHierarchy 2013-03-22 13:02:11 2013-03-22 13:02:11 Henry (455) Henry (455) 6 Henry (455) Definition msc 68Q15 PH