polynomial hierarchy

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

We let ${\mathrm{\Sigma }}_0^p={\mathrm{\Pi }}_0^p={\mathrm{\Delta }}_0^p=𝒫$, then ${\mathrm{\Delta }}_{i+1}^p={𝒫}^{{\mathrm{\Sigma }}_i^p}$ and ${\mathrm{\Sigma }}_{i+1}^p=𝒩{𝒫}^{{\mathrm{\Sigma }}_i^p}$. Define ${\mathrm{\Pi }}_i^p$ to be $co{\mathrm{\Sigma }}_i^p$.

For instance ${\mathrm{\Sigma }}_1^p=𝒩{𝒫}^{𝒫}=𝒩𝒫$.

The complexity class $𝒫\mathscr{H}={\bigcup }_{i\in \mathbb{N}}{\mathrm{\Sigma }}_i^p$.

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
Canonical name	PolynomialHierarchy
Date of creation	2013-03-22 13:02:11
Last modified on	2013-03-22 13:02:11
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	6
Author	Henry (455)
Entry type	Definition
Classification	msc 68Q15
Defines	PH