## You are here

Homeanalytic hierarchy

## Primary tabs

# analytic hierarchy

The *analytic hierarchy* is a hierarchy of either (depending on context) formulas or relations similar to the arithmetical hierarchy. It is essentially the second order equivalent. Like the arithmetical hierarchy, the relations in each level are exactly the relations defined by the formulas of that level.

The first level can be called $\Delta^{1}_{0}$, $\Delta^{1}_{1}$, $\Sigma^{1}_{0}$, or $\Pi^{1}_{0}$, and consists of the arithmetical formulas or relations.

A formula $\phi$ is $\Sigma^{1}_{n}$ if there is some arithmetical formula $\psi$ such that:

$\phi(\vec{k})=\exists X_{1}\forall X_{2}\cdots QX_{n}\psi(\vec{k},\vec{X}_{n})$ |

$\text{ where }Q\text{ is either }\forall\text{ or }\exists\text{, whichever % maintains the pattern of alternating quantifiers, and each }X_{i}\text{ is a % set variable (that is, second order)}$ |

Similarly, a formula $\phi$ is $\Pi^{1}_{n}$ if there is some arithmetical formula $\psi$ such that:

$\phi(\vec{k})=\forall X_{1}\exists X_{2}\cdots QX_{n}\psi(\vec{k},\vec{X}_{n})$ |

$\text{ where }Q\text{ is either }\forall\text{ or }\exists\text{, whichever % maintains the pattern of alternating quantifiers, and each }X_{i}\text{ is a % set variable (that is, second order)}$ |

Related:

ArithmeticalHierarchy

Synonym:

analytical hierarchy

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

03B15*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith