arithmetical hierarchy

The arithmetical hierarchy is a hierarchy of either (depending on the context) formulas or relations. The relations of a particular level of the hierarchy are exactly the relations defined by the formulas of that level, so the two uses are essentially the same.

The first level consists of formulas with only bounded quantifiers, the corresponding relations are also called the Primitive Recursive relations (this definition is equivalent to the definition from computer science). This level is called any of ${\mathrm{\Delta }}_0^0$, ${\mathrm{\Sigma }}_0^0$ and ${\mathrm{\Pi }}_0^0$, depending on context.

A formula $\varphi$ is ${\mathrm{\Sigma }}_n^0$ if there is some ${\mathrm{\Delta }}_0^0$ formula $\psi$ such that $\varphi$ can be written:

$$\varphi (\overrightarrow{k})=\exists x_1\forall x_2\mathrm{\cdots }Q{x}_n\psi (\overrightarrow{k},\overrightarrow{x})$$

$$\text{ where }Q\text{ is either }\forall \text{ or }\exists \text{, whichever maintains the pattern of alternating quantifiers}$$

The ${\mathrm{\Sigma }}_1^0$ relations are the same as the Recursively Enumerable relations.

Similarly, $\varphi$ is a ${\mathrm{\Pi }}_n^0$ relation if there is some ${\mathrm{\Delta }}_0^0$ formula $\psi$ such that:

$$\varphi (\overrightarrow{k})=\forall x_1\exists x_2\mathrm{\cdots }Q{x}_n\psi (\overrightarrow{k},\overrightarrow{x})$$

$$\text{ where }Q\text{ is either }\forall \text{ or }\exists \text{, whichever maintains the pattern of alternating quantifiers}$$

A formula is ${\mathrm{\Delta }}_n^0$ if it is both ${\mathrm{\Sigma }}_n^0$ and ${\mathrm{\Pi }}_n^0$. Since each ${\mathrm{\Sigma }}_n^0$ formula is just the negation of a ${\mathrm{\Pi }}_n^0$ formula and vice-versa, the ${\mathrm{\Sigma }}_n^0$ relations are the complements of the ${\mathrm{\Pi }}_n^0$ relations.

The relations in ${\mathrm{\Delta }}_1^0={\mathrm{\Sigma }}_1^0\cap {\mathrm{\Pi }}_1^0$ are the Recursive relations.

Higher levels on the hierarchy correspond to broader and broader classes of relations. A formula or relation which is ${\mathrm{\Sigma }}_n^0$ (or, equivalently, ${\mathrm{\Pi }}_n^0$) for some integer $n$ is called arithmetical.

The superscript $0$ is often omitted when it is not necessary to distinguish from the analytic hierarchy.

Functions can be described as being in one of the levels of the hierarchy if the graph of the function is in that level.

Title	arithmetical hierarchy
Canonical name	ArithmeticalHierarchy
Date of creation	2013-03-22 12:55:11
Last modified on	2013-03-22 12:55:11
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	19
Author	CWoo (3771)
Entry type	Definition
Classification	msc 03B10
Synonym	arithmetic hierarchy
Synonym	arithmetic
Synonym	arithmetical
Synonym	arithmetic formula
Synonym	arithmetical formulas
Related topic	AnalyticHierarchy
Defines	sigma n
Defines	sigma-n
Defines	pi n
Defines	pi-n
Defines	delta n
Defines	delta-n
Defines	recursive
Defines	recursively enumerable
Defines	delta-0
Defines	delta 0
Defines	delta-1
Defines	delta 1
Defines	arithmetical