- Alpha recursion theory
In

recursion theory , the mathematical theory of computability,**alpha recursion**(often written**α recursion**) is a generalisation ofrecursion theory to subsets ofadmissible ordinal s $alpha$. An admissible ordinal is closed under $Sigma\_1(L\_alpha)$ functions. Admissible ordinals are models ofKripke–Platek set theory . In what follows $alpha$ is considered to be fixed.The objects of study in $alpha$ recursion are subsets of $alpha$. A is said to be

**$alpha$ recursively enumerable**if it is $Sigma\_1$ definable over $L\_alpha$. A is recursive if both A and $alpha\; /\; A$ (its complement in $alpha$) are recursively enumerable.Members of $L\_alpha$ are called $alpha$ finite and play a similar role to the finite numbers in classical recursion theory.

We say R is a reduction procedure if it is recursively enumerable and every member of R is of the form $langle\; H,J,K\; angle$ where "H", "J", "K" are all α-finite.

"A" is said to be α-recusive in "B" if there exist $R\_0,R\_1$ reduction procedures such that:

: $K\; subseteq\; A\; leftrightarrow\; exists\; H:\; exists\; J:\; [,j,k>\; in\; R\_0\; wedge\; H\; subseteq\; B\; wedge\; J\; subseteq\; alpha\; /\; B\; ]\; ,$

: $K\; subseteq\; alpha\; /\; A\; leftrightarrow\; exists\; H:\; exists\; J:\; [,j,k>\; in\; R\_1\; wedge\; H\; subseteq\; B\; wedge\; J\; subseteq\; alpha\; /\; B\; ]\; .$

If "A" is recursive in "B" this is written $scriptstyle\; A\; le\_alpha\; B$. By this definition "A" is recursive in $scriptstylevarnothing$ (the

empty set ) if and only if "A" is recursive. However it should be noted that A being recursive in B is not equivalent to A being $Sigma\_1(L\_alpha\; [B]\; )$.We say "A" is regular if $forall\; eta\; in\; alpha:\; A\; cap\; eta\; in\; L\_alpha$ or in other words if every initial portion of "A" is α-finite.

**Results in $alpha$ recursion**Shore's splitting theorem: Let A be $alpha$ recursively enumerable and regular. There exist $alpha$ recursively enumerable $B\_0,B\_1$ such that $A=B\_0\; cup\; B\_1\; wedge\; B\_0\; cap\; B\_1\; =\; varnothing\; wedge\; A\; otle\_alpha\; B\_i\; (i<2).$

Shore's density theorem: Let "A", "C" be α-regular recursively enumerable sets such that $scriptstyle\; A\; <\_alpha\; C$ then there exists a regular α-recursively enumerable set "B" such that $scriptstyle\; A\; <\_alpha\; B\; <\_alpha\; C$.

**References*** Gerald Sacks, "Higher recursion theory", Springer Verlag, 1990

* Robert Soare, "Recursively Enumerable Sets and Degrees", Springer Verlag, 1987

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