Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals $$\alpha$$. An admissible set is closed under $$\Sigma_1(L_\alpha)$$ functions, where $$L_\xi$$ denotes a rank of Godel's constructible hierarchy. $$\alpha$$ is an admissible ordinal if $$L_{\alpha}$$ is a model of Kripke–Platek set theory. In what follows $$\alpha$$ is considered to be fixed.

Definitions
The objects of study in $$\alpha$$ recursion are subsets of $$\alpha$$. These sets are said to have some properties:
 * A set $$A\subseteq\alpha$$ is said to be $$\alpha$$-recursively-enumerable if it is $$ \Sigma_1$$ definable over $$L_\alpha$$, possibly with parameters from $$L_\alpha$$ in the definition.
 * A is $$\alpha$$-recursive if both A and $$\alpha \setminus A$$ (its relative complement in $$\alpha$$) are $$\alpha$$-recursively-enumerable. It's of note that $$\alpha$$-recursive sets are members of $$L_{\alpha+1}$$ by definition of $$L$$.
 * Members of $$L_\alpha$$ are called $$\alpha$$-finite and play a similar role to the finite numbers in classical recursion theory.
 * Members of $$L_{\alpha+1}$$ are called $$\alpha$$-arithmetic.

There are also some similar definitions for functions mapping $$\alpha$$ to $$\alpha$$:
 * A partial function from $$\alpha$$ to $$\alpha$$ is $$\alpha$$-recursively-enumerable, or $$\alpha$$-partial recursive, iff its graph is $$\Sigma_1$$-definable on $$(L_\alpha,\in)$$.
 * A partial function from $$\alpha$$ to $$\alpha$$ is $$\alpha$$-recursive iff its graph is $$\Delta_1$$-definable on $$(L_\alpha,\in)$$. Like in the case of classical recursion theory, any total $$\alpha$$-recursively-enumerable function $$f:\alpha\rightarrow\alpha$$ is $$\alpha$$-recursive.
 * Additionally, a partial function from $$\alpha$$ to $$\alpha$$ is $$\alpha$$-arithmetical iff there exists some $$n\in\omega$$ such that the function's graph is $$\Sigma_n$$-definable on $$(L_\alpha,\in)$$.

Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:
 * The functions $$\Delta_0$$-definable in $$(L_\alpha,\in)$$ play a role similar to those of the primitive recursive functions.

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

A is said to be α-recursive in B if there exist $$R_0,R_1$$ reduction procedures such that:


 * $$K \subseteq A \leftrightarrow \exists H: \exists J:[\langle H,J,K \rangle \in R_0 \wedge H \subseteq B \wedge J \subseteq \alpha / B ],$$


 * $$K \subseteq \alpha / A \leftrightarrow \exists H: \exists J:[\langle H,J,K \rangle \in R_1 \wedge H \subseteq B \wedge J \subseteq \alpha / B ].$$

If A is recursive in B this is written $$\scriptstyle A \le_\alpha B$$. By this definition A is recursive in $$\scriptstyle\varnothing$$ (the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being $$\Sigma_1(L_\alpha[B])$$.

We say A is regular if $$\forall \beta \in \alpha: A \cap \beta \in L_\alpha$$ or in other words if every initial portion of A is α-finite.

Work in α recursion
Shore's splitting theorem: Let A be $$\alpha$$ recursively enumerable and regular. There exist $$\alpha$$ recursively enumerable $$B_0,B_1$$ such that $$A=B_0 \cup B_1 \wedge B_0 \cap B_1 = \varnothing \wedge A \not\le_\alpha B_i (i<2).$$

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 $$\scriptstyle A <_\alpha B <_\alpha C$$.

Barwise has proved that the sets $$\Sigma_1$$-definable on $$L_{\alpha^+}$$ are exactly the sets $$\Pi_1^1$$-definable on $$L_\alpha$$, where $$\alpha^+$$ denotes the next admissible ordinal above $$\alpha$$, and $$\Sigma$$ is from the Levy hierarchy.

There is a generalization of limit computability to partial $$\alpha\to\alpha$$ functions.

A computational interpretation of $$\alpha$$-recursion exists, using "$$\alpha$$-Turing machines" with a two-symbol tape of length $$\alpha$$, that at limit computation steps take the limit inferior of cell contents, state, and head position. For admissible $$\alpha$$, a set $$A\subseteq\alpha$$ is $$\alpha$$-recursive iff it is computable by an $$\alpha$$-Turing machine, and $$A$$ is $$\alpha$$-recursively-enumerable iff $$A$$ is the range of a function computable by an $$\alpha$$-Turing machine.

A problem in α-recursion theory which is open (as of 2019) is the embedding conjecture for admissible ordinals, which is whether for all admissible $$\alpha$$, the automorphisms of the $$\alpha$$-enumeration degrees embed into the automorphisms of the $$\alpha$$-enumeration degrees.

Relationship to analysis
Some results in $$\alpha$$-recursion can be translated into similar results about second-order arithmetic. This is because of the relationship $$L$$ has with the ramified analytic hierarchy, an analog of $$L$$ for the language of second-order arithmetic, that consists of sets of integers.

In fact, when dealing with first-order logic only, the correspondence can be close enough that for some results on $$L_\omega=\textrm{HF}$$, the arithmetical and Levy hierarchies can become interchangeable. For example, a set of natural numbers is definable by a $$\Sigma_1^0$$ formula iff it's $$\Sigma_1$$-definable on $$L_\omega$$, where $$\Sigma_1$$ is a level of the Levy hierarchy. More generally, definability of a subset of ω over HF with a $$\Sigma_n$$ formula coincides with its arithmetical definability using a $$\Sigma_n^0$$ formula.