Alpha recursion theory

In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals ${\displaystyle \alpha }$. An admissible set is closed under ${\displaystyle \Sigma _{1}(L_{\alpha })}$ functions, where ${\displaystyle L_{\xi }}$ denotes a rank of Godel's constructible hierarchy. ${\displaystyle \alpha }$ is an admissible ordinal if ${\displaystyle L_{\alpha }}$ is a model of Kripke–Platek set theory. In what follows ${\displaystyle \alpha }$ is considered to be fixed.

Definitions

The objects of study in ${\displaystyle \alpha }$ recursion are subsets of ${\displaystyle \alpha }$. These sets are said to have some properties:

• A set ${\displaystyle A\subseteq \alpha }$ is said to be ${\displaystyle \alpha }$-recursively-enumerable if it is ${\displaystyle \Sigma _{1}}$ definable over ${\displaystyle L_{\alpha }}$, possibly with parameters from ${\displaystyle L_{\alpha }}$ in the definition.^[1]
• A is ${\displaystyle \alpha }$-recursive if both A and ${\displaystyle \alpha \setminus A}$ (its relative complement in ${\displaystyle \alpha }$) are ${\displaystyle \alpha }$-recursively-enumerable. It's of note that ${\displaystyle \alpha }$-recursive sets are members of ${\displaystyle L_{\alpha +1}}$ by definition of ${\displaystyle L}$.
• Members of ${\displaystyle L_{\alpha }}$ are called ${\displaystyle \alpha }$-finite and play a similar role to the finite numbers in classical recursion theory.
• Members of ${\displaystyle L_{\alpha +1}}$ are called ${\displaystyle \alpha }$-arithmetic. ^[2]

There are also some similar definitions for functions mapping ${\displaystyle \alpha }$ to ${\displaystyle \alpha }$:^[3]

• A partial function from ${\displaystyle \alpha }$ to ${\displaystyle \alpha }$ is ${\displaystyle \alpha }$-recursively-enumerable, or ${\displaystyle \alpha }$-partial recursive,^[4] iff its graph is ${\displaystyle \Sigma _{1}}$-definable on ${\displaystyle (L_{\alpha },\in )}$.
• A partial function from ${\displaystyle \alpha }$ to ${\displaystyle \alpha }$ is ${\displaystyle \alpha }$-recursive iff its graph is ${\displaystyle \Delta _{1}}$-definable on ${\displaystyle (L_{\alpha },\in )}$. Like in the case of classical recursion theory, any total ${\displaystyle \alpha }$-recursively-enumerable function ${\displaystyle f:\alpha \rightarrow \alpha }$ is ${\displaystyle \alpha }$-recursive.
• Additionally, a partial function from ${\displaystyle \alpha }$ to ${\displaystyle \alpha }$ is ${\displaystyle \alpha }$-arithmetical iff there exists some ${\displaystyle n\in \omega }$ such that the function's graph is ${\displaystyle \Sigma _{n}}$-definable on ${\displaystyle (L_{\alpha },\in )}$.

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 ${\displaystyle \Delta _{0}}$-definable in ${\displaystyle (L_{\alpha },\in )}$ play a role similar to those of the primitive recursive functions.^[3]

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

A is said to be α-recursive in B if there exist ${\displaystyle R_{0},R_{1}}$ reduction procedures such that:

${\displaystyle K\subseteq A\leftrightarrow \exists H:\exists J:[\langle H,J,K\rangle \in R_{0}\wedge H\subseteq B\wedge J\subseteq \alpha /B],}$

${\displaystyle K\subseteq \alpha /A\leftrightarrow \exists H:\exists J:[\langle H,J,K\rangle \in R_{1}\wedge H\subseteq B\wedge J\subseteq \alpha /B].}$

If A is recursive in B this is written ${\displaystyle \scriptstyle A\leq _{\alpha }B}$. By this definition A is recursive in ${\displaystyle \scriptstyle \varnothing }$ (the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being ${\displaystyle \Sigma _{1}(L_{\alpha }[B])}$.

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

Work in α recursion

Shore's splitting theorem: Let A be ${\displaystyle \alpha }$ recursively enumerable and regular. There exist ${\displaystyle \alpha }$ recursively enumerable ${\displaystyle B_{0},B_{1}}$ such that ${\displaystyle A=B_{0}\cup B_{1}\wedge B_{0}\cap B_{1}=\varnothing \wedge A\not \leq _{\alpha }B_{i}(i<2).}$

Shore's density theorem: Let A, C be α-regular recursively enumerable sets such that ${\displaystyle \scriptstyle A<_{\alpha }C}$ then there exists a regular α-recursively enumerable set B such that ${\displaystyle \scriptstyle A<_{\alpha }B<_{\alpha }C}$.

Barwise has proved that the sets ${\displaystyle \Sigma _{1}}$-definable on ${\displaystyle L_{\alpha ^{+}}}$ are exactly the sets ${\displaystyle \Pi _{1}^{1}}$-definable on ${\displaystyle L_{\alpha }}$, where ${\displaystyle \alpha ^{+}}$ denotes the next admissible ordinal above ${\displaystyle \alpha }$, and ${\displaystyle \Sigma }$ is from the Levy hierarchy.^[5]

There is a generalization of limit computability to partial ${\displaystyle \alpha \to \alpha }$ functions.^[6]

A computational interpretation of ${\displaystyle \alpha }$-recursion exists, using "${\displaystyle \alpha }$-Turing machines" with a two-symbol tape of length ${\displaystyle \alpha }$, that at limit computation steps take the limit inferior of cell contents, state, and head position. For admissible ${\displaystyle \alpha }$, a set ${\displaystyle A\subseteq \alpha }$ is ${\displaystyle \alpha }$-recursive iff it is computable by an ${\displaystyle \alpha }$-Turing machine, and ${\displaystyle A}$ is ${\displaystyle \alpha }$-recursively-enumerable iff ${\displaystyle A}$ is the range of a function computable by an ${\displaystyle \alpha }$-Turing machine. ^[7]

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 ${\displaystyle \alpha }$, the automorphisms of the ${\displaystyle \alpha }$-enumeration degrees embed into the automorphisms of the ${\displaystyle \alpha }$-enumeration degrees.^[8]

Relationship to analysis

Some results in ${\displaystyle \alpha }$-recursion can be translated into similar results about second-order arithmetic. This is because of the relationship ${\displaystyle L}$ has with the ramified analytic hierarchy, an analog of ${\displaystyle L}$ for the language of second-order arithmetic, that consists of sets of integers.^[9]

In fact, when dealing with first-order logic only, the correspondence can be close enough that for some results on ${\displaystyle L_{\omega }={\textrm {HF}}}$, the arithmetical and Levy hierarchies can become interchangeable. For example, a set of natural numbers is definable by a ${\displaystyle \Sigma _{1}^{0}}$ formula iff it's ${\displaystyle \Sigma _{1}}$-definable on ${\displaystyle L_{\omega }}$, where ${\displaystyle \Sigma _{1}}$ is a level of the Levy hierarchy.^[10] More generally, definability of a subset of ω over HF with a ${\displaystyle \Sigma _{n}}$ formula coincides with its arithmetical definability using a ${\displaystyle \Sigma _{n}^{0}}$ formula.^[11]

References

• Gerald Sacks, Higher recursion theory, Springer Verlag, 1990 https://projecteuclid.org/euclid.pl/1235422631
• Robert Soare, Recursively Enumerable Sets and Degrees, Springer Verlag, 1987 https://projecteuclid.org/euclid.bams/1183541465
• Keith J. Devlin, An introduction to the fine structure of the constructible hierarchy (p.38), North-Holland Publishing, 1974
• J. Barwise, Admissible Sets and Structures. 1975

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