Alpha recursion theory: Difference between revisions

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

==Definitions==

The objects of study in <math>\alpha</math> recursion are subsets of <math>\alpha</math>. These sets are said to have some properties:

*A set <math>A\subseteq\alpha</math> is said to be <math>\alpha</math>-recursively-enumerable if it is <math> \Sigma_1</math> definable over <math>L_\alpha</math>, possibly with parameters from <math>L_\alpha</math> in the definition.<ref>P. Koepke, B. Seyfferth, [https://www.math.uni-bonn.de/people/koepke/Preprints/Ordinal_machines_and_admissible_recursion_theory.pdf Ordinal machines and admissible recursion theory (preprint)] (2009, p.315). Accessed October 12, 2021</ref>

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There are also some similar definitions for functions mapping <math>\alpha</math> to <math>\alpha</math>:<ref name="relconstr">Srebrny, Marian, [http://matwbn.icm.edu.pl/ksiazki/fm/fm96/fm96114.pdf Relatively constructible transitive models] (1975, p.165). Accessed 21 October 2021.</ref>

*A [[partial function]] mappingfrom <math>\alpha</math> to <math>\alpha</math> is '''<math>\alpha</math>-recursively-enumerable''', or '''<math>\alpha</math>-partial recursive''',<ref>W. Richter, P. Aczel, "[https://www.duo.uio.no/bitstream/handle/10852/44063/1973-13.pdf Inductive Definitions and Reflecting Properties of Admissible Ordinals]" (1974), p.30. Accessed 7 February 2023.</ref> iff its graph is <math>\Sigma_1</math>-definable inon <math>(L_\alpha,\in)</math>.

*A partial function mappingfrom <math>\alpha</math> to <math>\alpha</math> is '''<math>\alpha</math>-recursive''' iff its graph is <math>\Delta_1</math>-definable inon <math>(L_\alpha,\in)</math>. Like in the case of classical recursion theory, any ''total'' <math>\alpha</math>-recursively-enumerable function <math>f:\alpha\rightarrow\alpha</math> is <math>\alpha</math>-recursive.

*Additionally, a partial function mappingfrom <math>\alpha</math> to <math>\alpha</math> is '''<math>\alpha</math>-arithmetical''' iff there exists some <math>n\in\omega</math> such that the function's graph is <math>\Sigma_n</math>-definable inon <math>(L_\alpha,\in)</math>.

Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:

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

==ResultsWork in α recursion==

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

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Barwise has proved that the sets <math>\Sigma_1</math>-definable on <math>L_{\alpha^+}</math> are exactly the sets <math>\Pi_1^1</math>-definable on <math>L_\alpha</math>, where <math>\alpha^+</math> denotes the next admissible ordinal above <math>\alpha</math>, and <math>\Sigma</math> is from the [[Levy hierarchy]].<ref>T. Arai, [https://www.sciencedirect.com/science/article/pii/S0168007203000204 Proof theory for theories of ordinals - I: recursively Mahlo ordinals] (1998). p.2</ref>

There is a generalization of [[Computability in the limit|limit computability]] to partial <math>\alpha\to\alpha</math> functions.<math>\alpha</math>.<ref>S. G. Simpson, "Degree Theory on Admissible Ordinals", pp.170--171. Appearing in J. Fenstad, P. Hinman, ''Generalized Recursion Theory: Proceedings of the 1972 Oslo Symposium'' (1974), ISBN 0 7204 22760.</ref>

A computational interpretation of <math>\alpha</math>-recursion exists, using "<math>\alpha</math>-Turing machines" with a two-symbol tape of length <math>\alpha</math>, that at limit computation steps take the [[limit inferior]] of cell contents, state, and head position. For admissible <math>\alpha</math>, a set <math>A\subseteq\alpha</math> is <math>\alpha</math>-recursive iff it is computable by an <math>\alpha</math>-Turing machine, and <math>A</math> is <math>\alpha</math>-recursively-enumerable iff <math>A</math> is the range of a function computable by an <math>\alpha</math>-Turing machine. <ref>P. Koepke, B. Seyfferth, "[https://www.math.uni-bonn.de/people/koepke/Preprints/Ordinal_machines_and_admissible_recursion_theory.pdf Ordinal machines and admissible recursion theory]". Annals of Pure and Applied Logic vol. 160 (2009), pp.310--318.</ref>

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 <math>\alpha</math>, the automorphisms of the <math>\alpha</math>-enumeration degrees embed into the automorphisms of the <math>\alpha</math>-enumeration degrees.<ref>D. Natingga, ''Embedding Theorem for the automorphism group of the α-enumeration degrees'' (p.155), PhD thesis, 2019.</ref>

==Relationship to analysis==