Alpha recursion theory - Wikiwand
In recursion theory, α recursion theory is a generalisation of recursion theory to subsets of admissible ordinals . An admissible set is closed under
functions, where
denotes a rank of Godel's constructible hierarchy.
is an admissible ordinal if
is a model of Kripke–Platek set theory. In what follows
is considered to be fixed.
The objects of study in recursion are subsets of
. These sets are said to have some properties:
There are also some similar definitions for functions mapping to
:[3]
Additional connections between recursion theory and α recursion theory can be drawn, although explicit definitions may not have yet been written to formalize them:
We say R is a reduction procedure if it is recursively enumerable and every member of R is of the form
where H, J, K are all α-finite.
A is said to be α-recursive in B if there exist reduction procedures such that:
If A is recursive in B this is written . By this definition A is recursive in
(the empty set) if and only if A is recursive. However A being recursive in B is not equivalent to A being
.
We say A is regular if or in other words if every initial portion of A is α-finite.
Shore's splitting theorem: Let A be recursively enumerable and regular. There exist
recursively enumerable
such that
Shore's density theorem: Let A, C be α-regular recursively enumerable sets such that then there exists a regular α-recursively enumerable set B such that
.
Barwise has proved that the sets -definable on
are exactly the sets
-definable on
, where
denotes the next admissible ordinal above
, and
is from the Levy hierarchy.[5]
There is a generalization of limit computability to partial functions.[6]
A computational interpretation of -recursion exists, using "
-Turing machines" with a two-symbol tape of length
, that at limit computation steps take the limit inferior of cell contents, state, and head position. For admissible
, a set
is
-recursive iff it is computable by an
-Turing machine, and
is
-recursively-enumerable iff
is the range of a function computable by an
-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 , the automorphisms of the
-enumeration degrees embed into the automorphisms of the
-enumeration degrees.[8]
Some results in -recursion can be translated into similar results about second-order arithmetic. This is because of the relationship
has with the ramified analytic hierarchy, an analog of
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 , the arithmetical and Levy hierarchies can become interchangeable. For example, a set of natural numbers is definable by a
formula iff it's
-definable on
, where
is a level of the Levy hierarchy.[10] More generally, definability of a subset of ω over HF with a
formula coincides with its arithmetical definability using a
formula.[11]
- 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