{\alpha} degrees as an automorphism base for the {\alpha}-enumeration degrees
D\'avid Natingga

TL;DR
This paper generalizes Selman's Theorem from classical Computability Theory to the setting of lpha-Computability Theory, establishing conditions under which lpha-degrees form an automorphism base, extending known results to broader ordinal contexts.
Contribution
The paper extends Selman's Theorem to lpha-Computability Theory, providing a full generalization for regular cardinals and partial results for admissible ordinals.
Findings
lpha-degrees form an automorphism base under certain conditions
Full generalization for regular cardinals lpha
Partial results for admissible ordinals lpha
Abstract
Selman's Theorem in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: . This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the -Computability Theory to obtain the full generalization when is a regular cardinal and partial results for a general admissible ordinal .
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · semigroups and automata theory
degrees as an automorphism base for the -enumeration degrees
Dávid Natingga
Abstract
Selman’s Theorem [10] in classical Computability Theory gives a characterization of the enumeration reducibility for arbitrary sets in terms of the enumeration reducibility on the total sets: . This theorem implies directly that the Turing degrees are an automorphism base of the enumeration degrees. We lift the classical proof to the setting of the -Computability Theory to obtain the full generalization when is a regular cardinal and partial results for a general admissible ordinal .
1 -Computability Theory
-Computability Theory is the study of the definability theory over Gödel’s where is an admissible ordinal. One can think of equivalent definitions on Turing machines with a transfinite tape and time [3] [4] [5] [6] or on generalized register machines [7]. Recommended references for this section are [9], [1], [8] and [2].
Classical Computability Theory is -Computability Theory where .
1.1 Gödel’s Constructible Universe
Definition 1.1**.**
(Gödel’s Constructible Universe)
Define Gödel’s constructible universe as where , is a limit ordinal and:
,
and is first-order definable over ,
.
1.2 Admissibility
Definition 1.2**.**
(Admissible ordinal[1])
An ordinal is admissible (admissible for short) iff is a limit ordinal and satisfies -collection: where is the -th level of the Gödel’s Constructible Hierarchy (1.1).
Example 1.3**.**
(Examples of admissible ordinals [1] [12])
- •
- Church-Kleene , the first non-computable ordinal
- •
every stable ordinal (i.e. ), e.g. - the least ordinal which is not an order type of a subset of , \nth1 stable ordinal
- •
every infinite cardinal in a transitive model of
1.3 Basic concepts
Definition 1.4**.**
A set is -finite iff .
Definition 1.5**.**
(-computability and computable enumerability)
- •
A function is -computable iff is definable.
- •
A set is -computably enumerable (-c.e.) iff .
- •
A set is -computable iff iff and .
Proposition 1.6**.**
[1] There exists a -definable bijection . ∎
Let denote an -finite set . The next proposition establishes that we can also index pairs and other finite vectors from by an index in .
Proposition 1.7**.**
[8] For every , there is a -definable bijection : (n-fold product). ∎
Similarly, we can index -c.e., -computable sets by an index in . Let denote an -c.e. set with an index .
Proposition 1.8**.**
(-finite union of -finite sets111From [9] p162.)
-finite union of -finite sets is -finite, i.e. if , then . ∎
1.4 Enumeration reducibility
The generalization of the enumeration reducibility corresponds to two different notions - weak -enumeration reducibility and -enumeration reducibility.
Definition 1.9**.**
(Weak -enumeration reducibility)
is weakly -enumeration reducible to denoted as iff st . The set is called a weak -enumeration operator.
Definition 1.10**.**
(-enumeration reducibility)
is -enumeration reducible to denoted as iff st .
Denote the fact that reduces to via as .
Fact 1.11**.**
(Transitivity)
The -enumeration reducibility is transitive. But in general the weak -enumeration reducibility is not transitive.
1.5 Properties of -enumeration operator
Fact 1.12**.**
If , then .
Fact 1.13**.**
(Monotonicity)
. ∎
Proposition 1.14**.**
(Witness property)
If , then .
Proof.
Note . Thus if , then st and so . Taking to be concludes the proof. ∎
1.6 Totality
Definition 1.15**.**
222From [1] p8.
The computable join of sets denoted is defined to be
.
The computable join satisfies the usual properties of the case .
The generalization of the Turing reducibility corresponds to two different notions - weak reducibility and reducibility.
Definition 1.16**.**
(Total reducibilities)
- •
is -reducible to denoted as iff .
- •
is weakly -reducible to denoted as iff .
Definition 1.17**.**
(Total set)
A subset is total iff iff .
1.7 Degree Theory
Definition 1.18**.**
(Degrees)
- •
is a set of -degrees.
- •
is a set of -enumeration degrees.
Induce on and by and respectively.
Fact 1.19**.**
(Embedding of the total degrees)
embeds into via , .
Definition 1.20**.**
(Total degrees)
Let be the embedding from above. The total -enumeration degrees are the image of , i.e. .
1.8 Megaregularity
Megaregularity of a set measures the amount of the admissibility of a structure structure , i.e. a structure extended by a predicate with an access to .
Note 1.21**.**
(Formula with a positive/negative parameter)
- •
Let denote a set, its enumeration, the enumeration of its complement .
- •
Denote by the class of formulas with a parameter or in .
- •
A formula is iff occurs in only positively, i.e. there is no negation up the formula tree above the literal .
- •
Similarly, a formula is iff occurs in only negatively.
Definition 1.22**.**
(Megaregularity)
Let and add as a predicate to the language for the structure .
- •
Then is -megaregular iff is admissible iff the structure is admissible, i.e. every definable function satisfies the replacement axiom: .
- •
is positively -megaregular iff is -megaregular.
- •
is negatively -megaregular iff is -megaregular.
If clear from the context, we just say megaregular instead of -megaregular.
A person familiar with the notion of hyperregularity shall note that a set is megaregular iff it is regular and hyperregular.
Fact 1.23**.**
(Megaregularity and definability)
- •
is megaregular,
- •
is megaregular.
Proposition 1.24**.**
Let be megaregular and let . Then: iff and is bounded by some .
Proof.
direction is clear. For the other direction, assume that and for some . WLOG let and let . Define a function by . Since , the function is definable. By the megaregularity of , we have that as required. ∎
Corollary 1.25**.**
(Megaregularity closure and degree invariance)
i) If and megaregular, then megaregular.
ii) If , then megaregular iff megaregular .
iii) If and megaregular, then megaregular.
iv) If , then megaregular iff megaregular . ∎
Proposition 1.26**.**
(Correspondence between the -enumeration reducibilities)
We have the following implication diagram:
{A\leq_{w\alpha e}B}$${A\leq_{\alpha e}B}if megaregularalways
∎
2 Selman’s theorem
We generalize the theorem of Selman present in [10].
Definition 2.1**.**
(Odd enumeration and -finite part)
- •
Let . The total function is an odd enumeration of iff .
- •
odd -finite part is a function for st .
- •
Let denote the order type of , i.e. .
If is an initial segment of , we have . If is a function, then and so . If is a odd -finite part, then there is an odd enumeration of st .
Lemma 2.2**.**
Let be an odd enumeration of . Then .
Proof.
We have via . ∎
Lemma 2.3**.**
Assume that . Then .
Proof.
Note . Let via . Then via . ∎
Definition 2.4**.**
(Weak halting set)
The weak halting set is defined as .
Proposition 2.5**.**
333Generalized from [11] for .
Let and . Assume that is megaregular. Then there exists an odd enumeration of st and .
Proof.
Construction
Note that and as . We construct a sequence of odd -finite parts in many stages st:
In the end, the desired function is defined as .
Let . If is a limit ordinal, then let . Now assume that has been constructed, then at the stage construct :
- •
Stage :
Set where and is the concatenation of [math] and . E.g. if , then .
- •
Stage :
Use and to define set as
is a odd -finite part and .
As , so . Thus we have two cases:
- –
Case : Then let be the minimal from . Note for some where is a odd -finite part.
- –
Case : Then let for some .
Verification
By the two cases above we have for all :
[TABLE]
Note that is an odd enumeration of . This is ensured by stages . If , then a pair is added to for some at the stage at latest.
Moreover, . For suppose not, then . Hence by lemma 2.3 and so there is st and thus:
[TABLE]
Consider the stage . Let . Note .
- •
Case 1: using statement 2. By the witness property of an -enumeration operator there exists odd -finite part of st . So . Hence . Also by statement 1 we have . This is a contradiction.
- •
Case 2: using statement 2. By the monotonicity of an -enumeration operator there is no odd -finite part of st . So . Hence . By statement 1 we have . This is a contradiction.
Hence in any case as required.
Let be defined by where . During the construction we use the oracle , hence . We show that is well-defined and that is -finite at a limit stage . Let be a limit stage. Then since , and by the megaregularity of the oracle . Hence is -finite as required. Therefore as needed.
Note that since the construction of uses the oracle . By that and the megaregularity of , also must by megaregular. By lemma 2.2 we have . By the megaregularity of , we have as required. ∎
Theorem 2.6**.**
(Selman’s theorem for admissible ordinals)
Let and let be megaregular. Then:
Part of the following proof is adapted from the classical case present in [11].
Proof.
direction is by the transitivity of .
For the direction assume that . We want to show that . Assume not, then and so as is megaregular. Then by 2.5 and the megaregularity of there exists a total function st , but . But is total and so which is a contradiction to the statement . Hence as required. ∎
Corollary 2.7**.**
(Selman’s theorem for regular cardinals)
Let be a regular cardinal. Then for any we have:
Proof.
If is a regular cardinal, then every subset of is megaregular. Hence is megaregular. The remaining proof of the corollary follows from 2.6. ∎
3 as an automorphism base for
Definition 3.1**.**
(Automorphism base)
TFAE:
- •
The subset is an automorphism base of the model .
- •
- •
Theorem 3.2**.**
Assume is a regular cardinal. The total degrees are an automorphism base for .
Proof.
-
Assume is a regular cardinal.
-
by [math] and 2.7.
-
by 1.
-
Assume .
-
by 3.
-
Assume .
-
Assume .
-
by 4.
-
by 5, 7.
-
by 2, 8.
-
by 6, 9.
-
is an automorphism base for by 3, 5, 10. ∎
4 Acknowledgements
The author would like to thank Mariya Soskova for the explanation of the proof in the classical case, i.e. .
The author was supported by Hausdorff Research Institute for Mathematics during Hausdorff Trimester Program Types, Sets and Constructions.
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