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Weak Truth Table Degrees of Structures David Belanger 1 April 2012 at UWMadison EMAIL : dbelanger@math.cornell.edu Department of Mathematics Cornell University David Belanger wtt Degrees of Structures Preliminaries David Belanger wtt

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Weak Truth Table Degrees of Structures

David Belanger 1 April 2012 at UW–Madison EMAIL: dbelanger@math.cornell.edu Department of Mathematics Cornell University

David Belanger wtt Degrees of Structures

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Preliminaries

David Belanger wtt Degrees of Structures

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Preliminaries

Recall: Definition

1 A set X ⊆ N is Turing reducible to a second set Y ⊆ N if

there is an algorithm that can use Y to decide membership in X.

2 The Turing degree degT(X) of a set X is the class of all

subsets of N that are mutually Turing reducible with X.

3 A set X is weak truth table reducible to a second set Y if

there is an algorithm that can use a computably-bounded piece of Y to decide membership in X.

4 The weak truth table degree degwtt(X) of a set X is defined

in the analogous way.

David Belanger wtt Degrees of Structures

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Preliminaries

Definition

1 A structure is a first-order structure, with universe N, on a

finite or countable alphabet (R0, R1, R2, . . .) of relations. The arities of Rk are computable as a function of k. We identify a structure A with its atomic diagram D(A) = {k, a1, a2, . . . , an : A | = Rk(a1, . . . , an)}. Note that this is a subset of N.

2 The Turing degree of A, written degT(A), is the Turing

degree of D(A).

3 The wtt degree of A is defined similarly. David Belanger wtt Degrees of Structures

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Preliminaries

We defined degT(A) as the Turing degree of the atomic diagram

f A. Typically, there is a second structure B, isomorphic to A,

such that degT(B) = degT(A). Definition

1 The Turing degree spectrum of A is the family of all Turing

degrees of isomorphic copies of A. specT(A) = {degT(B) : B ∼ = A}.

2 The wtt degree spectrum of A is

specwtt(A) = {degwtt(B) : B ∼ = A}.

David Belanger wtt Degrees of Structures

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Some motivating examples from the Turing case

Theorem (Knight 86) If specT(A) is contained in a countable union

n Cn of upward

cones, then specT(A) is contained in a particular Cn0. Theorem (Hirschfeldt–Khoussainov–Shore–Slinko 02) If A is a nontrivial structure, then there exists a graph G with universe N such that specT(G) = specT(A). Theorem (Knight 86)

1 specT(A) is a singleton if and only if A is trivial. 2 specT(A) is upward closed in the Turing degrees if and only if

A is not trivial.

David Belanger wtt Degrees of Structures

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Some motivating examples from the Turing case

Theorem (Knight 86) If specT(A) is contained in a countable union

n Cn of upward

cones, then specT(A) is contained in a particular Cn0. Theorem (Hirschfeldt–Khoussainov–Shore–Slinko 02) If A is a nontrivial structure, then there exists a graph G with universe N such that specT(G) = specT(A). Theorem (Knight 86)

1 specT(A) is a singleton if and only if A is trivial. 2 specT(A) is upward closed in the Turing degrees if and only if

A is not trivial. A structure A with universe N is trivial if there exists a finite subset S ⊂ N such that any permutation of N fixing S pointwise is an automorphism of A.

David Belanger wtt Degrees of Structures

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Big questions

Questions

I. What can be said about specwtt(A) as a family of wtt

degrees?

David Belanger wtt Degrees of Structures

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Big questions

Questions

I. What can be said about specwtt(A) as a family of wtt

degrees?

II. What classes of reals can be written as (specwtt(A)) for a

structure A?

David Belanger wtt Degrees of Structures

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Big questions

Questions

I. What can be said about specwtt(A) as a family of wtt

degrees?

II. What classes of reals can be written as (specwtt(A)) for a

structure A?

III. Just how is a wtt degree spectrum different from a Turing

degree spectrum? Furthermore, what happens when we narrow the class of structures A that are allowed?

David Belanger wtt Degrees of Structures

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A result on wtt degree spectra

When we classify the possible Turing degree spectra, the following dichotomy is a good start. Theorem (Knight 86)

1 specT(A) is a singleton if and only if A is trivial. 2 specT(A) is upward closed in the Turing degrees if and only if

A is not trivial.

David Belanger wtt Degrees of Structures

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A result on wtt degree spectra

When we classify the possible Turing degree spectra, the following dichotomy is a good start. Theorem (Knight 86)

1 specT(A) is a singleton if and only if A is trivial. 2 specT(A) is upward closed in the Turing degrees if and only if

A is not trivial. Theorem

1 specwtt(A) is a singleton if and only if A is trivial. David Belanger wtt Degrees of Structures

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A result on wtt degree spectra

When we classify the possible Turing degree spectra, the following dichotomy is a good start. Theorem (Knight 86)

1 specT(A) is a singleton if and only if A is trivial. 2 specT(A) is upward closed in the Turing degrees if and only if

A is not trivial. Theorem

1 specwtt(A) is a singleton if and only if A is trivial. 2 specwtt(A) avoids an upward cone if and only if A is w-trivial. 3 specwtt(A) contains an upward cone if and only if A is not

w-trivial.

David Belanger wtt Degrees of Structures

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Is the wtt case really distinct?

As subsets of 2N, it is easy to see that the inequality

specwtt(A) ⊆
specT(A)

holds.

David Belanger wtt Degrees of Structures

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Is the wtt case really distinct?

As subsets of 2N, it is easy to see that the inequality

specwtt(A) ⊆
specT(A)
holds. There are plenty of examples where the two sets are equal:

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). In fact, A can be a graph. We’d like to be sure that this is not always the case.

David Belanger wtt Degrees of Structures

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Is the wtt case really distinct?

As subsets of 2N, it is easy to see that the inequality

specwtt(A) ⊆
specT(A)
holds. There are plenty of examples where the two sets are equal:

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). In fact, A can be a graph. We’d like to be sure that this is not always the case. Proposition

1 If A is trivial, and its Turing degree consists of more than one

wtt degree, then the inclusion is strict.

2 For any wtt degree b, we can construct a B, with infinite

signature, such that specwtt(B) = Dwtt(≥ b).

David Belanger wtt Degrees of Structures

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Is the wtt case really distinct?

As subsets of 2N, it is easy to see that the inequality

specwtt(A) ⊆
specT(A)
holds. There are plenty of examples where the two sets are equal:

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). In fact, A can be a graph. We’d like to be sure that this is not always the case. Proposition

1 If A is trivial, and its Turing degree consists of more than one

wtt degree, then the inclusion is strict.

2 For any wtt degree b, we can construct a B, with infinite

signature, such that specwtt(B) = Dwtt(≥ b).

3 There exists a nontrivial structure C with finite signature

where the inclusion is strict.

David Belanger wtt Degrees of Structures

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Structures with finite signature

Theorem (H–K–S–S 2002) If B is a nontrivial structure, then there exists a graph G such that specT(G) = specT(B). We say that graphs are universal for Turing degree spectra.

David Belanger wtt Degrees of Structures

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Structures with finite signature

Theorem (H–K–S–S 2002) If B is a nontrivial structure, then there exists a graph G such that specT(G) = specT(B). We say that graphs are universal for Turing degree spectra. Fact If A is a structure with finite signature and A is w-trivial, then A is trivial. In particular, graphs are not similarly universal for wtt degree spectra.

David Belanger wtt Degrees of Structures

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Structures with finite signature

Theorem (H–K–S–S 2002) If B is a nontrivial structure, then there exists a graph G such that specT(G) = specT(B). We say that graphs are universal for Turing degree spectra. Fact If A is a structure with finite signature and A is w-trivial, then A is trivial. In particular, graphs are not similarly universal for wtt degree spectra. Question Is there an interesting class of structures (for example, graphs) that is universal for wtt degree spectra for models of finite signature?

David Belanger wtt Degrees of Structures

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When is specwtt(A) upward closed?

Recall: Theorem (Knight 86) specT(A) is upward closed if and only if A is not trivial. It is fairly easy to show that the wtt degree spectrum is upward closed for ‘nice’ types of structure.

1 Nontrivial equivalence relations 2 Nontrivial graphs with infinitely many components 3 Groups, and so on David Belanger wtt Degrees of Structures

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When is specwtt(A) upward closed?

Recall: Theorem (Knight 86) specT(A) is upward closed if and only if A is not trivial. It is fairly easy to show that the wtt degree spectrum is upward closed for ‘nice’ types of structure.

1 Nontrivial equivalence relations 2 Nontrivial graphs with infinitely many components 3 Groups, and so on

This may call for a precise, novel definition of ‘nice’:

1 Nontrivial graphs? David Belanger wtt Degrees of Structures

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When is specwtt(A) upward closed?

Recall: Theorem (Knight 86) specT(A) is upward closed if and only if A is not trivial. It is fairly easy to show that the wtt degree spectrum is upward closed for ‘nice’ types of structure.

1 Nontrivial equivalence relations 2 Nontrivial graphs with infinitely many components 3 Groups, and so on

This may call for a precise, novel definition of ‘nice’:

1 Nontrivial graphs?

Question If specwtt(A) contains a cone (i.e., if it is not w-trivial), must it be upward closed?

David Belanger wtt Degrees of Structures

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph.

David Belanger wtt Degrees of Structures

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). David Belanger wtt Degrees of Structures

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. David Belanger wtt Degrees of Structures

SLIDE 27

A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

David Belanger wtt Degrees of Structures

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A

SLIDE 30

A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A

SLIDE 31

A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A · · ·

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A · · ·

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A quick construction

Proposition For any nontrivial B, there is an A such that specwtt(A) = specT(B). We may assume that B is a graph. It suffices to build an A satisfying:

1 specT(A) = specT(B). 2 specwtt(A) is upward closed. 3 If X is Turing-above a copy of A, then X is wtt-above a copy

f A.

The following transformation does the trick: B A · · ·

David Belanger wtt Degrees of Structures

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Questions

Question If specwtt(A) contains a cone, must it be upward closed? . . . Or is there some other nice dichotomy to be found? Question Is there an interesting class ∆ of structures such that, for each A with finite signature, there is a B ∈ ∆ with the same wtt degree spectrum? . . . for each A with a single binary relation symbol . . .? Question Can we characterize the structures A such that specT(A) = specwtt(A)?

David Belanger wtt Degrees of Structures

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Questions?

Question If specwtt(A) contains a cone, must it be upward closed? . . . Or is there some other nice dichotomy to be found? Question Is there an interesting class ∆ of structures such that, for each A with finite signature, there is a B ∈ ∆ with the same wtt degree spectrum? . . . for each A with a single binary relation symbol . . .? Question Can we characterize the structures A such that specT(A) = specwtt(A)?

Thank you!

David Belanger wtt Degrees of Structures

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Questions?

Question If specwtt(A) contains a cone, must it be upward closed? . . . Or is there some other nice dichotomy to be found? Question Is there an interesting class ∆ of structures such that, for each A with finite signature, there is a B ∈ ∆ with the same wtt degree spectrum? . . . for each A with a single binary relation symbol . . .? Question Can we characterize the structures A such that specT(A) = specwtt(A)?

Thank you!

David Belanger wtt Degrees of Structures

SLIDE 37

Questions?

Question If specwtt(A) contains a cone, must it be upward closed? . . . Or is there some other nice dichotomy to be found? Question Is there an interesting class ∆ of structures such that, for each A with finite signature, there is a B ∈ ∆ with the same wtt degree spectrum? . . . for each A with a single binary relation symbol . . .? Question Can we characterize the structures A such that specT(A) = specwtt(A)?

Thank you!

David Belanger wtt Degrees of Structures

SLIDE 38

Questions?

Question If specwtt(A) contains a cone, must it be upward closed? . . . Or is there some other nice dichotomy to be found? Question Is there an interesting class ∆ of structures such that, for each A with finite signature, there is a B ∈ ∆ with the same wtt degree spectrum? . . . for each A with a single binary relation symbol . . .? Question Can we characterize the structures A such that specT(A) = specwtt(A)?

Thank you!

David Belanger wtt Degrees of Structures

SLIDE 39

Gloss

Definition A structure A with universe A is trivial if there exists a finite subset S ⊂ A such that any permutation of A fixing S pointwise is an automorphism of A. In this case, we say that S witnesses the triviality of A. Definition A structure A with universe A and relations (R0, R1, . . .) is w-trivial if, for each total computable function f , there is a finite set S witnessing the triviality of the reduct of A to the language (R0, R1, . . . , Rf (|S|)).

David Belanger wtt Degrees of Structures