Definability in the Computably Enumerable Sets

41
Definability in the Computably Enumerable Sets Rachel Epstein Department of Mathematics Harvard University October 29, 2010 Harvard Logic Colloquium 1

Transcript of Definability in the Computably Enumerable Sets

Page 1: Definability in the Computably Enumerable Sets

Definability in the Computably EnumerableSets

Rachel Epstein

Department of MathematicsHarvard University

October 29, 2010Harvard Logic Colloquium

1

Page 2: Definability in the Computably Enumerable Sets

Godel [1931] defined his diagonal Klasse K .

This is the Π01 complement of our familiar Σ0

1 class K , whichfirst defined by Kleene in 1936.

Kleene showed K is c.e. and noncomputable.

Kleene-Post [1954] defined the jump: A′ = K A

A′ is c.e. in and above A.

All sets and degrees are c.e.

2

Page 3: Definability in the Computably Enumerable Sets

DefinitionL1 = {d | d′ = 0′}.

H1 = {d | d′ = 0′′}.

Definition

Ln = {d | d(n) = 0(n)}

Hn = {d | d(n) = 0(n+1)}

Sacks proved the Jump Inversion Theorem, which led tothe following corollary:

Corollary0 = L0 ( L1 ( L2 ( L3 ( . . ., and

0′ = H0 ( H1 ( H2 ( H3 . . .3

Page 4: Definability in the Computably Enumerable Sets

4

Page 5: Definability in the Computably Enumerable Sets

Myhill [1956] studied the lattice E of c.e. sets underinclusion.

E = {{We}e∈ω,∪,∩, ω, ∅}

We define E∗ = E/F , where F = {We |We finite}.

5

Page 6: Definability in the Computably Enumerable Sets

DefinitionA set A is maximal if A∗ is a coatom of E∗, i.e. if for all e,

A ⊂We =⇒ We =∗ A or We =∗ ω.

Myhill asked if there exists a maximal set.

Friedberg [1958]: There exists a maximal set; E∗ is notdense.

Sacks [1964]: There is an incomplete maximal set.

Yates [1965]: There is a complete maximal set.

Theorem (Martin, 1966)H1 = the degrees of maximal sets.

6

Page 7: Definability in the Computably Enumerable Sets

Definability has played a major role in the field, for thestructures D, R, and E .

Shore and Slaman [1999] wrote: “The overarching goal ofthese [many years of] investigations has been thedefinition of the (Turing) jump operator”.

DefinitionWe say a class of degrees C is definable ifC = {deg(W ) |W ∈ S} where S is a class of sets definablein E .

QuestionWhich jump classes of degrees are definable in E?

7

Page 8: Definability in the Computably Enumerable Sets

DefinitionA set A is atomless if it is not contained in any maximal set.

Lachlan [1968]: The atomless sets are contained in theclass L2.

Shoenfield [1976]: Every degree in L2 contains anatomless set.

Thus, L2 = {deg(A) | A atomless}.

8

Page 9: Definability in the Computably Enumerable Sets

Which Jump Classes are Definable?

Red = Definable Blue = Not definable

L0= {0}: Definable by {deg(∅)}.

L0= {d | d > 0}: Definable by {deg(W ) |W /∈ E}.

H0= {0′}: Definable because the creative sets aredefinable [Harrington, 1986].

H1= {d | d′ = 0′′}: Definable by {deg(W ) |W maximal} byMartin.

L2= {d | d′′ > 0′′}: Definable by {deg(W ) |W atomless}by Lachlan and Shoenfield.

9

Page 10: Definability in the Computably Enumerable Sets

DefinitionA class of sets S ⊆ E is invariant if it is closed under Aut(E). Aclass of degrees C is invariant if C = {deg(W ) |W ∈ S}, whereS is invariant.

Definable classes are invariant.

To show a class is not definable, we show it is noninvariant.

For the c.e. degrees R, we don’t know any nontrivialautomorphisms.

10

Page 11: Definability in the Computably Enumerable Sets

DefinitionA c.e. set A is prompt if there is an enumeration {As} of A and acomputable function p such that for all s, p(s) ≥ s, and for all e,

We infinite =⇒ (∃x) (∃s) [x ∈We, at s & As�x 6= Ap(s)�x ].

Theorem (Harrington-Soare, 1996)

For all prompt sets A, there exists B ≡T 0′ such that A ' B.

Cholak, Downey, and Stob [1992] showed this for promptlysimple sets.

There is a low prompt degree. Hence, every set in thatdegree is automorphic to a complete set.

Thus, the downward closed jump classes {Ln}n>0 and{Hn}n≥0 are noninvariant, and thus not definable.

11

Page 12: Definability in the Computably Enumerable Sets

12

Page 13: Definability in the Computably Enumerable Sets

Upward Closed Jump Classes

Theorem (Cholak-Harrington, 2002)

For n ≥ 2, Hn and Ln are definable.

Corollary (Lachlan-Shoenfield)

L2 is definable.

Nies, Shore, and Slaman showed that in the c.e. degrees(R, <T ), Hn and Ln are all definable except possibly L1.

13

Page 14: Definability in the Computably Enumerable Sets

Red = Definable Blue = Not definable

Upward Closed Downward Closed

nonlown highn lown nonhighn

L1 H1 L1 H1

L2 H2 L2 H2

L3 H3 L3 H3

......

......

The only remaining class is L1.For the c.e. degrees R, the definability of L1 is unknown.

Conjecture (Harrington-Soare, 1996)

L1 is noninvariant.

14

Page 15: Definability in the Computably Enumerable Sets

Theorem (Epstein)

L1 is noninvariant, and thus not definable.

Red = Definable Blue = Not definable

Upward Closed

nonlown highn

L1 H1

L2 H2

L3 H3

......

15

Page 16: Definability in the Computably Enumerable Sets

Theorem (Epstein)There exists a nonlow D such that for all A ≤T D, there exists alow set B such that A ' B.

Corollary (Epstein)The nonlow degrees are noninvariant, and thus not definable.

Proof: Let d = deg(D). Then d is an L1 degree such that allsets in d are automorphic to low sets.

D must be L2.

We will focus on a single set A = ΨD.

16

Page 17: Definability in the Computably Enumerable Sets

17

Page 18: Definability in the Computably Enumerable Sets

Building an automorphism

Given an enumeration {Un}n∈ω of the c.e. sets, whereU0 = A.

Build an enumeration {Un}n∈ω of the c.e. sets. Let B = U0.

We build Un so that Θ : Un 7→ Un is an automorphism.

18

Page 19: Definability in the Computably Enumerable Sets

19

Page 20: Definability in the Computably Enumerable Sets

20

Page 21: Definability in the Computably Enumerable Sets

21

Page 22: Definability in the Computably Enumerable Sets

Recall:

Theorem (Harrington-Soare, 1996)

For all prompt sets A, there exists B ≡T 0′ such that A ' B

To achieve K ≤T B, as n enters K , enumerate Γn into B.

22

Page 23: Definability in the Computably Enumerable Sets

Theorem (Cholak 1995, Harrington-Soare 1996)Every noncomputable c.e. set is automorphic to a high set.

These theorems move sets up in degree. We move sets down.

The Harrington-Soare machinery is inflexible.

It does not allow us to restrain elements from falling into A.

23

Page 24: Definability in the Computably Enumerable Sets

We must build an automorphism taking A ≤T D down to alow set B.

We restrain B to make it low, so we must also restrain A.

This is the first automorphism theorem that uses restraint.

We divide the theorem into two phases because we needtwo sets of machinery.

24

Page 25: Definability in the Computably Enumerable Sets

25

Page 26: Definability in the Computably Enumerable Sets

Phase 1

DefinitionLet E(X ) = {We ∩ X |We ∈ E}

Theorem (Soare, 1982)

For all coinfinite low B, E(B) ∼= E .

Thus, to make A automorphic to B low, we must have E(A) ∼= E .

26

Page 27: Definability in the Computably Enumerable Sets

27

Page 28: Definability in the Computably Enumerable Sets

28

Page 29: Definability in the Computably Enumerable Sets

29

Page 30: Definability in the Computably Enumerable Sets

Phase 2: E(A) ∼= E(B)

Elements enter the Phase 2 when they enter A or B.

They may already be enumerated into Un or Un.

Difficulty: Un ∩ A finite, Un ∩ B infinite.

We can’t let Phase 1 do anything it wants.Bad situation:

30

Page 31: Definability in the Computably Enumerable Sets

The states that have infinitely many elements enter intothem as they enter A or B are the gateway states.

We make the gateway states equal

31

Page 32: Definability in the Computably Enumerable Sets

Summary of automorphism construction:

Phase 1: Ensure E(A) ∼= E(B), while matching gatewaystates, and

Phase 2: Ensure E(A) ∼= E(B).

We extend the map Θ′ : E(A)→ E(B) to an automorphismΘ of E .

32

Page 33: Definability in the Computably Enumerable Sets

33

Page 34: Definability in the Computably Enumerable Sets

Building D

There are infinitary positive requirements to make Dnonlow.

We also restrain D to keep elements in A.

This causes conflict between the positive and negativerequirements on D.

34

Page 35: Definability in the Computably Enumerable Sets

We can build automorphisms on a tree of strategies.

We use one tree to keep track of positive andautomorphism requirements.

Negative requirements are built in to automorphism nodes.

35

Page 36: Definability in the Computably Enumerable Sets

36

Page 37: Definability in the Computably Enumerable Sets

37

Page 38: Definability in the Computably Enumerable Sets

38

Page 39: Definability in the Computably Enumerable Sets

Conclusion

L1 is the only upward closed jump class that is not definable.

Upward Closed Downward Closed

nonlown highn lown nonhighn

L1 H1 L1 H1

L2 H2 L2 H2

L3 H3 L3 H3

......

......

39

Page 40: Definability in the Computably Enumerable Sets

Open Questions

For all L2 degrees a, does there exists A ∈ a such that A isautomorphic to a low set?

For all c.e. sets A < 0′ and noncomputable c.e. sets C, is Aautomorphic to a c.e. set B, C 6≤T B? What if A is L2?

40

Page 41: Definability in the Computably Enumerable Sets

For Further Reading

R. Epstein,The Structure and Applications of the ComputablyEnumerable Sets, (2010), PhD thesis.

R. Epstein,Definability and Automorphisms of the ComputablyEnumerable Sets, ip.

L. Harrington and R. I. Soare,The ∆0

3 automorphism method and noninvariant classes ofdegrees,Jour. Amer. Math. Soc., 9 (1996), 617–666.

www.math.harvard.edu/∼ repstein

Thank you for listening.

41