Goodness and Jump Inversion in the Enumeration Degrees. Charles M. - PowerPoint PPT Presentation

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Goodness and Jump Inversion in the Enumeration Degrees. Charles M. Harris Department Of Mathematics University of Leeds CiE Sofia 2011 Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Outline The local enumeration degrees 1 Jump Interpolation 2 A special Σ 0 2 enumeration degree 3 Cuppable and noncuppable enumeration degrees 4 A special ∆ 0 5 2 enumeration degree The Boundaries of Goodness 6 Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Section Guide 1 The local enumeration degrees 2 Jump Interpolation A special Σ 0 3 2 enumeration degree Cuppable and noncuppable enumeration degrees 4 A special ∆ 0 2 enumeration degree 5 The Boundaries of Goodness 6 Other Results 7 Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other What is enumeration reducibility? Definition (Intuitive) A ≤ e B if there exists an effective procedure that, given any enumeration of B , computes an enumeration A . Definition (Formal) A ≤ e B if there exists a c.e. set W e such that for all x ∈ ω x ∈ A iff ∃ u [ � x , u � ∈ W e & D u ⊆ B ] This is written A = Φ e ( B ) . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Enumeration Degrees • For X , Y ⊆ ω , X ≡ e Y iff X ≤ e Y and Y ≤ e X . • This is an equivalence relation (whose equivalence classes we call enumeration degrees). Notation • �D e , ≤� , �D T , ≤� denote the structures of the enumeration degrees and Turing degrees. (Or more simply D e and D T ). • deg e ( A ) denotes the degree of the set A . • We say, for Γ ∈ { Σ , Π , ∆ } that the degree x is Γ 0 n if it contains a Γ 0 n set. Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Enumeration Degrees • For X , Y ⊆ ω , X ≡ e Y iff X ≤ e Y and Y ≤ e X . • This is an equivalence relation (whose equivalence classes we call enumeration degrees). Notation • �D e , ≤� , �D T , ≤� denote the structures of the enumeration degrees and Turing degrees. (Or more simply D e and D T ). • deg e ( A ) denotes the degree of the set A . • We say, for Γ ∈ { Σ , Π , ∆ } that the degree x is Γ 0 n if it contains a Γ 0 n set. Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Enumeration Degrees • 0 e is the class of c.e . sets. • deg e ( X ⊕ Y ) is the least upper bound of deg e ( X ) and deg e ( Y ) . • There exist enumeration degrees x and y which have no greatest lower bound. • Consequence: D e is an upper semilattice. Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Enumeration Degrees ( D e ). Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Totality • Y is total if Y ≤ e Y . ( X ⊕ X is total for any X and if f is total then G f is total.) • If Y is total then, for any X : X ≤ e Y iff X is c.e. in Y . • Consequence: A ≤ T B iff A ⊕ A ≤ e B ⊕ B . • ι : D T → D e induced by Z �→ Z ⊕ Z is an u.s.l. embedding. Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Jumps Notation We assume a standard effective listing of Turing machines { ϕ e } e ∈ ω , c.e . sets { W e } e ∈ ω (where W e = { x | ϕ e ( x ) ↓ } ), and enumeration operators { Φ e } e ∈ ω . (i) K denotes the halting set { e | ϕ e ( e ) ↓ } . (ii) For any Z , define : I Z = { e | e ∈ Φ Z e } . • J Z = I Z ⊕ I Z is total. • If Z is Σ 0 2 then I Z is total. (In fact, if Z has a good approximation , as defined below, then I Z is total). Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Totality and Jumps I X = { x | x ∈ Φ x ( X ) } • z ′ = deg e ( J Z ) some/any Z ∈ z . • K ≡ 1 I ∅ . So K ≡ e J ∅ ( = def I ∅ ⊕ I ∅ ). • 0 ′ e = deg e ( J ∅ ) = deg e ( K ) . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Totality, Jumps and Local Enumeration Degrees • x is Σ 0 2 iff x ≤ 0 ′ e (Cooper). • In fact: X is Σ 0 2 iff X ≤ e K . • Equivalently: X is Σ 0 2 iff X is c.e. in K . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Local Enumeration Degrees Notation e � denotes the structure of the Σ 0 �D e , ≤ 0 ′ 2 enumeration degrees. • �D e , ≤ 0 ′ e � is an upper semilattice. • 0 e and 0 ′ e are the bottom and top elements of �D e , ≤ 0 ′ e � . • �D e , ≤ 0 ′ e � is dense (Cooper 1984). • �D T , ≤ 0 ′ T � is a substructure (u.s.l.) of �D e , ≤ 0 ′ e � . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The Canonical Embedding ι . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Jump Classes • x is low n if x n = 0 n e . • x is high n if x n = 0 n + 1 . e • We also say that x is low / high in the case when n = 1. • L n = { x | x is low n } . • H n = { x | x is high n } . • I = { x | ∀ n [ 0 n < x n < 0 n + 1 ] } . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other The High/Low Jump Hierarchy Lemma For every n ≥ 0 there exist total degrees x , y ≤ 0 ′ e such that x ∈ H n + 1 − H n and y ∈ L n + 1 − L n . There also exist total z ≤ 0 ′ e such that z ∈ I (the class of intermediate degrees). Proof. Apply the equivalent results (Sacks 1963, 1967) proved in the context of the Σ 0 1 Turing degrees in conjunction with the jump preservation properties of the embedding ι : D T → D e . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Lowness and proper Σ 0 2 -ness Proposition (Cooper and McEvoy) x is low iff every X ∈ x is ∆ 0 2 . Definition e is properly Σ 0 2 if x contains no ∆ 0 x < 0 ′ 2 set and downwards properly Σ 0 2 if every y ∈ { z | 0 e < z ≤ y } is properly Σ 0 2 . Definition x < 0 ′ e is cuppable if there exists y < 0 ′ e such that x ∪ y = 0 ′ e . Otherwise x is noncuppable . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Good Approximations Definition (Lachlan and Shore 1992) A uniformly computable enumeration of finite sets { X s } s ∈ ω is said to be a good approximation to the set X if: (1) ∀ s ( ∃ t ≥ s )[ X t ⊆ X ] (2) ∀ x [ x ∈ X iff ∃ t ( ∀ s ≥ t )[ X s ⊆ X ⇒ x ∈ X s ] ] . Lemma (Jockusch 1968) X is Σ 0 2 iff X has a good Σ 0 2 approximation. I.e . a good approximation with(2 ′ ) replacing (2). (2 ′ ) ∀ x [ x ∈ X iff ∃ t ( ∀ s ≥ t )[ x ∈ X s ] ] . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Some Essential Notation. Notation X [ e ] denotes the set { � e , x � | � e , x � ∈ X } . Goodness and Jump Inversion. Charles M. Harris

A Special Σ 0 A Special ∆ 0 Local Degrees Interpolation 2 Degree Cuppable Degrees 2 Degree Goodness Other Section Guide 1 The local enumeration degrees 2 Jump Interpolation A special Σ 0 3 2 enumeration degree Cuppable and noncuppable enumeration degrees 4 A special ∆ 0 2 enumeration degree 5 The Boundaries of Goodness 6 Other Results 7 Goodness and Jump Inversion. Charles M. Harris