Presenting the effectively closed Medvedev degrees requires 0 Paul - PowerPoint PPT Presentation

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Presenting the effectively closed Medvedev degrees requires 0 ′′′ Paul Shafer Appalachian State University shaferpe@appstate.edu http://www.appstate.edu/~shaferpe/ ASL 2012 North American Annual Meeting Madison, WI March 31, 2012

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to mass problems A mass problem is a set A ⊆ 2 ω . Think of the mass problem A as representing the problem of finding a member of A .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to mass problems A mass problem is a set A ⊆ 2 ω . Think of the mass problem A as representing the problem of finding a member of A . The mass problem A is closed if it is closed in the usual (product) topology on 2 ω . Equivalently, A is closed if A = [ T ] for some tree T ⊆ 2 <ω .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to mass problems A mass problem is a set A ⊆ 2 ω . Think of the mass problem A as representing the problem of finding a member of A . The mass problem A is closed if it is closed in the usual (product) topology on 2 ω . Equivalently, A is closed if A = [ T ] for some tree T ⊆ 2 <ω . The mass problem A is effectively closed if A = [ T ] for some computable tree T ⊆ 2 <ω . Equivalently, A is effectively closed if it is a Π 0 1 class.

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } . • D s , cl = { deg s ( A ) | A is closed in 2 ω } .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } . • D s , cl = { deg s ( A ) | A is closed in 2 ω } . • E s = { deg s ( A ) | A is effectively closed in 2 ω } \ { deg s ( ∅ ) } .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } . • D s , cl = { deg s ( A ) | A is closed in 2 ω } . • E s = { deg s ( A ) | A is effectively closed in 2 ω } \ { deg s ( ∅ ) } . B ⊆ A ⇒ A ≤ s B (by the identity functional).

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } . • D s , cl = { deg s ( A ) | A is closed in 2 ω } . • E s = { deg s ( A ) | A is effectively closed in 2 ω } \ { deg s ( ∅ ) } . B ⊆ A ⇒ A ≤ s B (by the identity functional). D s and D s , cl have 0 = deg s (2 ω ) and 1 = deg s ( ∅ ).

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Welcome to the Medvedev degrees Definition • A ≤ s B iff there is a Turing functional Φ such that Φ( B ) ⊆ A . • A ≡ s B iff A ≤ s B and B ≤ s A . • deg s ( A ) = {B | B ≡ s A} . • D s = { deg s ( A ) | A ⊆ 2 ω } . • D s , cl = { deg s ( A ) | A is closed in 2 ω } . • E s = { deg s ( A ) | A is effectively closed in 2 ω } \ { deg s ( ∅ ) } . B ⊆ A ⇒ A ≤ s B (by the identity functional). D s and D s , cl have 0 = deg s (2 ω ) and 1 = deg s ( ∅ ). E s has 0 = deg s (2 ω ) and 1 = deg s (complete consistent extensions of PA ).

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! D s , D s , cl , and E s are distributive lattices For mass problems A and B , let • A + B = { f ⊕ g | f ∈ A ∧ g ∈ B} ;

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! D s , D s , cl , and E s are distributive lattices For mass problems A and B , let • A + B = { f ⊕ g | f ∈ A ∧ g ∈ B} ; • A × B = 0 � A ∪ 1 � B .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! D s , D s , cl , and E s are distributive lattices For mass problems A and B , let • A + B = { f ⊕ g | f ∈ A ∧ g ∈ B} ; • A × B = 0 � A ∪ 1 � B . Then • deg s ( A ) + deg s ( B ) = deg s ( A + B ); • deg s ( A ) × deg s ( B ) = deg s ( A × B ).

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! D s , D s , cl , and E s are distributive lattices For mass problems A and B , let • A + B = { f ⊕ g | f ∈ A ∧ g ∈ B} ; • A × B = 0 � A ∪ 1 � B . Then • deg s ( A ) + deg s ( B ) = deg s ( A + B ); • deg s ( A ) × deg s ( B ) = deg s ( A × B ). D s is a Brouwer algebra (for every a and b there is a least c with a + c ≥ b ). Neither D s , cl (Lewis, Shore, Sorbi) nor E s (Higuchi) is a Brouwer algebra.

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Complexity in D s , D s , cl , and E s The Medvedev degrees and its substructures are as complicated as possible.

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Complexity in D s , D s , cl , and E s The Medvedev degrees and its substructures are as complicated as possible. Theorem (S) • Th( D s ) ≡ 1 Th 3 ( N ) (independently by Lewis, Nies, & Sorbi). • Th( D s , cl ) ≡ 1 Th 2 ( N ) . • Th( E s ) ≡ 1 Th( N ) .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Complexity in D s , D s , cl , and E s The Medvedev degrees and its substructures are as complicated as possible. Theorem (S) • Th( D s ) ≡ 1 Th 3 ( N ) (independently by Lewis, Nies, & Sorbi). • Th( D s , cl ) ≡ 1 Th 2 ( N ) . • Th( E s ) ≡ 1 Th( N ) . Today’s theorem: Theorem (S) The degree of E s is 0 ′′′ . That is, 0 ′′′ computes a presentation of E s , and every presentation of E s computes 0 ′′′ .

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Presentations of E s Definition A presentation of E s is a pair of functions + , × : ω × ω → ω such that the structure ( ω ; + , × ) is isomorphic to E s . The degree of a presentation is deg T (+ ⊕ × ).

Encoding 0 ′′′ Decoding 0 ′′′ Mass problems and Medvedev degrees Complexity in the Medvedev degrees The end! Presentations of E s Definition A presentation of E s is a pair of functions + , × : ω × ω → ω such that the structure ( ω ; + , × ) is isomorphic to E s . The degree of a presentation is deg T (+ ⊕ × ). That 0 ′′′ computes a presentation follows from the fact that the relation [ T i ] ≤ s [ T j ] (where T i and T j are primitive recursive subtrees of 2 <ω with indices i and j ) is a Σ 0 3 property of � i , j � : [ T i ] ≤ s [ T j ] ⇔ ∃ e ∀ n ∃ s ( ∀ σ ∈ 2 s )( σ ∈ T j → Φ e ( σ ) ↾ n ∈ T i )