The Uniform Martin Conjecture and Wadge Degrees Takayuki Kihara - PowerPoint PPT Presentation

The Uniform Martin Conjecture and Wadge Degrees Takayuki Kihara Joint Work with Antonio Montalb´ an Department of Mathematics, University of California, Berkeley, USA Algorithmic Randomness Interacts with Analysis and Ergodic Theory, Oaxaca, Mexico, Dec 8, 2016 Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Main Theorem (K. and Montalb´ an) ( AD + ) Let Q be BQO. There is an isomorphism between the “ natural ” many-one degrees of Q -valued functions on ω and the Wadge degrees of Q -valued functions on ω ω . Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Main Theorem (K. and Montalb´ an) ( AD + ) Let Q be BQO. There is an isomorphism between the “ natural ” many-one degrees of Q -valued functions on ω and the Wadge degrees of Q -valued functions on ω ω . ( Q = 2 ) The natural many-one degrees are exactly the Wadge degrees. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Main Theorem (K. and Montalb´ an) ( AD + ) Let Q be BQO. There is an isomorphism between the “ natural ” many-one degrees of Q -valued functions on ω and the Wadge degrees of Q -valued functions on ω ω . ( Q = 2 ) The natural many-one degrees are exactly the Wadge degrees. The assumption AD + can be slightly weakened as: ZF + DC + AD + “All subsets of ω ω are completely Ramsey (that is, every subset of ω ω has the Baire property w.r.t. the Ellentuck topology)”. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Definition Let A , B ⊆ ω . A is many-one reducible to B if 1 there is a computable function Φ : ω → ω such that ( ∀ n ∈ ω ) n ∈ A ⇐ ⇒ Φ( n ) ∈ B . Let A , B ⊆ ω ω . A is Wadge reducible to B if 2 there is a continuous function Ψ : ω ω → ω ω such that ( ∀ x ∈ ω ω ) x ∈ A ⇐ ⇒ Ψ( x ) ∈ B . Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Definition Let A , B ⊆ ω . A is many-one reducible to B if 1 there is a computable function Φ : ω → ω such that ( ∀ n ∈ ω ) n ∈ A ⇐ ⇒ Φ( n ) ∈ B . Let A , B ⊆ ω ω . A is Wadge reducible to B if 2 there is a continuous function Ψ : ω ω → ω ω such that ( ∀ x ∈ ω ω ) x ∈ A ⇐ ⇒ Ψ( x ) ∈ B . Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Di ff erence Hierarchy Ershov Hierarchy (Hausdor ff -Kuratowski) 3-c.e. d-c.e. open c.e. co-c.e. closed computable clopen ; ! ! ; ! Many-one degrees versus Wadge degrees Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the many-one degrees is very complicated: There are continuum-size antichains, every countable distributive lattice is isomorphic to an initial segment, etc. (Nerode-Shore 1980) The theory of the many-one degrees is computably isomorphic to the true second-order arithmetic. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 ; the α -th level in the diff. hierarchy = Σ α ; Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 ; the α -th level in the diff. hierarchy = Σ α ; 0 F σ ( Σ 2 ) = Σ ω 1 ∼ Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 ; the α -th level in the diff. hierarchy = Σ α ; 0 0 F σ ( Σ 2 ) = Σ ω 1 ; G δ ( Π 2 ) = Π ω 1 ∼ ∼ Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 ; the α -th level in the diff. hierarchy = Σ α ; 0 0 0 F σ ( Σ 2 ) = Σ ω 1 ; G δ ( Π 2 ) = Π ω 1 ; G δσ ( Σ 3 ) = Σ ω ω 1 ∼ ∼ ∼ 1 Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

Ershov Hierarchy Di ff erence Hierarchy (Hausdor ff -Kuratowski) 3-c.e. intermediate d-c.e. intermediate open c.e. co-c.e. closed intermediate computable clopen ; ; ! ! ! Many-one degrees versus Wadge degrees The structure of the Wadge degrees is very clear: one can assign names to each Wadge degree using an ordinal < Θ and a symbol from { ∆ , Σ , Π } clopen = ∆ 1 ; open = Σ 1 ; the α -th level in the diff. hierarchy = Σ α ; 0 0 0 0 F σ ( Σ 2 ) = Σ ω 1 ; G δ ( Π 2 ) = Π ω 1 ; G δσ ( Σ 3 ) = Σ ω ω 1 1 ; F σδ ( Π 3 ) = Π ω ω 1 1 . ∼ ∼ ∼ ∼ Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

A natural degree should be relativizable and degree invariant. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

A natural degree should be relativizable and degree invariant. In other words, it is induced by a homomorphism f from ≡ T to ≡ T , that is, X ≡ T Y implies f ( X ) ≡ T f ( Y ) . Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

A natural degree should be relativizable and degree invariant. In other words, it is induced by a homomorphism f from ≡ T to ≡ T , that is, X ≡ T Y implies f ( X ) ≡ T f ( Y ) . (AD) The Martin measure µ is defined on ≡ T -invariant sets in 2 ω by:  1 if ( ∃ x )( ∀ y ≥ T x ) y ∈ A ,   µ ( A ) =   0 otherwise.   For homomorphisms f , g from ≡ T to ≡ T , define f ≤ ▽ T g ⇐ ⇒ f ( x ) ≤ T g ( x ) , µ -a.e. The Martin Conjecture (1960’s) For every homomorphism f from ≡ T to ≡ T 1 either f maps a µ -conull set into a single ≡ T -class or f is increasing, that is, f ( x ) ≥ T x , µ -a.e. 2 The increasing homomorphisms from ≡ T to ≡ T are well-ordered by ≤ ▽ T , and each successor rank is given by the Turing jump. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture

length Θ analytic (rank ! 1 ) O 0 000 (rank 3) (rank ! ! 1 G δσ 1 ) 0 00 (rank 2) F σ (rank ! 1 ) 0 0 open (rank 1) (rank 1) 0 clopen Natural Turing degrees and Wadge degrees (Steel, Slaman-Steel 80’s) The Martin conjecture is true for uniform homomorphisms! In particular, increasing uniform homomorphisms are well-ordered, and each successor rank is given by the Turing jump. (Becker 1988) Indeed, increasing uniform homomorphisms form a well-order of type Θ . Takayuki Kihara and Antonio Montalb´ an The Uniform Martin Conjecture