The Uniform Martins Conjecture and the Wadge Degrees Takayuki - PowerPoint PPT Presentation

The Uniform Martin’s Conjecture and the Wadge Degrees Takayuki Kihara Joint Work with Antonio Montalb´ an Department of Mathematics, University of California, Berkeley, USA Computability Theory and Foundations of Mathematics 2016, Waseda University, Sep 21, 2016 Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture

Under some set theoretic hypothesis, we show that: There is a natural one-to-one correspondence between the “ natural ” many-one degrees and the Wadge degrees. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s 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’s Conjecture

Is there a “ natural ” intermediate c.e. Turing degree? Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture

Is there a “ natural ” intermediate c.e. Turing degree? Natural degrees should be relativizable and degree invariant: (Relativizability) It is a function f : 2 ω → 2 ω . (Degree-Invariance) X ≡ T Y implies f ( X ) ≡ T f ( Y ) . (Sacks 1963) Is there a degree invariant c.e. operator which always gives an intermediate Turing degree? Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture

Is there a “ natural ” intermediate c.e. Turing degree? Natural degrees should be relativizable and degree invariant: (Relativizability) It is a function f : 2 ω → 2 ω . (Degree-Invariance) X ≡ T Y implies f ( X ) ≡ T f ( Y ) . (Sacks 1963) Is there a degree invariant c.e. operator which always gives an intermediate Turing degree? (Lachlan 1975) There is no uniformly degree invariant c.e. operator which always gives an intermediate Turing degree. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture

Is there a “ natural ” intermediate c.e. Turing degree? Natural degrees should be relativizable and degree invariant: (Relativizability) It is a function f : 2 ω → 2 ω . (Degree-Invariance) X ≡ T Y implies f ( X ) ≡ T f ( Y ) . (Sacks 1963) Is there a degree invariant c.e. operator which always gives an intermediate Turing degree? (Lachlan 1975) There is no uniformly degree invariant c.e. operator which always gives an intermediate Turing degree. (The Martin Conjecture) There is no intermediate natural Turing degree at each level in the following sense: Every Degree invariant functions function is either constant or increasing. Degree invariant increasing functions are well-ordered, and each successor rank is given by the Turing jump. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture

Is there a “ natural ” intermediate c.e. Turing degree? Natural degrees should be relativizable and degree invariant: (Relativizability) It is a function f : 2 ω → 2 ω . (Degree-Invariance) X ≡ T Y implies f ( X ) ≡ T f ( Y ) . (Sacks 1963) Is there a degree invariant c.e. operator which always gives an intermediate Turing degree? (Lachlan 1975) There is no uniformly degree invariant c.e. operator which always gives an intermediate Turing degree. (The Martin Conjecture) There is no intermediate natural Turing degree at each level in the following sense: Every Degree invariant functions function is either constant or increasing. Degree invariant increasing functions are well-ordered, and each successor rank is given by the Turing jump. (Steel 1982; Slaman-Steel 1988) The Martin Conjecture holds true for uniformly degree invariant functions. Takayuki Kihara and Antonio Montalb´ an The Uniform Martin’s Conjecture