Ordinal Ranks on the Baire and non-Baire class functions Takayuki Kihara Nagoya University, Japan Second Workshop on Mathematical Logic and its Applications, Kanazawa Mar 6th, 2018 Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
There are various notions of reducibility for decision problems; e.g. many-one ( m ), truth-table ( tt ), and Turing ( T ) reducibility Day-Downey-Westrick (DDW) recently introduced m -, tt -, and T -reducibility for real-valued functions. We give a full description of the structures of DDW’s m - and T -degrees of real-valued functions. Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
There are various notions of reducibility for decision problems; e.g. many-one ( m ), truth-table ( tt ), and Turing ( T ) reducibility Day-Downey-Westrick (DDW) recently introduced m -, tt -, and T -reducibility for real-valued functions. We give a full description of the structures of DDW’s m - and T -degrees of real-valued functions. Caution: Without mentioning, we always assume AD (axiom of determinacy) . But, of course: If we restrict our attention to Borel sets and Baire functions, every result presented in this talk is provable within ZFC. If we restrict our attention to projective sets and functions, every result presented in this talk is provable within ZF+DC+PD. = AD + , assuming that there are arbitrarily large We even have L ( R ) | Woodin cardinals. Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Day-Downey-Westrick’s m -reducibility For f , g : 2 ω → R , say f is m -reducible to g (written f ≤ m g ) if given p < q , there are r < s and continuous θ : 2 ω → 2 ω s.t. If f ( x ) ≤ p then g ( θ ( x )) ≤ r . If f ( x ) ≥ q then g ( θ ( x )) ≥ s . θ f f ≥ q g f g ≥ s g f g ≤ r g f f ≤ p g Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Definition (Bourgain 1980) Let f : X → R , p < q , and S ⊆ X . Define the ( f , p , q ) -derivative D f , p , q ( S ) of S as follows. ∪ { x ∈ S : ( ∃ U ∋ x ) f [ U ] ⊆ ( −∞ , q ) or f [ U ] ⊆ ( p , ∞ ) } , S \ where U ranges over open sets. Consider the derivation procedure ∩ f , p , q = X , P ξ + 1 f , p , q = D f , p , q ( P ξ P ξ P 0 f , p , q ) , P λ f , p , q = f , p , q for λ limit ξ<λ The Bourgain rank α ( f ) is defined as follows: α ( f , p , q ) = min { α : P α f , p , q = ∅} ; α ( f ) = sup p < q α ( f , p , q ) . α ( f ) = 1 ⇐ ⇒ f is continuous. The rank α ( f ) exists ⇐ ⇒ f is a Baire-one function. ( f is Baire-one iff it is a pointwise limit of continuous functions) ( Bourgain introduced this notion to analyze the Odell-Rosenthal theorem in Banach space theory ) Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Definition (Day-Downey-Westrick) Let f : X → R be a Baire-one function. f is two-sided if ∃ p < q s.t. α ( f , p , q ) = α ( f ) and 1 ∀ ν < α ( f ) ∃ x , y rank > ν s.t. f ( x ) ≤ p < q ≤ f ( y ) . If f is not two-sided, it is called one-sided . 2 f is left-sided if ∀ p < q with α ( f , p , q ) = α ( f ) , 3 ∃ ν < α ( f ) ∀ x rank > ν , f ( x ) < p . f is right-sided if ∀ p < q with α ( f , p , q ) = α ( f ) , 4 ∃ ν < α ( f ) ∀ x rank > ν , f ( x ) > q . f is free one-sided if it is one-, but neither left- nor right-sided. 5 Rank 2 and left-sided ⇐ ⇒ lower semicontinuous Rank 2 and right-sided ⇐ ⇒ upper semicontinuous Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Theorem (Day-Downey-Westrick) For Baire-one functions, α ( f ) ≤ α ( g ) implies f ≤ m g . The α -rank 1 consists of two m -degrees. Each successor α -rank > 1 consists of four m -degrees. Each limit α -rank consists of a single m -degree. α ( f ) = 2 α ( f ) = 3 α ( f ) = ! + 1 α ( f ) = 1 α ( f ) = ! lower semicontinuous two-sided one-sided upper semicontinuous continuous right-sided left-sided constant This gives a full description of the m -degrees of the Baire-one functions. Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
1st Main Theorem (K.) The structure of the DDW- m -degrees of real-valued functions looks like the following figure: Σ 1 Σ 2 Σ ! Σ ! 1 ∆ jr ∆ jr ∆ jr 1 ∆ 1 2 ∆ 2 ! Π 1 Π 2 Π ! Π ! 1 The DDW- m -degrees form a semi-well-order of height Θ . ( Θ = the least ordinal α > 0 s.t. there is no surjection from R onto α .) For a limit ordinal ξ < Θ and finite n < ω , the DDW- m -rank ξ + 3 n + c ξ consists of two incomparable degrees each of the other ranks consists of a single degree. Here, c ξ = 2 if ξ = 0 ; c ξ = 1 if cf ( ξ ) = ω ; and c ξ = 0 if cf ( ξ ) ≥ ω 1 . Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Theorem (K.-Montalb´ an; 201x) The Wadge degrees ≈ the “natural” many-one degrees. DDW defined T -reducibility for R -valued functions as parallel continuous (strong) Weihrauch reducibility ( f ≤ T g iff f ≤ c sW � g ): f ≤ T g ⇐ ⇒ ( ∃ H , K )( ∀ x ) K ( x , ( g ( H ( n , x )) n ∈ N ) = f ( x ) . 2nd Main Theorem (K.) The DDW T -degrees ≈ the “natural” Turing degrees. (Steel ’82; Becker ’88) The “natural” Turing degrees form a well-order of type Θ . Hence, the DDW T -degrees (of nonconst. functions) form a well-order of type Θ . (The DDW T -rank of a Baire class function coincides with 2 + its Baire rank) More Theorems... (with Westrick) There are many other characterizations of DDW T -degrees, e.g., relative computability w.r.t. point-open topology on the space R ( 2 ω ) . Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
The 1st Main Theorem Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Pointclass: Γ ⊆ ω ω Γ = { ω ω \ A : A ∈ Γ } . Dual: ˇ A pointclass Γ is selfdual iff Γ = ˇ Γ . For A , B ⊆ ω ω , A is Wadge reducible to B ( A ≤ w B ) if ( ∃ θ continuous )( ∀ X ∈ ω ω ) X ∈ A ⇐ ⇒ θ ( X ) ∈ B . A ⊆ ω ω is selfdual if A ≡ w ω ω \ A . A ⊆ ω ω is selfdual iff Γ A = { B ∈ ω ω : B ≤ w A } is selfdual. i i i ∆ α is selfdual, but Σ α and Π α are nonselfdual. ∼ ∼ ∼ Theorem (Wadge; Martin-Monk 1970s) The Wadge degrees are semi-well-ordered. In particular, nonselfdual pairs are well-ordered, say (Γ α , ˇ Γ α ) α< Θ where Θ is the height of the Wadge degrees. Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
A pointclass Γ has the separation property if ⇒ ( ∃ C ∈ Γ ∩ ˇ ( ∀ A , B ∈ Γ) [ A ∩ B = ∅ = Γ) A ⊆ C & B ∩ C = ∅ ] A B C Example (Lusin 1927, Novikov 1935, and others) 0 Π α has the separation property for any α < ω 1 . ∼ 1 1 Σ 1 and Π 2 have the separation property. ∼ ∼ 1 1 (PD) Σ 2 n + 1 and Π 2 n + 2 have the separation property. ∼ ∼ Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Nonselfdual pairs are well-ordered, say (Γ α , ˇ Γ α ) α< Θ . Theorem (Van Wasep 1978; Steel 1981) Exactly one of Γ α and ˇ Γ α has the separation property. Π α : the one which has the separation property Σ α : the other one ∆ α = Σ α ∩ Π α Σ 1 Σ 2 Σ ! Σ ! 1 Σ α ∆ 1 ∆ 2 ∆ ! ∆ λ Π 1 Π 2 Π ! Π ! 1 (cf( λ ) = ! ) Π α (cf( α ) > ! ) Example 0 0 0 ∆ 1 = clopen ( ∆ 1 ); Σ 1 = open ( Σ 1 ); Π 1 = closed ( Π 1 ); ∼ ∼ ∼ ∆ α , Σ α , Π α ( α < ω 1 ) : the α th level of the Hausdorff difference hierarchy 0 0 Σ ω 1 = F σ ( Σ 2 ); Π ω 1 = G δ ( Π 2 ) ∼ ∼ Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Example 0 0 Σ 2 , Π 2 : Wadge-rank ω 1 . ∼ ∼ 3 : Wadge-rank ω ω 1 0 0 Σ 3 , Π 1 . ∼ ∼ 0 n : Wadge-rank ω 1 ↑↑ n (the n th level of the superexp hierarchy) 0 Σ n , Π ∼ ∼ ε 0 [ ω 1 ] := lim n →∞ ( ω 1 ↑↑ n ) : Its cofinality is ω . Hence, the class of rank ε 0 [ ω 1 ] is selfdual. 0 Moreover, ∆ ε 0 [ ω 1 ] is far smaller than ∆ ω . ∼ (Wadge) ε ω 1 [ ω 1 ] : the ω th 1 fixed point of the exp. of base ω 1 . 0 0 ω , Π ω : Wadge-rank ε ω 1 [ ω 1 ] . Σ ∼ ∼ Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
Example (Wadge) The Veblen hierarchy of base ω 1 : φ α ( γ ) : the γ th ordinal closed under + , sup n ∈ ω , and ( φ β ) β<α . 1 , . . . , ω ω + 1 , ω ω + 2 φ 0 enumerates 1 , ω 1 , ω 2 1 , ω 3 , . . . 1 1 φ 1 enumerates 1 , ε ω 1 [ ω 1 ] , . . . 0 0 ω α : Wadge-rank φ α ( 1 ) ( 0 < α < ω 1 ) . Σ ω α , Π ∼ ∼ 1 1 1 : Wadge-rank sup ξ<ω 1 φ ξ ( 1 ) . Σ 1 , Π ∼ ∼ Takayuki Kihara (Nagoya) Ordinal Ranks on the Baire and non-Baire class functions
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