. Two Keywords . . . The Shore-Slaman Join Theorem (1999) 1 It - PowerPoint PPT Presentation

. An Application of Turing Degree Theory to the ω -Decomposability Problem on Borel Functions . . . Takayuki Kihara Japan Advanced Institute of Science and Technology (JAIST) Japan Society for the Promotion of Science (JSPS) research fellow PD Feb. 18, 2013 Computability Theory and Foundation of Mathematics 2013 Takayuki Kihara Decomposability Problem on Borel Functions

. Two Keywords . . . The Shore-Slaman Join Theorem (1999) 1 It was proved by using Kumabe-Slaman forcing. It was used to show the first-order definability of the Turing jump in D T . . . The Decomposability Problem of Borel Functions 2 The original decomposability problem was proposed by Luzin (191?) and negatively answered by Keldis (1934). The modified decomposability problem was proposed by Andretta (2007), Semmes (2009), Pawlikowski-Sabok (2012), Motto Ros (201?). . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . Decomposing a hard function F into easy functions . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . Decomposing a discontinuous function F into easy functions . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . Decomposing a discontinuous function F into continuous functions . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . Decomposing a discontinuous function F into continuous functions . . . F Takayuki Kihara Decomposability Problem on Borel Functions

. Decomposing a discontinuous function F into continuous functions . . . F G 1 G 2 G 0 I I I 0 1 2 .  G 0 ( x ) if x ∈ I 0      F ( x ) =  G 1 ( x ) if x ∈ I 1     G 2 ( x )  if x ∈ I 2  . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . . Decomposing a discontinuous function into continuous functions . . F Takayuki Kihara Decomposability Problem on Borel Functions

. . . . Decomposing a discontinuous function into continuous functions . . F G 0 Takayuki Kihara Decomposability Problem on Borel Functions

. . . . Decomposing a discontinuous function into continuous functions . . F x 7! 0 P 1 Takayuki Kihara Decomposability Problem on Borel Functions

. . Decomposing a discontinuous function into continuous functions . . F G 0 P 1 .  G 0 ( x ) if x � P 1   F ( x ) =   0 if x ∈ P 1   . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . Decomposing a discontinuous function into continuous functions . . . n →∞ cos 2 n ( m ! π x ) Dirichlet ( x ) = lim m →∞ lim = ⇒  1 , if x ∈ Q .   Dirichlet ( x ) =   if x ∈ R \ Q .  0 ,  . . . Takayuki Kihara Decomposability Problem on Borel Functions

. If F is a Borel measurable function on R , then can it be presented by using a countable partition { P n } n ∈ ω of dom ( F ) and a countable list { G n } n ∈ ω of continuous functions as follows?  G 0 ( x ) if x ∈ P 0       G 1 ( x ) if x ∈ P 1        G 2 ( x ) if x ∈ P 2 F ( x ) =      G 3 ( x ) if x ∈ P 3      . .   . .  . .   . . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . . . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . . . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? = ⇒ No! (Keldis 1934) An indecomposable Borel function exists! . . . Takayuki Kihara Decomposability Problem on Borel Functions

. . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? = ⇒ No! (Keldis 1934) An indecomposable Borel function exists! . . . . Example . . The Turing jump TJ : 2 ω → 2 ω is Σ 0 2 -measurable, but it is indecomposable! . . . Takayuki Kihara Decomposability Problem on Borel Functions

. Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? = ⇒ No! (Keldis 1934) An indecomposable Borel function exists! . . . . Example . . The Turing jump TJ : 2 ω → 2 ω is Σ 0 2 -measurable, but it is indecomposable! . . . . Question . . Which Borel function is decomposable into countably many continuous functions? . . . Takayuki Kihara Decomposability Problem on Borel Functions

. ∪ Σ 0 Borel = α . . . α<ω 1 . Definition . . . A function F : X → Y is Borel if 1 ∪ ∪ Σ 0 ⇒ F − 1 [ A ] ∈ Σ 0 A ∈ α ( Y ) = α ( X ) . α<ω 1 α<ω 1 . . A function F : X → Y is Σ 0 α -measurable if 2 A ∈ Σ 0 ⇒ F − 1 [ A ] ∈ Σ 0 1 ( Y ) = α ( X ) . . . A function F : X → Y is Σ α,β if 3 A ∈ Σ 0 ⇒ F − 1 [ A ] ∈ Σ 0 α ( Y ) = β ( X ) . . . . Takayuki Kihara Decomposability Problem on Borel Functions

. A function F : X → Y is Σ α,β if A ∈ Σ 0 ⇒ F − 1 [ A ] ∈ Σ 0 α ( Y ) = β ( X ) . . . . � � � � � � 1 ; 1 1 ; 2 1 ; 3 1 ; 4 1 ; 5 1 ; 6 � � � � � 2 ; 2 2 ; 3 2 ; 4 2 ; 5 2 ; 6 � � � � 3 ; 3 3 ; 4 3 ; 5 3 ; 6 � � � 4 ; 4 4 ; 5 4 ; 6 � � 5 ; 5 5 ; 6 Takayuki Kihara Decomposability Problem on Borel Functions

. A function F : X → Y is Σ α,β if A ∈ Σ 0 ⇒ F − 1 [ A ] ∈ Σ 0 α ( Y ) = β ( X ) . . . . � � � � � � 1 ; 1 1 ; 2 1 ; 3 1 ; 4 1 ; 5 1 ; 6 Con ti. � � � � � 2 ; 2 2 ; 3 2 ; 4 2 ; 5 2 ; 6 � � � � 3 ; 3 3 ; 4 3 ; 5 3 ; 6 Heviside's fun tion � � � 4 ; 4 4 ; 5 4 ; 6 Diri hlet's fun tion Thomae's fun tion � � 5 ; 5 5 ; 6 Leb esgue's fun tion Takayuki Kihara Decomposability Problem on Borel Functions

. . . . Definition . . F ∈ dec ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions . . . . (Keldis 1934) Σ 1 ,α + 1 ⊈ dec ( Σ α ) i.e., there is a Σ 0 α + 1 -measurable function which is not decomposable into countably many Σ 0 α -measurable functions! The α -th Turing jump x �→ x ( α ) is such a function. . . . Takayuki Kihara Decomposability Problem on Borel Functions

. Definition . . F ∈ dec ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions . . . . (Keldis 1934) Σ 1 ,α + 1 ⊈ dec ( Σ α ) i.e., there is a Σ 0 α + 1 -measurable function which is not decomposable into countably many Σ 0 α -measurable functions! The α -th Turing jump x �→ x ( α ) is such a function. . . . . Problem . . Given ( α, β, γ ) ∈ ( ω 1 ) 3 , determine whether or not ✞ ☎ ✞ ☎ dec ( Σ γ ) Σ α,β ⊆ . . . ✝ ✆ ✝ ✆ Takayuki Kihara Decomposability Problem on Borel Functions

. . ， . . Definition . . F : a function from a top. sp. X into a top. sp. Y . F ∈ dec ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions. F ∈ dec β ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions with Π 0 β domains, that is, there are a list { P n } n ∈ ω of Π 0 β subsets of X with n P n and a list { G n } n ∈ ω of Σ 0 X = ∪ α -measurable functions such that F ↾ P n = G n ↾ P n holds for all n ∈ ω . . . . Takayuki Kihara Decomposability Problem on Borel Functions

. Definition . . F : a function from a top. sp. X into a top. sp. Y . F ∈ dec ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions. F ∈ dec β ( Σ α ) if it is decomposable into countably many Σ 0 α -measurable functions with Π 0 β domains, that is, there are a list { P n } n ∈ ω of Π 0 β subsets of X with n P n and a list { G n } n ∈ ω of Σ 0 X = ∪ α -measurable functions such that F ↾ P n = G n ↾ P n holds for all n ∈ ω . . . . . The Jayne-Rogers Theorem 1982 . . X , Y : metric separable ， X : analytic For the class of all functions from X into Y , ✞ ☎ ✞ ☎ Σ 2 , 2 = dec 1 ( Σ 1 ) . . . ✝ ✆ ✝ ✆ Takayuki Kihara Decomposability Problem on Borel Functions

. Borel Functions and Decomposability . . . 1 2 3 4 5 6 1 Σ 1 Σ 2 Σ 3 Σ 4 Σ 5 Σ 6 2 – ? ? ? ? dec 1 Σ 1 3 – – ? ? ? ? 4 – – – ? ? ? 5 – – – – ? ? 6 – – – – – ? . The Jayne-Rogers Theorem 1982 . . X , Y : metric separable ， X : analytic For the class of all functions from X into Y , ✞ ☎ ✞ ☎ Σ 2 , 2 = dec 1 ( Σ 1 ) . . . ✝ ✆ ✝ ✆ Takayuki Kihara Decomposability Problem on Borel Functions