. From Classical Recursion Theory to Descriptive Set Theory via Computable Analysis . . . Takayuki Kihara Japan Advanced Institute of Science and Technology (JAIST) Japan Society for the Promotion of Science (JSPS) research fellow PD Jul. 8, 2013 Computability and Complexity in Analysis 2013 Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Main Theme . . Application of Recursion Theory to Descriptive Set Theory . . . . Which Result in Recursion Theory is applied? . . . . Which Problem in Descriptive Set Theory is solved? . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Main Theme . . Application of Recursion Theory to Descriptive Set Theory . . . . Which Result in Recursion Theory is applied? ⇒ The Shore-Slaman Join Theorem (1999) . . . . Which Problem in Descriptive Set Theory is solved? . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Main Theme . . Application of Recursion Theory to Descriptive Set Theory . . . . Which Result in Recursion Theory is applied? ⇒ The Shore-Slaman Join Theorem (1999) It was proved by using Kumabe-Slaman forcing. It was used to show that The Turing jump is first-order definable in D T . . . . . Which Problem in Descriptive Set Theory is solved? . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Main Theme . . Application of Recursion Theory to Descriptive Set Theory . . . . Which Result in Recursion Theory is applied? ⇒ The Shore-Slaman Join Theorem (1999) It was proved by using Kumabe-Slaman forcing. It was used to show that The Turing jump is first-order definable in D T . . . . . Which Problem in Descriptive Set Theory is solved? ⇒ The Decomposability Problem of Borel Functions . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Main Theme . . Application of Recursion Theory to Descriptive Set Theory . . . . Which Result in Recursion Theory is applied? ⇒ The Shore-Slaman Join Theorem (1999) It was proved by using Kumabe-Slaman forcing. It was used to show that The Turing jump is first-order definable in D T . . . . . Which Problem in Descriptive Set Theory is solved? ⇒ The Decomposability Problem of Borel Functions The original decomposability problem was proposed by Luzin, and negatively answered by Keldysh (1934). A partial positive result was given by Jayne-Rogers (1982). The modified decomposability problem was proposed by Andretta (2007), Semmes (2009), Pawlikowski-Sabok (2012), Motto Ros (2013). . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . Decomposing a hard function F into easy functions . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . Decomposing a discontinuous function F into easy functions . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . Decomposing a discontinuous function F into continuous functions . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . Decomposing a discontinuous function F into continuous functions . . . F Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. 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 From Recursion Theory to Descriptive Set Theory
. . . . Decomposing a discontinuous function into continuous functions . . F Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . . Decomposing a discontinuous function into continuous functions . . F G 0 Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . . Decomposing a discontinuous function into continuous functions . . F x 7! 0 P 1 Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . 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 From Recursion Theory to Descriptive Set Theory
. . 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 ) = 0 , if x ∈ R \ Q . . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. 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 From Recursion Theory to Descriptive Set Theory
. Definition (Baire 1899) . . Baire 0 = continuous. Baire α = the pointwise limit of a seq. of Baire < α functions. Baire function = Baire α for some α . The Baire functions = the smallest class closed under taking pointwise limit and containing all continuous functions. . . . . Definition (Borel 1904, Hausdorff 1913) . . 0 1 = open. Σ ∼ 0 0 α = the complement of a Σ Π α set. ∼ ∼ 0 0 Σ α = the countable union of a seq. of Π β sets for some β < α . ∼ ∼ 0 Borel set = Σ α for some α . ∼ The Borel sets = the smallest σ -algebra containing all open . . . sets. Takayuki Kihara From Recursion Theory to Descriptive Set Theory
Borel hierar h y 0 0 0 � � � 1 2 3 S S S C C C Borel = S 0 � �<! 1 � 0 0 0 � � � 1 2 3 ! 1 . Definition ( X , Y : topological spaces, B ⊆ P ( X ) ) . f : X → Y is B -measurable if f − 1 [ A ] ∈ B for every open A ⊆ Y . . . . . Lebesgue-Hausdorff-Banach Theorem . . ☛ ✟ ✄ � 0 Baire α = Σ α + 1 -measurable ✂ ✁ ∼ ✡ ✠ ✞ ☎ ✞ ☎ the Baire functions = the Borel measurable functions . . . ✝ ✆ ✝ ✆ Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . . Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? = ⇒ No! (Keldysh 1934) An indecomposable Baire 1 function exists! . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Luzin’s Problem (almost 100 years ago) . . Can every Borel function on R be decomposed into countably many continuous functions? = ⇒ No! (Keldysh 1934) An indecomposable Baire 1 function exists! . . . . Example . . The Turing jump TJ : 2 N → 2 N is: 1 , if the n -th Turing machine with oracle x halts TJ ( x )( n ) = 0 , otherwise Then, TJ is Baire 1 , but indecomposable! . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Example . . Turing jump TJ : 2 N → 2 N is indecomposable. . . . . Lemma . . For F : X → Y , the following are equivalent: . . F is decomposable into countably many continuous functions. 1 . . ( ∃ α ∈ 2 N )( ∀ x ∈ 2 N ) F ( x ) ≤ T x ⊕ α 2 Here, ( x ⊕ y )( 2 n ) = x ( n ) and ( x ⊕ y )( 2 n + 1 ) = y ( n ) . . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. Decomposable = ⇒ ( ∃ α )( ∀ x ) F ( x ) ≤ T x ⊕ α . . . . F is decomposable into continuous functions F i : X i → Y . (TTE) Since F i is continuous, it must be computable relative to an oracle α i ! Hence ( ∀ x ∈ X i ) F i ( x ) ≤ T x ⊕ α i ⊕ ( ∀ x ∈ X ) F ( x ) ≤ T x ⊕ i ∈ N α i ⊕ Put α = i ∈ N α i . □ . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. = ( ∃ α )( ∀ x ) F ( x ) ≤ T x ⊕ α Decomposable ⇐ . . . . Assume ( ∀ x ∈ X ) F ( x ) ≤ T x ⊕ α . Φ e : the e -th Turing machine ( ∀ x ∈ X )( ∃ e ∈ N ) Φ e ( x ⊕ α ) = F ( x ) e [ x ] : The least such e for x ∈ X . x �→ Φ e ( x ⊕ α ) is computable relative to α . (TTE) x �→ Φ e ( x ⊕ α ) is continuous. For X e = { x ∈ X : e [ x ] = e } the restriction F | X e = Φ e ( ∗ ⊕ α ) is continuous □ . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
. . . Hierarchy of Indecomposable Functions . . (Keldysh 1934) For every α there is a Baire α function which is not decomposable into countably many Baire < α functions! The α -th Turing jump x �→ x ( α ) is such a function. . . . Takayuki Kihara From Recursion Theory to Descriptive Set Theory
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