A Categorical Description of Relativization Kazuto Yoshimura Japan - PowerPoint PPT Presentation

A Categorical Description of Relativization Kazuto Yoshimura Japan Advanced Institute of Science and Technology School of Information Science February 18, 2013 Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 1 / 33

Outline Objective 1 Preliminaries 2 Settings 3 Main Results 4 Conclusion 5 Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 2 / 33

Objective Concept Non-Computability in Categories Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 3 / 33

Objective Concept Non-Computability in Categories How to deal with non-computability in computable analysis? −− > Relativizations to oracles (computability with oracles) Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 3 / 33

Objective Concept Non-Computability in Categories How to deal with non-computability in computable analysis? −− > Relativizations to oracles (computability with oracles) Objective To give a categorical description of “relativization to oracles” Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 3 / 33

Goal We propose to reformulate the following proposition on a categorical setting Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 4 / 33

Goal We propose to reformulate the following proposition on a categorical setting Proposition For a given represented space ( X , δ X ), if δ X is addmissible, then oracle co-r.e. closedness coincides with topological closedness for every subset of X Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 4 / 33

Preliminaries on TTE Type-2 Theory of Effectivity A framework of computable analysis It provides us “de facto standard” terminologies Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 5 / 33

Preliminaries on TTE: 1/5 Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 6 / 33

Preliminaries on TTE: 1/5 (Type-2) Computability is defined for partial functions on Cantor space Oracle computability is also defined Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 6 / 33

Preliminaries on TTE: 1/5 (Type-2) Computability is defined for partial functions on Cantor space Oracle computability is also defined Represented Space Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 6 / 33

�� Preliminaries on TTE: 1/5 (Type-2) Computability is defined for partial functions on Cantor space Oracle computability is also defined Represented Space a representation of a set X : a partial surjection from Cantor space to X X δ supp( δ ) ⊆ 2 ω Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 6 / 33

�� Preliminaries on TTE: 1/5 (Type-2) Computability is defined for partial functions on Cantor space Oracle computability is also defined Represented Space a representation of a set X : a partial surjection from Cantor space to X X δ supp( δ ) ⊆ 2 ω a represented space: a set equipped with a representation Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 6 / 33

Preliminaries on TTE: 2/5 Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 7 / 33

Preliminaries on TTE: 2/5 Example 1 Each u ⊆ 2 ω can be regarded as a represented space w.r.t. the representation δ u defined as follows:  p if p ∈ u   δ u ( p ) =   undefined otherwise   where p ∈ 2 ω Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 7 / 33

Preliminaries on TTE: 3/5 Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 8 / 33

Preliminaries on TTE: 3/5 Example 2 We define a representation δ Ω of 2 as follows:  if p ( i ) = 0 ( ∀ i ∈ ω ) 0   δ Ω ( p ) =   1 otherwise   where p ∈ 2 ω Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 8 / 33

Preliminaries on TTE: 4/5 ( X , δ X ), ( Y , δ Y ): represented spaces Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 9 / 33

� � Preliminaries on TTE: 4/5 ( X , δ X ), ( Y , δ Y ): represented spaces Relatively Computable Function Each f : X → Y is said to be computable w.r.t. δ X , δ Y if there is a computable partial function g on 2 ω which makes the following diagram commute f � Y X δ X δ Y � supp( δ Y ) supp( δ X ) ∃ g Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 9 / 33

� � Preliminaries on TTE: 4/5 ( X , δ X ), ( Y , δ Y ): represented spaces Relatively Computable Function Each f : X → Y is said to be computable w.r.t. δ X , δ Y if there is a computable partial function g on 2 ω which makes the following diagram commute f � Y X δ X δ Y � supp( δ Y ) supp( δ X ) ∃ g Oracle computability can also be extended in the same manner Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 9 / 33

Preliminaries on TTE: 5/5 ( X , δ X ): represented space u : a subset of X Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 10 / 33

Preliminaries on TTE: 5/5 ( X , δ X ): represented space u : a subset of X Co-r.e. Closedness We denote by ch u : X → 2 its characteristic function i.e. the unique function such that u = ch − 1 [ { 0 } ] u u is said to be (oracle) co-r.e. closed if ch u is (oracle) computable w.r.t. δ X , δ Ω Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 10 / 33

Preliminaries on TTE: 5/5 ( X , δ X ): represented space u : a subset of X Co-r.e. Closedness We denote by ch u : X → 2 its characteristic function i.e. the unique function such that u = ch − 1 [ { 0 } ] u u is said to be (oracle) co-r.e. closed if ch u is (oracle) computable w.r.t. δ X , δ Ω Topological Closedness One can think ( X , δ X ) as a topological space w.r.t. the quotient topology induced from Cantor topology via δ X u is said to be closed if it is closed w.r.t. the quotient topology Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 10 / 33

Preliminaries on Category Theory We introduce: three examples of categories one example of functors the notion of factorization system Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 11 / 33

Preliminaries on Category Theory We introduce: three examples of categories one example of functors the notion of factorization system Notations E : arbitrarily fixed category Iso E : the class of all isomorphisms Epi E : the class of all epimorphisms Mono E : the class of all monomorphisms Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 11 / 33

Preliminaries on Category Theory: 1/6 Example 1 Set object: small sets morphism: functions Kazuto Yoshimura (Japan Advanced Institute of Science and TechnologySchool of Information Science) A Categorical Description of Relativization February 18, 2013 12 / 33