Towards a better Understanding of the Scott Domain Louis Vuilleumier - PowerPoint PPT Presentation

Introduction Definitions Games on P( N ) Examples Results Towards a better Understanding of the Scott Domain Louis Vuilleumier joint work with Jacques Duparc Department of Information Systems University of Lausanne, Switzerland Jacques.Duparc@unil.ch Louis.Vuilleumier.1@unil.ch September 7, 2016 Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 1 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Classical Descriptive Set Theory Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 2 / 10

Introduction Definitions Games on P( N ) Examples Results Generalization Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 3 / 10

Introduction Definitions Games on P( N ) Examples Results Generalization Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 3 / 10

Introduction Definitions Games on P( N ) Examples Results Generalization Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 3 / 10

Introduction Definitions Games on P( N ) Examples Results Generalization Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 3 / 10

Introduction Definitions Games on P( N ) Examples Results Scott domain and continuous reduction Definition Consider the set P( N ) together with the topology — known as the Scott topology — generated by the basis B = {O F ∶ F finite subset of N } , where O F = { X ⊆ N ∶ F ⊆ X } . This space is the Scott domain . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

Introduction Definitions Games on P( N ) Examples Results Scott domain and continuous reduction Definition Consider the set P ( N ) together with the topology — known as the Scott topology — generated by the basis B = { O F ∶ F finite subset of N } , where O F = { X ⊆ N ∶ F ⊆ X } . This space is the Scott domain . Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π 0 2 ( P ( N )) . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

Introduction Definitions Games on P( N ) Examples Results Scott domain and continuous reduction Definition Consider the set P ( N ) together with the topology — known as the Scott topology — generated by the basis B = { O F ∶ F finite subset of N } , where O F = { X ⊆ N ∶ F ⊆ X } . This space is the Scott domain . Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π 0 2 ( P ( N )) . Definition Let A , B ⊆ P ( N ) . If there exists a continuous function f ∶ P ( N ) → P ( N ) such that f − 1 ( B ) = A , we say that A is continuously reducible or Wadge reducible to B , and we write A ≤ W B . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

Introduction Definitions Games on P( N ) Examples Results Scott domain and continuous reduction Definition Consider the set P ( N ) together with the topology — known as the Scott topology — generated by the basis B = { O F ∶ F finite subset of N } , where O F = { X ⊆ N ∶ F ⊆ X } . This space is the Scott domain . Theorem (de Brecht, 2013) A space is quasi-Polish if and only if it is homeomorphic to some A ∈ Π 0 2 ( P ( N )) . Definition Let A , B ⊆ P ( N ) . If there exists a continuous function f ∶ P ( N ) → P ( N ) such that f − 1 ( B ) = A , we say that A is continuously reducible or Wadge reducible to B , and we write A ≤ W B . ≤ W induces a quasi-order relation on the subsets of the Scott domain. Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 4 / 10

Introduction Definitions Games on P( N ) Examples Results Games on the Scott domain Definition Let A , B ⊆ P ( N ) . We define a game G ∞ ( A , B ) . I II X 0 Y 0 ⊆ X n ∈ P <∞ ( N ) , Y n ∈ P ( N ) X 1 ⊆ Y 1 ⊆ X 2 ⊆ Y 2 ⊆ II wins if and only if ⋮ ⋮ ( X ∈ A ↔ Y ∈ B ) . X = ⋃ n ∈ N X n , Y = ⋃ n ∈ N Y n Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 5 / 10

Introduction Definitions Games on P( N ) Examples Results Strategy for II Definition An ultrapositional strategy for II in a game G ∞ ( A , B) is an increasing function σ ∶ P <∞ ( N ) → P( N ) , i.e. such that for all X 0 , X 1 ∈ P <∞ ( N ) with X 0 ⊆ X 1 , we have σ ( X 0 ) ⊆ σ ( X 1 ) . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 6 / 10

Introduction Definitions Games on P( N ) Examples Results Strategy for II Definition An ultrapositional strategy for II in a game G ∞ ( A , B) is an increasing function σ ∶ P <∞ ( N ) → P( N ) , i.e. such that for all X 0 , X 1 ∈ P <∞ ( N ) with X 0 ⊆ X 1 , we have σ ( X 0 ) ⊆ σ ( X 1 ) . Definition An ultrapositional strategy is winning for II if, following this ultrapositional strategy and whatever I plays, II wins the game G ∞ (A , B) . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 6 / 10

Introduction Definitions Games on P( N ) Examples Results Strategy for II Definition An ultrapositional strategy for II in a game G ∞ ( A , B) is an increasing function σ ∶ P <∞ ( N ) → P( N ) , i.e. such that for all X 0 , X 1 ∈ P <∞ ( N ) with X 0 ⊆ X 1 , we have σ ( X 0 ) ⊆ σ ( X 1 ) . Definition An ultrapositional strategy is winning for II if, following this ultrapositional strategy and whatever I plays, II wins the game G ∞ (A , B) . Proposition Let A , B ⊆ P( N ) . The following are equivalent. 1 A ≤ W B , 2 II has a winning ultrapositional strategy in G ∞ (A , B) . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 6 / 10

Introduction Definitions Games on P( N ) Examples Results Examples Consider G ∞ ({ N } , P ∞ ( N )) . σ ∶ P <∞ ( N ) → P ( N ) X ↦ ⋃ { 0 , . . . , n } . n ∈ N { 0 ,..., n }⊆ X It is a ultrapositional winning strategy. Hence, { N } ≤ W P ∞ ( N ) . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 7 / 10

Introduction Definitions Games on P( N ) Examples Results Examples Consider G ∞ ({{ 0 }} , {{ 0 } , { 0 , 1 , 2 }}) . Consider G ∞ ({ N } , P ∞ ( N )) . σ ∶ P <∞ ( N ) → P ( N ) σ ∶ P <∞ ( N ) → P ( N ) ∅ ↦ ∅ , X ↦ ⋃ { 0 , . . . , n } . { 0 } ↦ { 0 } , n ∈ N { 0 ,..., n }⊆ X X ↦ { 0 , 1 } otherwise. It is a ultrapositional winning strategy. It is a ultrapositional winning strategy. Hence, { N } ≤ W P ∞ ( N ) . Hence, {{ 0 }} ≤ W {{ 0 } , { 0 , 1 , 2 }} . Louis Vuilleumier UNIL Highlights 2016 September 7, 2016 7 / 10

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 }

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 }

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 } ∅

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 } ∅ { 0 , 1 , 2 }

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 } ∅ { 0 , 1 , 2 } { 0 }

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 } ∅ { 0 , 1 , 2 } { 0 } { 0 , 1 , 2 }

Introduction Definitions Games on P( N ) Examples Results Examples Consider the game G ∞ ({ N } , {{ 0 }}) . I II ∅ ∅ { 0 } ∅ { 0 , 1 } ∅ { 0 , 1 , 2 } { 0 } { 0 , 1 , 2 } { 0 }