The Weihrauch degree of Ramseys Theorem for two colors Tahina - PowerPoint PPT Presentation

Introduction Variants Idempotency and Parallelization The Weihrauch degree of Ramsey’s Theorem for two colors Tahina Rakotoniaina Department of Mathematics & Applied Mathematics University of Cape Town, South Africa Faculty of Computer Science Universit¨ at der Bundeswehr M¨ unchen, Germany CCA, Nancy 2013 T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Puspose of the study Use Weihrauch degrees to classify mathematical theorems according to their computational content. Idea Regard a theorem as a map: Example ◮ A Π 2 theorem: “( ∀ x ∈ X )( ∃ y ∈ Y )( x , y ) ∈ A ” can be seen as a multivalued map f : x �→ { y : ( x , y ) ∈ A } . T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Contents ◮ Introduction to Weihrauch Degrees ◮ Variants of Ramsey’s Theorem ◮ Idempotency and Parallelization T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Contents ◮ Introduction to Weihrauch Degrees ◮ Variants of Ramsey’s Theorem ◮ Idempotency and Parallelization T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Represented Sets and Realizers ✲ ✲ g f ✲ Z ✲ V X U T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Represented Sets and Realizers N N N N N N N N δ X δ Y δ U δ V ❄ ❄ ❄ ❄ ✲ ✲ g f ✲ Z ✲ V X U T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Represented Sets and Realizers ✲ ✲ F G N N N N N N N N δ X δ Y δ U δ V ❄ ❄ ❄ ❄ ✲ ✲ g f ✲ Z ✲ V X U T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Represented Sets and Realizers ✲ ✲ F G N N N N N N N N δ X δ Y δ U δ V ❄ ❄ ❄ ❄ ✲ ✲ g f ✲ Z ✲ V X U ◮ ( X , δ X ) is a represented set if δ X : ⊆ N N → X is surjective ◮ F is a realizer of f if for all p ∈ dom ( f δ X ) we get δ Y F ( p ) ∈ f δ X ( p ) (noted by F ⊢ f ) If δ ( p ) = x then we say p is a name of the object x . T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Weihrauch Degree ✲ ✲ F G N N N N N N N N δ X δ Y δ U δ V ❄ ❄ ❄ ❄ ✲ g ✲ f ✲ Z ✲ V X U ◮ f is strongly Weihrauch reducible to g if there exist two computable functions H and K such that H ◦ G ◦ K ⊢ f for all G ⊢ g (noted be f ≤ sW g ) ◮ f is (weakly) Weihrauch reducible to g if there exist two computable functions H and K such that H � id , G ◦ K � ⊢ f for all G ⊢ g (noted be f ≤ W g ) T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Invariance Under Representations Definition If we have two representations δ 1 and δ 2 of a set X then δ 1 is said reducible to δ 2 , noted by δ 1 ≤ δ 2 , if there is a computable function Φ : ⊆ N N → N N such that δ 1 ( p ) = δ 2 Φ( p ) for all p ∈ dom( δ 1 ) Lemma Weihrauch degrees are invariant under equivalent representations. T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Tupling Functions and the Limit Map Definition Let ( p i ) i ∈ N be a sequence in Baire space. We define the following: ◮ � p i , p j � (2 n ) = p i ( n ) and � p i , p j � (2 n + 1) = p j ( n ) ◮ � p 0 , p 1 , ..., p n � = � p 0 , � p 1 , ..., p n �� ◮ � p 0 , p 1 , ... �� n , k � = p n ( k ) ◮ lim : ⊆ N N → N N ; lim � p 0 , p 1 , ... � ( n ) = lim i →∞ p i ( n ) T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Operators Let f : ⊆ ( X , δ X ) ⇒ ( Y , δ Y ) be a multivalued function. Then we define ◮ the parallelization � f : ⊆ ( X N , δ N X ) ⇒ ( Y N , δ N Y ) of f by � f ( x i ) i ∈ N := × ∞ i =0 f ( x i ) for all ( x i ) ∈ X N , where δ N : ⊆ N N → X N is defined by δ N � p 0 , p 1 , ... � := ( δ ( p i )) i ∈ N ◮ the jump f ′ : ⊆ ( X , δ ′ X ) ⇒ ( Y , δ Y ) of f by f ′ ( x ) = f ( x ) and δ ′ := δ ◦ lim ◮ for n ≥ 1; f n : ⊆ ( X n , δ n ) ⇒ ( Y n , δ n ) where δ n � p 0 , ..., p n � = ( δ ( p 0 ) , ..., δ ( p n )) T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Facts Let f and g be multivalued functions on represented spaces. Then ◮ f ≤ W � f ⇒ � ◮ f ≤ W g = f ≤ W � g f ≡ W � ◮ � � f ◮ f ≤ sW f ′ ⇒ f ′ ≤ sW g ′ ◮ f ≤ sW g = T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Invariance Principles Lemma Let f and g be multivalued functions on represented spaces such that f ≤ W g. Let n ∈ N . ◮ (Computable Invariance Principle) If g has a realizer that maps computable inputs to computable outputs, then f has a realizer that maps computable inputs to computable outputs. T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Contents ◮ Introduction to Weihrauch Degrees ◮ Variants of Ramsey’s Theorem ◮ Idempotency and Parallelization T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Ramsey Theory Definition Given l ≥ 1 and k ≥ 2 we define ◮ [ N ] l := { size l subsets of N } • [ N ] 1 = {{ 0 } , { 1 } , { 2 } , { 3 } , ... } • [ N ] 2 = {{ 0 , 1 } , { 0 , 2 } , { 1 , 2 } , { 0 , 3 } , { 1 , 3 } , { 2 , 3 } , { 0 , 4 } , ... } ◮ a coloring c : [ N ] l → { 0 , 1 , 2 , ..., k − 1 } Theorem (Ramsey’s Theorem) Given l , k ≥ 1 and a coloring c, there is an infinite subset M of N on which c is constant on [ M ] l Such sets M will be called homogeneous and we write c ( M ) = x if x is the constant value of c on M . T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Ramsey’s Theorem as a Map Definition We define the following: ◮ C l , k denotes the set of all c : [ N ] l → { 0 , 1 , 2 , ..., k − 1 } ◮ RT l , k : C l , k ⇒ 2 N ; c �→ { M : M is homogeneous for c } Sets are represented by their characteristic function and C l , k can be represented in the following way: δ C l , k ( p ) = c if for all { i 1 , ..., i l } ∈ [ N ] l we have c { i 1 , ..., i l } = x iff p � i 1 , ..., i l � = x T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Ramsey’s Theorem as a Map Definition We define the following: ◮ C l , k denotes the set of all c : [ N ] l → { 0 , 1 , 2 , ..., k − 1 } ◮ RT l , k : C l , k ⇒ 2 N ; c �→ { M : M is homogeneous for c } Sets are represented by their characteristic function and C l , k can be represented in the following way: δ C l , k ( p ) = c if for all { i 1 , ..., i l } ∈ [ N ] l we have c { i 1 , ..., i l } = x iff p � i 1 , ..., i l � = x The following maps are also very interesting ◮ MRT l , k : C l , k ⇒ 2 N ; c �→ { M : M is a maximal homogeneous set for c } ◮ CRT l , k : C l , k ⇒ N × 2 N ; c �→ { ( x , M ) : M is an homogeneous set with c ( M ) = x } T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Finite Intersection Lemma Given n ∈ N and c 1 , ..., c n in C l , k , we get ∩ n i =1 RT l , k ( c i ) � = ∅ . Proof idea. We construct a map t : ( C l , k ) n → C l , k n ; ( c 1 , ..., c n ) �→ c such that RT l , k n ( c ) = ∩ n i =1 RT l , k ( c i ). And we apply Ramsey’s Theorem itself. Definition ∩ n RT l , k : ( C l , k ) n ⇒ 2 N ; ( c 1 , ..., c n ) �→ { M : M is homogeneous for each c i } T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Bolzano-Weierstrass and Ramsey Theorems Definition We define the Bolzano-Weierstrass map for { 0 , 1 } as the following: BWT 2 : { 0 , 1 } N ⇒ { 0 , 1 } ; p �→ { x : ( ∃ ∞ n ) p ( n ) = x } Lemma ◮ BWT 2 ≡ W RT 1 , 2 ≡ W CRT 1 , 2 ≡ W MRT 1 , 2 ◮ BWT 2 | sW RT 1 , 2 ◮ BWT 2 < sW CRT 1 , 2 and RT 1 , 2 < sW CRT 1 , 2 ◮ CRT 1 , 2 < sW MRT 1 , 2 ◮ MRT 1 , 2 ≡ sW id × RT 1 , 2 T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem

Introduction Variants Idempotency and Parallelization Strong Reducibility MRT 1 , 2 ≡ id × RT 1 , 2 ❄ CRT 1 , 2 ✡ ❏ ✡ ❏ ✢ ✡ ❏ ❫ BWT 2 RT 1 , 2 T. Rakotoniaina UniBw & UCT Weihrauch Degrees & Ramsey’s Theorem