Computable Functions Definition 3 A representation of a set X is a surjective function δ : ⊆ N N → X . Definition 4 A function f : ⊆ X → Y is called ( δ, δ ′ ) –computable , if there exists a computable function F : ⊆ N N → N N such that δ ′ F ( p ) = fδ ( p ) for all p ∈ dom( fδ ) . ✲ F N N N N δ ′ δ ❄ ❄ ✲ X Y f Definition 5 If δ, δ ′ are representations of X, Y , respectively, then there is a canonical representation [ δ → δ ′ ] of the set of ( δ, δ ′ ) –continuous functions f : X → Y . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 21
Admissible Representations Definition 6 A representation δ of a topological space X is called admissible , if δ is continuous and if the identity id : X → X is ( δ ′ , δ ) –continuous for any continuous representation δ ′ of X . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 22
Admissible Representations Definition 6 A representation δ of a topological space X is called admissible , if δ is continuous and if the identity id : X → X is ( δ ′ , δ ) –continuous for any continuous representation δ ′ of X . Definition 7 If δ, δ ′ are admissible representations of (sequential) topological spaces X, Y , then [ δ → δ ′ ] is a representation of C ( X, Y ) := { f : X → Y : f continuous } . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 23
Admissible Representations Definition 6 A representation δ of a topological space X is called admissible , if δ is continuous and if the identity id : X → X is ( δ ′ , δ ) –continuous for any continuous representation δ ′ of X . Definition 7 If δ, δ ′ are admissible representations of (sequential) topological spaces X, Y , then [ δ → δ ′ ] is a representation of C ( X, Y ) := { f : X → Y : f continuous } . • The representation [ δ → δ ′ ] just includes sufficiently much information on operators T in order to evaluate them effectively. • A computable description of an operator T with respect to [ δ → δ ′ ] corresponds to a “program” of T . • The underlying topology induced on C ( X, Y ) is typically the compact-open topology. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 24
The Category of Admissibly Represented Spaces Theorem 8 (Schr¨ oder) The category of admissibly represented sequential T 0 –spaces is cartesian closed. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 25
The Category of Admissibly Represented Spaces Theorem 8 (Schr¨ oder) The category of admissibly represented sequential T 0 –spaces is cartesian closed. (weak) limit spaces topological spaces sequential T 0 –spaces Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 26
Computable Metric Spaces Definition 9 A tuple ( X, d, α ) is called a computable metric space , if 1. d : X × X → R is a metric on X , 2. α : N → X is a sequence which is dense in X , 3. d ◦ ( α × α ) : N 2 → R is a computable (double) sequence in R . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 27
Computable Metric Spaces Definition 9 A tuple ( X, d, α ) is called a computable metric space , if 1. d : X × X → R is a metric on X , 2. α : N → X is a sequence which is dense in X , 3. d ◦ ( α × α ) : N 2 → R is a computable (double) sequence in R . Definition 10 Let ( X, d, α ) be a computable metric space. The Cauchy representation δ X : ⊆ N N → X of X is defined by δ X ( p ) := lim i →∞ αp ( i ) for all p such that ( αp ( i )) i ∈ N converges and d ( αp ( i ) , αp ( j )) < 2 − i for all j > i (and undefined for all other input sequences). Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 28
Examples of Computable Metric Spaces Example 11 The following are computable metric spaces: 1. ( R n , d R n , α R n ) with the Euclidean metric pP n i =1 | x i − y i | 2 d R n ( x, y ) := and a standard numbering α R n of Q n . 2. ( K ( R n ) , d K , α K ) with the set K ( R n ) of non-empty compact subsets of R n and the Hausdorff metric ˘ ¯ d K ( A, B ) := max sup a ∈ A inf b ∈ B d R n ( a, b ) , sup b ∈ B inf a ∈ A d R n ( a, b ) and a standard numbering α K of the non-empty finite subsets of Q n . 3. ( C ( R n ) , d C , α C ) with the set C ( R n ) of continuous functions f : R n → R , sup x ∈ [ − i,i ] n | f ( x ) − g ( x ) | d C ( f, g ) := P ∞ i =0 2 − i − 1 1+sup x ∈ [ − i,i ] n | f ( x ) − g ( x ) | and a standard numbering α C of Q [ x 1 , ..., x n ] . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 29
Kreitz-Weihrauch Representation Theorem Theorem 12 Let X, Y be computable metric spaces and let f : ⊆ X → Y be a function. Then the following are equivalent: 1. f is continuous, 2. f admits a continuous realizer F : ⊆ N N → N N . ✲ F N N N N δ X δ Y ❄ ❄ ✲ X Y f Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 30
Kreitz-Weihrauch Representation Theorem Theorem 12 Let X, Y be computable metric spaces and let f : ⊆ X → Y be a function. Then the following are equivalent: 1. f is continuous, 2. f admits a continuous realizer F : ⊆ N N → N N . ✲ F N N N N δ X δ Y ❄ ❄ ✲ X Y f Question: Can this theorem be generalized to Borel measurable functions? Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 31
Borel Hierarchy • Σ 0 1 ( X ) is the set of open subsets of X , • Π 0 1 ( X ) is the set of closed subsets of X , • Σ 0 2 ( X ) is the set of F σ subsets of X , • Π 0 2 ( X ) is the set of G δ subsets of X , etc. • ∆ 0 k ( X ) := Σ 0 k ( X ) ∩ Π 0 k ( X ) . ❳❳❳❳❳❳❳❳❳ ✘ ✘✘✘✘✘✘✘✘✘ . Σ 0 Π 0 . 5 5 . ❳ ❳ ✘ ❳ Σ 0 ✘ Π 0 ❳ ❳ ✘ 4 ❳ ✘ 4 ✘ ❳ ✘ ❳ ✘ ❳ ✘ ❳ ✘ ✘ ❳ ❳❳❳❳❳❳❳❳❳ ✘ ✘✘✘✘✘✘✘✘✘ Σ 0 Π 0 3 3 ❳ ❳ ✘ ❳ Σ 0 ✘ Π 0 ❳ ❳ ✘ 2 ❳ ✘ 2 ❳ ✘ ✘ ❳ ✘ ❳ ✘ ❳ ✘ ✘ ❳ ❳❳❳❳❳❳❳❳❳ ✘ ✘✘✘✘✘✘✘✘✘ Σ 0 Π 0 1 1 ❳ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 32
Representations of Borel Classes Definition 13 Let ( X, d, α ) be a separable metric space. We define k ( X ) of Σ 0 k ( X ) of Π 0 representations δ Σ 0 k ( X ) , δ Π 0 k ( X ) and δ ∆ 0 k ( X ) of ∆ 0 k ( X ) for k ≥ 1 as follows: • δ Σ 0 � 1 ( X ) ( p ) := B ( α ( i ) , j ) , � i,j �∈ range( p ) • δ Π 0 k ( X ) ( p ) := X \ δ Σ 0 k ( X ) ( p ) , ∞ • δ Σ 0 k +1 ( X ) � p 0 , p 1 , ... � := � k ( X ) ( p i ) , δ Π 0 i =0 • δ ∆ 0 k ( X ) � p, q � = δ Σ 0 k ( X ) ( p ) : ⇐ ⇒ δ Σ 0 k ( X ) ( p ) = δ Π 0 k ( X ) ( q ) , for all p, p i , q ∈ N N . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 33
Effective Closure Properties of Borel Classes Proposition 14 Let X, Y be computable metric spaces. The following operations are computable for any k ≥ 1 : 1. Σ 0 → Σ 0 k +1 , Σ 0 → Π 0 k +1 , Π 0 → Σ 0 k +1 , Π 0 → Π 0 k +1 , A �→ A (injection) k ֒ k ֒ k ֒ k ֒ k , A �→ A c := X \ A (complement) 2. Σ 0 k → Π 0 k , Π 0 k → Σ 0 3. Σ 0 k × Σ 0 k → Σ 0 k , Π 0 k × Π 0 k → Π 0 k , ( A, B ) �→ A ∪ B (union) 4. Σ 0 k × Σ 0 k → Σ 0 k , Π 0 k × Π 0 k → Π 0 k , ( A, B ) �→ A ∩ B (intersection) k ) N → Σ 0 k , ( A n ) n ∈ N �→ S ∞ 5. ( Σ 0 n =0 A n (countable union) k ) N → Π 0 k , ( A n ) n ∈ N �→ T ∞ 6. ( Π 0 n =0 A n (countable intersection) 7. Σ 0 k ( X ) × Σ 0 k ( Y ) → Σ 0 k ( X × Y ) , ( A, B ) �→ A × B (product) k ( X )) N → Π 0 8. ( Π 0 k ( X N ) , ( A n ) n ∈ N �→ × ∞ n =0 A n (countable product) 9. Σ 0 k ( X × N ) → Σ 0 k ( X ) , A �→ { x ∈ X : ( ∃ n )( x, n ) ∈ A } (countable projection) 10. Σ 0 k ( X × Y ) × Y → Σ 0 k ( X ) , ( A, y ) �→ A y := { x ∈ X : ( x, y ) ∈ A } (section) Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 34
Borel Measurable Operations Definition 15 Let X, Y be separable metric spaces. An operation f : X → Y is called • Σ 0 k –measurable , if f − 1 ( U ) ∈ Σ 0 k ( X ) for any U ∈ Σ 0 1 ( Y ) , Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 35
Borel Measurable Operations Definition 15 Let X, Y be separable metric spaces. An operation f : X → Y is called • Σ 0 k –measurable , if f − 1 ( U ) ∈ Σ 0 k ( X ) for any U ∈ Σ 0 1 ( Y ) , • effectively Σ 0 k –measurable or Σ 0 k –computable , if the map Σ 0 k ( f − 1 ) : Σ 0 1 ( Y ) → Σ 0 k ( X ) , U �→ f − 1 ( U ) is computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 36
Borel Measurable Operations Definition 15 Let X, Y be separable metric spaces. An operation f : X → Y is called • Σ 0 k –measurable , if f − 1 ( U ) ∈ Σ 0 k ( X ) for any U ∈ Σ 0 1 ( Y ) , • effectively Σ 0 k –measurable or Σ 0 k –computable , if the map Σ 0 k ( f − 1 ) : Σ 0 1 ( Y ) → Σ 0 k ( X ) , U �→ f − 1 ( U ) is computable. Definition 16 Let X, Y be separable metric spaces. We define k ( X → Y ) of Σ 0 k ( X → Y ) by representations δ Σ 0 k ( X ) ]( p ) = Σ 0 k ( f − 1 ) k ( X → Y ) ( p ) = f : ⇐ ⇒ [ δ Σ 0 1 ( Y ) → δ Σ 0 δ Σ 0 for all p ∈ N N , f : X → Y and k ≥ 1 . Let δ Σ 0 k ( X → Y ) denote the restriction to Σ 0 k ( X → Y ) . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 37
Effective Closure Properties of Borel Measurable Operations Proposition 17 Let W, X, Y and Z be computable metric spaces. The following operations are computable for all n, k ≥ 1 : 1. Σ 0 n ( Y → Z ) × Σ 0 k ( X → Y ) → Σ 0 n + k − 1 ( X → Z ) , ( g, f ) �→ g ◦ f (composition) 2. Σ 0 k ( X → Y ) × Σ 0 k ( X → Z ) → Σ 0 k ( X → Y × Z ) , ( f, g ) �→ ( x �→ f ( x ) × g ( x )) (juxtaposition) 3. Σ 0 k ( X → Y ) × Σ 0 k ( W → Z ) → Σ 0 k ( X × W → Y × Z ) , ( f, g ) �→ f × g (product) 4. Σ 0 k ( X → Y N ) → Σ 0 k ( X × N → Y ) , f �→ f ∗ (evaluation) 5. Σ 0 k ( X × N → Y ) → Σ 0 k ( X → Y N ) , f �→ [ f ] (transposition) k ( X N → Y N ) , f �→ f N (exponentiation) 6. Σ 0 k ( X → Y ) → Σ 0 7. Σ 0 k ( X × N → Y ) → Σ 0 k ( X → Y ) N , f �→ ( n �→ ( x �→ f ( x, n ))) (sequencing) k ( X → Y ) N → Σ 0 8. Σ 0 k ( X × N → Y ) , ( f n ) n ∈ N �→ (( x, n ) �→ f n ( x )) (de-sequencing) Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 38
Representation Theorem Theorem 18 Let X, Y be computable metric spaces, k ≥ 1 and let f : X → Y be a total function. Then the following are equivalent: 1. f is (effectively) Σ 0 k –measurable, k –measurable realizer F : ⊆ N N → N N . 2. f admits an (effectively) Σ 0 Proof. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 39
Representation Theorem Theorem 18 Let X, Y be computable metric spaces, k ≥ 1 and let f : X → Y be a total function. Then the following are equivalent: 1. f is (effectively) Σ 0 k –measurable, k –measurable realizer F : ⊆ N N → N N . 2. f admits an (effectively) Σ 0 Proof. ✲ F N N N N δ X δ Y ❄ ❄ ✲ X Y f The proof is based on effective versions of the • Kuratowski-Ryll-Nardzewski Selection Theorem, ✷ • Bhattacharya-Srivastava Selection Theorem. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 40
Weihrauch Reducibility of Functions Definition 19 Let X, Y, U, V be computable metric spaces and consider functions f : ⊆ X → Y and g : ⊆ U → V . We say that • f is reducible to g , for short f � t g , if there are continuous functions A : ⊆ X × V → Y and B : ⊆ X → U such that f ( x ) = A ( x, g ◦ B ( x )) for all x ∈ dom( f ) , • f is computably reducible to g , for short f � c g , if there are computable A, B as above. • The corresponding equivalences are denoted by ∼ = t and ∼ = c . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 41
Weihrauch Reducibility of Functions Definition 19 Let X, Y, U, V be computable metric spaces and consider functions f : ⊆ X → Y and g : ⊆ U → V . We say that • f is reducible to g , for short f � t g , if there are continuous functions A : ⊆ X × V → Y and B : ⊆ X → U such that f ( x ) = A ( x, g ◦ B ( x )) for all x ∈ dom( f ) , • f is computably reducible to g , for short f � c g , if there are computable A, B as above. • The corresponding equivalences are denoted by ∼ = t and ∼ = c . Proposition 20 The following holds for all k ≥ 1 : 1. f � t g and g is Σ 0 ⇒ f is Σ 0 k –measurable = k –measurable, 2. f � c g and g is Σ 0 ⇒ f is Σ 0 k –computable = k –computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 42
Completeness Theorem for Baire Space Definition 21 For any k ∈ N we define C k : N N → N N by if ( ∃ n k )( ∀ n k − 1 ) ... p � n, n 1 , ..., n k � � = 0 0 C k ( p )( n ) := 1 otherwise for all p ∈ N N and n ∈ N . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 43
Completeness Theorem for Baire Space Definition 21 For any k ∈ N we define C k : N N → N N by if ( ∃ n k )( ∀ n k − 1 ) ... p � n, n 1 , ..., n k � � = 0 0 C k ( p )( n ) := 1 otherwise for all p ∈ N N and n ∈ N . Theorem 22 Let k ∈ N . For any function F : ⊆ N N → N N we obtain: ⇒ F is Σ 0 1. F � t C k ⇐ k +1 –measurable, ⇒ F is Σ 0 2. F � c C k ⇐ k +1 –computable. Proof. Employ the Tarski-Kuratowski Normal Form in the appropriate way. ✷ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 44
Realizer Reducibility Definition 23 Let X, Y, U, V be computable metric spaces and consider functions f : X → Y and g : U → V . We define f � t g : ⇐ ⇒ fδ X � t g δ U and we say that f is realizer reducible to g , if this holds. Analogously, we define f � c g with � c instead of � t . The corresponding equivalences ≈ t and ≈ c are defined straightforwardly. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 45
Realizer Reducibility Definition 23 Let X, Y, U, V be computable metric spaces and consider functions f : X → Y and g : U → V . We define f � t g : ⇐ ⇒ fδ X � t g δ U and we say that f is realizer reducible to g , if this holds. Analogously, we define f � c g with � c instead of � t . The corresponding equivalences ≈ t and ≈ c are defined straightforwardly. Theorem 24 Let X, Y be computable metric spaces and let k ∈ N . For any function f : X → Y we obtain: ⇒ f is Σ 0 1. f � t C k ⇐ k +1 –measurable, ⇒ f is Σ 0 2. f � c C k ⇐ k +1 –computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 46
Characterization of Realizer Reducibility Definition 25 Let X, Y, U, V be computable metric spaces, let F be a set of functions F : X → Y and let G be a set of functions G : U → V . We define F � t G : ⇐ ⇒ ( ∃ A, B computable )( ∀ G ∈ G )( ∃ F ∈ F ) ( ∀ x ∈ dom( F )) F ( x ) = A ( x, GB ( x )) , where A : ⊆ X × V → Y and B : ⊆ X → U . Analogously, one can define � c with computable A, B . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 47
Characterization of Realizer Reducibility Definition 25 Let X, Y, U, V be computable metric spaces, let F be a set of functions F : X → Y and let G be a set of functions G : U → V . We define F � t G : ⇐ ⇒ ( ∃ A, B computable )( ∀ G ∈ G )( ∃ F ∈ F ) ( ∀ x ∈ dom( F )) F ( x ) = A ( x, GB ( x )) , where A : ⊆ X × V → Y and B : ⊆ X → U . Analogously, one can define � c with computable A, B . Proposition 26 Let X, Y, U, V be computable metric spaces and let f : X → Y and g : U → V be functions. Then f � c g ⇐ ⇒ { F : F ⊢ f } � c { G : G ⊢ g } . An analogous statement holds with respect to � t and � t . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 48
Completeness of the Limit Proposition 27 Let X be a computable metric space and consider c := { ( x n ) n ∈ N ∈ X N : ( x n ) n ∈ N ∈ X N converges } as computable metric subspace of X N . The ordinary limit map lim : c → X, ( x n ) n ∈ N �→ lim n →∞ x n is Σ 0 2 –computable and it is even Σ 0 2 –complete, if there is a computable embedding ι : { 0 , 1 } N ֒ → X . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 49
Completeness of the Limit Proposition 27 Let X be a computable metric space and consider c := { ( x n ) n ∈ N ∈ X N : ( x n ) n ∈ N ∈ X N converges } as computable metric subspace of X N . The ordinary limit map lim : c → X, ( x n ) n ∈ N �→ lim n →∞ x n is Σ 0 2 –computable and it is even Σ 0 2 –complete, if there is a computable embedding ι : { 0 , 1 } N ֒ → X . Proof. On the one hand, Σ 0 2 –computability follows from � ∞ � X n × B ( x, r − 2 − n ) N lim − 1 ( B ( x, r )) = � ∩ c ∈ Σ 0 2 ( c ) n =0 and on the other hand, Σ 0 2 –completeness follows from C 1 � c lim { 0 , 1 } N � c lim X . ✷ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 50
Lower Bounds for Unbounded Closed Linear Operators Theorem 28 Let X, Y be computable Banach spaces and let f : ⊆ X → Y be a closed linear and unbounded operator. Let ( e n ) n ∈ N be a computable sequence in dom( f ) whose linear span is dense in X and let f ( e n ) n ∈ N be computable in Y . Then C 1 � c f . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 51
Lower Bounds for Unbounded Closed Linear Operators Theorem 28 Let X, Y be computable Banach spaces and let f : ⊆ X → Y be a closed linear and unbounded operator. Let ( e n ) n ∈ N be a computable sequence in dom( f ) whose linear span is dense in X and let f ( e n ) n ∈ N be computable in Y . Then C 1 � c f . Corollary 29 (First Main Theorem of Pour-El and Richards) Under the same assumptions as above f maps some computable input x ∈ X to a non-computable output f ( x ) . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 52
Arithmetic Complexity of Points and the Invariance Theorem Definition 30 Let X be a computable metric space and let x ∈ X . Then x is called ∆ 0 n –computable , if there is a ∆ 0 n –computable p ∈ N N such that x = δ X ( p ) . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 53
Arithmetic Complexity of Points and the Invariance Theorem Definition 30 Let X be a computable metric space and let x ∈ X . Then x is called ∆ 0 n –computable , if there is a ∆ 0 n –computable p ∈ N N such that x = δ X ( p ) . Theorem 31 Let X, Y be computable metric spaces. • If f : X → Y is Σ 0 k –computable, then it maps ∆ 0 n –computable inputs x ∈ X to ∆ 0 n + k − 1 –computable outputs f ( x ) ∈ Y . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 54
Arithmetic Complexity of Points and the Invariance Theorem Definition 30 Let X be a computable metric space and let x ∈ X . Then x is called ∆ 0 n –computable , if there is a ∆ 0 n –computable p ∈ N N such that x = δ X ( p ) . Theorem 31 Let X, Y be computable metric spaces. • If f : X → Y is Σ 0 k –computable, then it maps ∆ 0 n –computable inputs x ∈ X to ∆ 0 n + k − 1 –computable outputs f ( x ) ∈ Y . • If f is even Σ 0 k –complete and k ≥ 2 , then there is some ∆ 0 n –computable input x ∈ X for any n ≥ 1 which is mapped to some ∆ 0 n + k − 1 –computable output f ( x ) ∈ Y which is not ∆ 0 n + k − 2 –computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 55
Arithmetic Complexity of Points and the Invariance Theorem Definition 30 Let X be a computable metric space and let x ∈ X . Then x is called ∆ 0 n –computable , if there is a ∆ 0 n –computable p ∈ N N such that x = δ X ( p ) . Theorem 31 Let X, Y be computable metric spaces. • If f : X → Y is Σ 0 k –computable, then it maps ∆ 0 n –computable inputs x ∈ X to ∆ 0 n + k − 1 –computable outputs f ( x ) ∈ Y . • If f is even Σ 0 k –complete and k ≥ 2 , then there is some ∆ 0 n –computable input x ∈ X for any n ≥ 1 which is mapped to some ∆ 0 n + k − 1 –computable output f ( x ) ∈ Y which is not ∆ 0 n + k − 2 –computable. Corollary 32 An Σ 0 2 –computable map f maps computable inputs x ∈ X to outputs f ( x ) that are computable in the halting problem ∅ ′ . If f is even Σ 0 2 –complete, then there is some computable x which is mapped to a non-computable f ( x ) . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 56
Completeness of Differentiation Proposition 33 (von Stein) Let C ( k ) [0 , 1] be the computable metric subspace of C [0 , 1] which contains the k –times continuously differentiable functions f : [0 , 1] → R . The operator of differentiation d k : C ( k ) [0 , 1] → C [0 , 1] , f �→ f ( k ) is Σ 0 k +1 –complete. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 57
Completeness of Differentiation Proposition 33 (von Stein) Let C ( k ) [0 , 1] be the computable metric subspace of C [0 , 1] which contains the k –times continuously differentiable functions f : [0 , 1] → R . The operator of differentiation d k : C ( k ) [0 , 1] → C [0 , 1] , f �→ f ( k ) is Σ 0 k +1 –complete. Corollary 34 The operator of differentiation d : C (1) [0 , 1] → C [0 , 1] is Σ 0 2 –complete. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 58
Completeness of Differentiation Proposition 33 (von Stein) Let C ( k ) [0 , 1] be the computable metric subspace of C [0 , 1] which contains the k –times continuously differentiable functions f : [0 , 1] → R . The operator of differentiation d k : C ( k ) [0 , 1] → C [0 , 1] , f �→ f ( k ) is Σ 0 k +1 –complete. Corollary 34 The operator of differentiation d : C (1) [0 , 1] → C [0 , 1] is Σ 0 2 –complete. Corollary 35 (Ho) The derivative f ′ : [0 , 1] → R of any computable and continuously differentiable function f : [0 , 1] → R is computable in the halting problem ∅ ′ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 59
Completeness of Differentiation Proposition 33 (von Stein) Let C ( k ) [0 , 1] be the computable metric subspace of C [0 , 1] which contains the k –times continuously differentiable functions f : [0 , 1] → R . The operator of differentiation d k : C ( k ) [0 , 1] → C [0 , 1] , f �→ f ( k ) is Σ 0 k +1 –complete. Corollary 34 The operator of differentiation d : C (1) [0 , 1] → C [0 , 1] is Σ 0 2 –complete. Corollary 35 (Ho) The derivative f ′ : [0 , 1] → R of any computable and continuously differentiable function f : [0 , 1] → R is computable in the halting problem ∅ ′ . Corollary 36 (Myhill) There exists a computable and continuously differentiable function f : [0 , 1] → R whose derivative f ′ : [0 , 1] → R is not computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 60
Survey 1. Basic Concepts • Computable Analysis • Computable Borel Measurability • The Representation Theorem 2. Classification of Topological Operations • Representations of Closed Subsets • Topological Operations 3. Classification of Theorems from Functional Analysis • Uniformity versus Non-Uniformity • Open Mapping and Closed Graph Theorem • Banach’s Inverse Mapping Theorem • Hahn-Banach Theorem Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 61
Some Topological Operations 1. Union: ∪ : A ( X ) × A ( X ) → A ( X ) , ( A, B ) �→ A ∪ B , 2. Intersection: ∩ : A ( X ) × A ( X ) → A ( X ) , ( A, B ) �→ A ∩ B , 3. Complement: c : A ( X ) → A ( X ) , A �→ A c , 4. Interior: i : A ( X ) → A ( X ) , A �→ A ◦ , 5. Difference: D : A ( X ) × A ( X ) → A ( X ) , ( A, B ) �→ A \ B , 6. Symmetric Difference: ∆ : A ( X ) × A ( X ) → A ( X ) , ( A, B ) �→ A ∆ B , 7. Boundary: ∂ : A ( X ) → A ( X ) , A �→ ∂A , 8. Derivative: d : A ( X ) → A ( X ) , A �→ A ′ . All results in the second part of the talk are based on joint work with Guido Gherardi, University of Siena, Italy. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 62
R.e. and Recursive Closed Subsets Definition 37 Let ( X, d, α ) be a computable metric space and let A ⊆ X a closed subset. Then • A is called r.e. closed , if { ( n, r ) ∈ N × Q : A ∩ B ( α ( n ) , r ) � = ∅} is r.e. • A is called co-r.e. closed , if there exists an r.e. set I ⊆ N × Q such that X \ A = � ( n,r ) ∈ I B ( α ( n ) , r ) . • A is called recursive closed , if A is r.e. and co-r.e. closed. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 63
R.e. and Recursive Closed Subsets Definition 38 Let ( X, d, α ) be a computable metric space and let A ⊆ X a closed subset. Then • A is called r.e. closed , if { ( n, r ) ∈ N × Q : A ∩ B ( α ( n ) , r ) � = ∅} is r.e. • A is called co-r.e. closed , if there exists an r.e. set I ⊆ N × Q such that X \ A = � ( n,r ) ∈ I B ( α ( n ) , r ) . • A is called recursive closed , if A is r.e. and co-r.e. closed. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 64
Some Hyperspace Representations Definition 39 Let ( X, d, α ) be a computable metric space. We define representations of A ( X ) := { A ⊆ X : A closed and non-empty } : 1. ψ + ( p ) = A : ⇐ ⇒ p is a “list” of all � n, k � with A ∩ B ( α ( n ) , k ) � = ∅ , ∞ 2. ψ − ( p ) = A : ⇐ ⇒ p is a “list” of � n i , k i � with X \ A = S B ( α ( n i ) , k i ) , i =0 3. ψ � p, q � = A : ⇐ ⇒ ψ + ( p ) = A and ψ − ( q ) = A , for all p, q ∈ N N and A ∈ A ( X ) . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 65
Some Hyperspace Representations Definition 39 Let ( X, d, α ) be a computable metric space. We define representations of A ( X ) := { A ⊆ X : A closed and non-empty } : 1. ψ + ( p ) = A : ⇐ ⇒ p is a “list” of all � n, k � with A ∩ B ( α ( n ) , k ) � = ∅ , ∞ 2. ψ − ( p ) = A : ⇐ ⇒ p is a “list” of � n i , k i � with X \ A = S B ( α ( n i ) , k i ) , i =0 3. ψ � p, q � = A : ⇐ ⇒ ψ + ( p ) = A and ψ − ( q ) = A , for all p, q ∈ N N and A ∈ A ( X ) . • The representation ψ + of A ( R n ) is admissible with respect Remark 40 to the lower Fell topology (with subbase elements { A : A ∩ U � = ∅} for any open U ). The computable points are exactly the r.e. closed subsets. • The representation ψ − of A ( R n ) is admissible with respect to the upper Fell topology (with subbase elements { A : A ∩ K = ∅} for any compact K ). The computable points are exactly the co-r.e. closed subsets. • The representation ψ of A ( R n ) is admissible with respect to the Fell topology. The computable points are exactly the recursive closed subsets. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 66
Borel Lattice of Closed Set Representations for Polish Spaces ③ ✲ ψ ψ − ✏ ✏ ✮ ✏ ✢ ✻ ✻ ✛ ✲ ψ + ≡ ψ dist ≡ ψ range ψ dist ψ dist + − ✒ ■ ✻ ✻ ✲ ψ = ψ > ■ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 67
Borel Lattice of Closed Set Representations for Polish Spaces ③ ✲ ψ ψ − ✏ ✏ ✮ ✏ ✢ ✻ ✻ ✛ ✲ ψ + ≡ ψ dist ≡ ψ range ψ dist ψ dist + − ✒ ■ ✻ ✻ ✲ ψ = ψ > ■ • Straight arrows stand for computable reductions. • Curved arrows stand for Σ 0 2 –computable reductions. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 68
Borel Lattice of Closed Set Representations for Polish Spaces ③ ✲ ψ ψ − ✏ ✏ ✮ ✏ ✢ ✻ ✻ ✛ ✲ ψ + ≡ ψ dist ≡ ψ range ψ dist ψ dist + − ✒ ■ ✻ ✻ ✲ ψ = ψ > ■ • Straight arrows stand for computable reductions. • Curved arrows stand for Σ 0 2 –computable reductions. • The Borel structure induced by the final topologies of all representations except ψ − is the Effros Borel structure. • If X is locally compact, then this also holds true for ψ − . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 69
Intersection Theorem 41 Let X be a computable metric space. Then intersection ∩ : A ( X ) × A ( X ) → A ( X ) , ( A, B ) �→ A ∩ B is 1. computable with respect to ( ψ − , ψ − , ψ − ) , 2. Σ 0 2 –computable with respect to ( ψ + , ψ + , ψ − ) , 3. Σ 0 2 –computable w.r.t. ( ψ − , ψ − , ψ ) , if X is effectively locally compact, 4. Σ 0 3 –computable w.r.t. ( ψ + , ψ + , ψ ) , if X is effectively locally compact, 5. Σ 0 3 –hard with respect to ( ψ + , ψ + , ψ + ) , if X is complete and perfect, 6. Σ 0 2 –hard with respect to ( ψ, ψ, ψ + ) , if X is complete and perfect, 7. not Borel measurable w.r.t. ( ψ, ψ, ψ + ) , if X is complete but not K σ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 70
Closure of the Complement Theorem 42 Let ( X, d ) be a computable metric space. Then the closure of the complement c : A ( X ) → A ( X ) , A �→ A c is 1. computable with respect to ( ψ − , ψ + ) , 2. Σ 0 2 –computable with respect to ( ψ + , ψ + ) and ( ψ − , ψ ) , 3. Σ 0 2 –complete with respect to ( ψ + , ψ + ) , if X is complete and perfect, 4. Σ 0 2 –complete with respect to ( ψ, ψ − ) , if X is complete, perfect and proper. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 71
Closure of the Complement Theorem 42 Let ( X, d ) be a computable metric space. Then the closure of the complement c : A ( X ) → A ( X ) , A �→ A c is 1. computable with respect to ( ψ − , ψ + ) , 2. Σ 0 2 –computable with respect to ( ψ + , ψ + ) and ( ψ − , ψ ) , 3. Σ 0 2 –complete with respect to ( ψ + , ψ + ) , if X is complete and perfect, 4. Σ 0 2 –complete with respect to ( ψ, ψ − ) , if X is complete, perfect and proper. Corollary 43 Let X be a computable, perfect and proper Polish space. Then there exists a recursive closed A ⊆ X such that A c is not co-r.e. closed, but A c is always co-r.e. closed in the halting problem ∅ ′ . There exists a r.e. closed A ⊆ X such that A c is not r.e. closed, but A c is always r.e. closed in the halting problem ∅ ′ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 72
Closure of the Interior Theorem 44 Let X be a computable metric space. Then the closure of the interior i : A ( X ) → A ( X ) , A �→ A ◦ is 1. Σ 0 2 –computable with respect to ( ψ − , ψ + ) , 2. Σ 0 3 –computable with respect to ( ψ + , ψ + ) and ( ψ − , ψ ) , 3. Σ 0 3 –complete with respect to ( ψ + , ψ + ) , if X is complete and perfect, 4. Σ 0 3 –complete with respect to ( ψ, ψ − ) , if X is complete, perfect and proper, 5. Σ 0 2 –complete with respect to ( ψ, ψ + ) , if X is complete, perfect and proper. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 73
Closure of the Interior Theorem 44 Let X be a computable metric space. Then the closure of the interior i : A ( X ) → A ( X ) , A �→ A ◦ is 1. Σ 0 2 –computable with respect to ( ψ − , ψ + ) , 2. Σ 0 3 –computable with respect to ( ψ + , ψ + ) and ( ψ − , ψ ) , 3. Σ 0 3 –complete with respect to ( ψ + , ψ + ) , if X is complete and perfect, 4. Σ 0 3 –complete with respect to ( ψ, ψ − ) , if X is complete, perfect and proper, 5. Σ 0 2 –complete with respect to ( ψ, ψ + ) , if X is complete, perfect and proper. Corollary 45 Let X be a computable, perfect and proper Polish space. Then there exists a recursive closed A ⊆ X such that A ◦ is not r.e. closed, but A ◦ is always r.e. closed in the halting problem ∅ ′ . There exists a recursive closed A ⊆ X such that A ◦ is not even co-r.e. closed in the halting problem ∅ ′ , but A ◦ is always co-r.e. closed in ∅ ′′ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 74
Boundary Theorem 46 Let X be a computable metric space. Then the boundary ∂ : A ( X ) → A ( X ) , A �→ ∂A is 1. computable with respect to ( ψ, ψ + ) , if X is effectively locally connected, 2. Σ 0 2 –computable with respect to ( ψ + , ψ + ) and ( ψ, ψ ) , if X is effectively locally connected, 3. Σ 0 2 –computable with respect to ( ψ − , ψ − ) , 4. Σ 0 3 –computable w.r.t. ( ψ − , ψ ) , if X is effectively locally compact, 5. Σ 0 2 –computable with respect to ( ψ − , ψ ) , if X is effectively locally connected and effectively locally compact, 6. Σ 0 2 –complete w.r.t. ( ψ, ψ − ) , if X is complete, perfect and proper, 7. Σ 0 3 –complete with respect to ( ψ, ψ + ) , if X = { 0 , 1 } N , 8. not Borel measurable with respect to ( ψ, ψ + ) , if X = N N . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 75
Boundary Theorem 46 Let X be a computable metric space. Then the boundary ∂ : A ( X ) → A ( X ) , A �→ ∂A is 1. computable with respect to ( ψ, ψ + ) , if X is effectively locally connected, 2. Σ 0 2 –computable with respect to ( ψ + , ψ + ) and ( ψ, ψ ) , if X is effectively locally connected, 3. Σ 0 2 –computable with respect to ( ψ − , ψ − ) , 4. Σ 0 3 –computable w.r.t. ( ψ − , ψ ) , if X is effectively locally compact, 5. Σ 0 2 –computable with respect to ( ψ − , ψ ) , if X is effectively locally connected and effectively locally compact, 6. Σ 0 2 –complete w.r.t. ( ψ, ψ − ) , if X is complete, perfect and proper, 7. Σ 0 3 –complete with respect to ( ψ, ψ + ) , if X = { 0 , 1 } N , 8. not Borel measurable with respect to ( ψ, ψ + ) , if X = N N . Corollary 47 Let X be a computable, perfect and proper Polish space. Then there exists a recursive closed A ⊆ X such that ∂A is not co-r.e. closed, but ∂A is always co-r.e. closed in the halting problem ∅ ′ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 76
Derivative Theorem 48 Let X be a computable metric space. Then the derivative d : A ( X ) → A ( X ) , A �→ A ′ is 1. Σ 0 2 –computable with respect to ( ψ + , ψ − ) , 2. Σ 0 3 –computable with respect to ( ψ + , ψ ) and ( ψ − , ψ − ) , if X is effectively locally compact, 3. Σ 0 2 –complete with respect to ( ψ, ψ − ) , if X is complete and perfect, 4. Σ 0 3 –hard with respect to ( ψ − , ψ − ) , if X is complete and perfect, 5. Σ 0 3 –hard with respect to ( ψ, ψ + ) , if X is complete and perfect, 6. not Borel measurable with respect to ( ψ, ψ + ) , if X is complete but not K σ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 77
Derivative Theorem 48 Let X be a computable metric space. Then the derivative d : A ( X ) → A ( X ) , A �→ A ′ is 1. Σ 0 2 –computable with respect to ( ψ + , ψ − ) , 2. Σ 0 3 –computable with respect to ( ψ + , ψ ) and ( ψ − , ψ − ) , if X is effectively locally compact, 3. Σ 0 2 –complete with respect to ( ψ, ψ − ) , if X is complete and perfect, 4. Σ 0 3 –hard with respect to ( ψ − , ψ − ) , if X is complete and perfect, 5. Σ 0 3 –hard with respect to ( ψ, ψ + ) , if X is complete and perfect, 6. not Borel measurable with respect to ( ψ, ψ + ) , if X is complete but not K σ . Corollary 49 Let X be a computable and perfect Polish space. Then there exists a recursive closed A ⊆ X such that A ′ is not r.e. closed in the halting problem ∅ ′ , but any such A ′ is co-r.e. closed in the halting problem ∅ ′ . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 78
Survey on Results R n { 0 , 1 } N N N [0 , 1] N R N C [0 , 1] [0 , 1] ℓ 2 N A ∪ B 1 1 1 1 1 1 1 1 1 A ∩ B ∞ ∞ ∞ ∞ 1 2 2 2 2 A c 1 2 2 2 2 2 2 2 2 A ◦ 1 3 3 3 3 3 3 3 3 A \ B 1 2 2 2 2 2 2 2 2 A ∆ B 1 2 2 2 2 2 2 2 2 1 3 ∞ 2 2 2 2 2 2 ∂A A ′ 1 3 ∞ 3 3 3 ∞ ∞ ∞ Degrees of computability with respect to ψ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 79
Survey 1. Basic Concepts • Computable Analysis • Computable Borel Measurability • The Representation Theorem 2. Classification of Topological Operations • Representations of Closed Subsets • Topological Operations 3. Classification of Theorems from Functional Analysis • Uniformity versus Non-Uniformity • Open Mapping and Closed Graph Theorem • Banach’s Inverse Mapping Theorem • Hahn-Banach Theorem Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 80
Uniform and Non-Uniform Computability f s X Y Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 81
Uniform and Non-Uniform Computability f s X Y • Uniform Computability: The function f : X → Y is computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 82
Uniform and Non-Uniform Computability f s X c Y c X Y • Uniform Computability: The function f : X → Y is computable. • Non-Uniform Computability: The function f maps computable elements to computable elements (i.e. f ( X c ) ⊆ f ( Y c ) ). Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 83
Banach’s Inverse Mapping Theorem Definition 50 A Banach space or a normed space X together with a dense sequence is called computable if the induced metric space is a computable metric space. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 84
Banach’s Inverse Mapping Theorem Definition 50 A Banach space or a normed space X together with a dense sequence is called computable if the induced metric space is a computable metric space. Theorem 51 Let X, Y be Banach spaces and let T : X → Y be a linear operator. If T is bijective and bounded, then T − 1 : Y → X is bounded. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 85
Banach’s Inverse Mapping Theorem Definition 50 A Banach space or a normed space X together with a dense sequence is called computable if the induced metric space is a computable metric space. Theorem 51 Let X, Y be Banach spaces and let T : X → Y be a linear operator. If T is bijective and bounded, then T − 1 : Y → X is bounded. Question: Given X and Y are computable Banach spaces, which of the following properties hold true under the assumptions of the theorem: 1. Non-uniform inversion problem: ⇒ T − 1 computable? T computable = 2. Uniform inversion problem: T �→ T − 1 computable? Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 86
Banach’s Inverse Mapping Theorem Definition 51 A Banach space or a normed space X together with a dense sequence is called computable if the induced metric space is a computable metric space. Theorem 52 Let X, Y be Banach spaces and let T : X → Y be a linear operator. If T is bijective and bounded, then T − 1 : Y → X is bounded. Question: Given X and Y are computable Banach spaces, which of the following properties hold true under the assumptions of the theorem: 1. Non-uniform inversion problem: ⇒ T − 1 computable? T computable = Yes! 2. Uniform inversion problem: T �→ T − 1 computable? No! Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 86
An Initial Value Problem Theorem 53 Let f 0 , ..., f n : [0 , 1] → R be computable functions with f n � = 0 . The solution operator L : C [0 , 1] × R n → C ( n ) [0 , 1] which maps each tuple ( y, a 0 , ..., a n − 1 ) ∈ C [0 , 1] × R n to the unique function x = L ( y, a 0 , ..., a n − 1 ) with n � f i ( t ) x ( i ) ( t ) = y ( t ) with x ( j ) (0) = a j for j = 0 , ..., n − 1 , i =0 is computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 87
An Initial Value Problem Theorem 53 Let f 0 , ..., f n : [0 , 1] → R be computable functions with f n � = 0 . The solution operator L : C [0 , 1] × R n → C ( n ) [0 , 1] which maps each tuple ( y, a 0 , ..., a n − 1 ) ∈ C [0 , 1] × R n to the unique function x = L ( y, a 0 , ..., a n − 1 ) with n � f i ( t ) x ( i ) ( t ) = y ( t ) with x ( j ) (0) = a j for j = 0 , ..., n − 1 , i =0 is computable. Proof. The following operator is linear and computable: � n � L − 1 : C ( n ) [0 , 1] → C [0 , 1] × R n , x �→ � f i x ( i ) , x (0) (0) , ..., x ( n − 1) (0) i =0 Computability follows since the i –th differentiation operator is computable. By the computable Inverse Mapping Theorem it follows that L is computable too. ✷ Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 88
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 89
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. • There exists no general algorithm which transfers any program of such an operator T into a program of T − 1 . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 90
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. • There exists no general algorithm which transfers any program of such an operator T into a program of T − 1 . • Thus, Banach’s Inverse Mapping Theorem admits only a non-uniform effective version. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 91
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. • There exists no general algorithm which transfers any program of such an operator T into a program of T − 1 . • Thus, Banach’s Inverse Mapping Theorem admits only a non-uniform effective version. • Since this effective version can also be applied to function spaces, it yields a simple proof method which guarantees the algorithmic solvability of certain uniform problems (e.g. differential equations). Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 92
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. • There exists no general algorithm which transfers any program of such an operator T into a program of T − 1 . • Thus, Banach’s Inverse Mapping Theorem admits only a non-uniform effective version. • Since this effective version can also be applied to function spaces, it yields a simple proof method which guarantees the algorithmic solvability of certain uniform problems (e.g. differential equations). • This method is highly non-constructive: the existence of algorithms is ensured without any hint how they could look like. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 93
Non-Constructive Existence Proofs of Algorithms • The inverse T − 1 : Y → X of any bijective and computable linear operator T : X → Y is computable. • There exists no general algorithm which transfers any program of such an operator T into a program of T − 1 . • Thus, Banach’s Inverse Mapping Theorem admits only a non-uniform effective version. • Since this effective version can also be applied to function spaces, it yields a simple proof method which guarantees the algorithmic solvability of certain uniform problems (e.g. differential equations). • This method is highly non-constructive: the existence of algorithms is ensured without any hint how they could look like. • In the finite dimensional case the method is even constructive: an algorithm of T − 1 can be effectively determined from an algorithm of T . Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 94
Operator Spaces in Computable Functional Analysis • It is known that the map Inv : B ( X, Y ) → B ( Y, X ) , T �→ T − 1 is continuous with respect to the operator norm || T || := sup || Tx || || x || =1 (Banach’s Uniform Inversion Theorem) Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 95
Operator Spaces in Computable Functional Analysis • It is known that the map Inv : B ( X, Y ) → B ( Y, X ) , T �→ T − 1 is continuous with respect to the operator norm || T || := sup || Tx || || x || =1 (Banach’s Uniform Inversion Theorem) • However, the space B ( X, Y ) of bounded linear operators is not separable in general and thus no admissible representation exists in general. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 96
Operator Spaces in Computable Functional Analysis • It is known that the map Inv : B ( X, Y ) → B ( Y, X ) , T �→ T − 1 is continuous with respect to the operator norm || T || := sup || Tx || || x || =1 (Banach’s Uniform Inversion Theorem) • However, the space B ( X, Y ) of bounded linear operators is not separable in general and thus no admissible representation exists in general. • A [ δ X → δ Y ] name of an operator T : X → Y does only contain lower information on || T || and some upper bound. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 97
Operator Spaces in Computable Functional Analysis • It is known that the map Inv : B ( X, Y ) → B ( Y, X ) , T �→ T − 1 is continuous with respect to the operator norm || T || := sup || Tx || || x || =1 (Banach’s Uniform Inversion Theorem) • However, the space B ( X, Y ) of bounded linear operators is not separable in general and thus no admissible representation exists in general. • A [ δ X → δ Y ] name of an operator T : X → Y does only contain lower information on || T || and some upper bound. • We consider the inversion Inv : ⊆ C ( X, Y ) → C ( Y, X ) , T �→ T − 1 with respect to [ δ X → δ Y ] (that is, with respect to the compact-open topology). In this sense, inversion is discontinuous. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 98
Operator Spaces in Computable Functional Analysis • It is known that the map Inv : B ( X, Y ) → B ( Y, X ) , T �→ T − 1 is continuous with respect to the operator norm || T || := sup || Tx || || x || =1 (Banach’s Uniform Inversion Theorem) • However, the space B ( X, Y ) of bounded linear operators is not separable in general and thus no admissible representation exists in general. • A [ δ X → δ Y ] name of an operator T : X → Y does only contain lower information on || T || and some upper bound. • We consider the inversion Inv : ⊆ C ( X, Y ) → C ( Y, X ) , T �→ T − 1 with respect to [ δ X → δ Y ] (that is, with respect to the compact-open topology). In this sense, inversion is discontinuous. • However, || || : ⊆ C ( X, Y ) → R , T �→ || T || is lower semi-computable. Vasco Brattka Department of Mathematics & Applied Mathematics · University of Cape Town 99
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