Scommettere aiuta: Certezze infinite ed errori consapevoli Guido - PowerPoint PPT Presentation

Scommettere aiuta: Certezze infinite ed errori consapevoli Guido Gherardi Joint work with Vasco Brattka and Rupert H¨ olzl Fakult¨ at f¨ ur Informatik Universit¨ at der Bundeswehr M¨ unchen Torino, 17.06.2015

Π 2 -Statements ( ∀ x ∈ X )( ∃ y ∈ Y ) R ( x , y )

The Computable Analysis Model ✶ 3 2581284527 5 67 32 8 99 576229 2118 0 25 13 ✲ M ✲ 93 15 52 38 182436 ✛ ✲ ❥ 0 28 7 ✲ Figure : A Turing machine working with infinite sequences.

Definition (Representations) A representation of a set X is a surjective function ρ X : ⊆ N N → X . The pair ( X , ρ X ) is called a represented space. Usually, admissible representations (e.g., Cauchy representartions) are used. x ∈ X is ρ X -computable if it has some computable ρ X -name p ∈ N N .

Definition (Representations) A representation of a set X is a surjective function ρ X : ⊆ N N → X . The pair ( X , ρ X ) is called a represented space. Usually, admissible representations (e.g., Cauchy representartions) are used. x ∈ X is ρ X -computable if it has some computable ρ X -name p ∈ N N .

r r r x 2 s r x 6 r x x 3 x 5 x 4 r x 1 r x 0 Figure : The Cauchy Representation δ X : for a separable metric space X , a point x ∈ X is encoded by a Cauchy sequence x 0 , x 1 , x 2 , ... of elements from a fixed dense countable set D ⊆ X that uniformly converges to x .

✬✩ ✗✔ ✎☞ ✖✕ ✍✌ ✬✩ ✫✪ ✛✘ ★ ★★★ ✫✪ ✚✙ ★★ ✛✘ ✗✔ ✓✏ ★✥ ✒✑ ✖✕ ✚✙ ✧✦ Figure : The Negative Representation ψ X : for a separable metric space X , a closed set A ⊆ X is encoded by a list of basic open balls exausting its complement.

� � � Definition (Realizers) Let ( X , ρ X ) , ( Y , ρ Y ) be represented spaces and let f : ⊆ X ⇒ Y be a multi-valued function. A function F : ⊆ N N → N N is a ( ρ X , ρ Y ) -realizer of f ( F ⊢ f ) if ρ Y ◦ F ( p ) ∈ f ( ρ X ( p )) for all p ∈ N N such that ρ X ( p ) ∈ dom ( f ) . F p ∈ N N F ( p ) ∈ N N ρ X ρ Y � y ∈ f ( x ) ⊆ Y x ∈ X f f is said to be ( ρ X , ρ Y ) -computable if it has a computable ( ρ X , ρ Y ) -realizer F .

� � � Definition (Realizers) Let ( X , ρ X ) , ( Y , ρ Y ) be represented spaces and let f : ⊆ X ⇒ Y be a multi-valued function. A function F : ⊆ N N → N N is a ( ρ X , ρ Y ) -realizer of f ( F ⊢ f ) if ρ Y ◦ F ( p ) ∈ f ( ρ X ( p )) for all p ∈ N N such that ρ X ( p ) ∈ dom ( f ) . F p ∈ N N F ( p ) ∈ N N ρ X ρ Y � y ∈ f ( x ) ⊆ Y x ∈ X f f is said to be ( ρ X , ρ Y ) -computable if it has a computable ( ρ X , ρ Y ) -realizer F .

Paradigm extensions: limit computability ✶ 3 2581284527 5 67 32 8 99 576229 2118 0 25 13 ✲ M ✲ 93 15 52 38 182436 ✛ ✲ ❥ 0 28 7 958847612316 Figure : A limit Turing machine.

Paradigm extensions: non deterministic computations A (multi-valued) function f : ⊆ X ⇒ Y is said to be non deterministic computable if the following conditions hold:

Succes oracles N N N N 2 N p ⊢ x ∈ dom ( f ) ⊆ X r S p dom ( F ) M Yes! Output q ⊢ y ∈ f ( x ) ⊆ Y

Failure recognition mechanism N N N N 2 N p ⊢ x ∈ dom ( f ) ⊆ X S p r dom ( F ) M No! Stop the computation Failure!

Las Vegas computability The closed set S p has moreover positive measure µ ( S p ) > 0: A µ ( A ) = µ ( 0102 N ) = 2 −| 010 | = 2 − 3 0 1 0 λ

Examples of theorems that are: ◮ computable: Urysohn Extension Lemma, Urysohn-Tietze Extension Lemma, Heine-Borel Theorem, Weierstrass Approximation Theorem, Baire Category Theorem,... ◮ finitely mind changes complete:, Banach Inverse Mapping Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Theorem, (contrapositive of) Baire Category Theorem... ◮ non deterministic complete: Hahn-Banach Extension Theorem, Brouwer Fixed Point Theorem,... ◮ limit complete: Monotone Convergence Theorem,...

Examples of theorems that are: ◮ computable: Urysohn Extension Lemma, Urysohn-Tietze Extension Lemma, Heine-Borel Theorem, Weierstrass Approximation Theorem, Baire Category Theorem,... ◮ finitely mind changes complete:, Banach Inverse Mapping Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Theorem, (contrapositive of) Baire Category Theorem... ◮ non deterministic complete: Hahn-Banach Extension Theorem, Brouwer Fixed Point Theorem,... ◮ limit complete: Monotone Convergence Theorem,...

Examples of theorems that are: ◮ computable: Urysohn Extension Lemma, Urysohn-Tietze Extension Lemma, Heine-Borel Theorem, Weierstrass Approximation Theorem, Baire Category Theorem,... ◮ finitely mind changes complete:, Banach Inverse Mapping Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Theorem, (contrapositive of) Baire Category Theorem... ◮ non deterministic complete: Hahn-Banach Extension Theorem, Brouwer Fixed Point Theorem,... ◮ limit complete: Monotone Convergence Theorem,...

Examples of theorems that are: ◮ computable: Urysohn Extension Lemma, Urysohn-Tietze Extension Lemma, Heine-Borel Theorem, Weierstrass Approximation Theorem, Baire Category Theorem,... ◮ finitely mind changes complete:, Banach Inverse Mapping Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Theorem, (contrapositive of) Baire Category Theorem... ◮ non deterministic complete: Hahn-Banach Extension Theorem, Brouwer Fixed Point Theorem,... ◮ limit complete: Monotone Convergence Theorem,...

Examples of theorems that are: ◮ computable: Urysohn Extension Lemma, Urysohn-Tietze Extension Lemma, Heine-Borel Theorem, Weierstrass Approximation Theorem, Baire Category Theorem,... ◮ finitely mind changes complete:, Banach Inverse Mapping Theorem, Open Mapping Theorem, Closed Graph Theorem, Uniform Boundedness Theorem, (contrapositive of) Baire Category Theorem... ◮ non deterministic complete: Hahn-Banach Extension Theorem, Brouwer Fixed Point Theorem,... ◮ limit complete: Monotone Convergence Theorem,...

Characterization of Las Vegas computable functions f is Weihrauch-reducible to g ( f ≤ W g ) if there are computable K : ⊆ N N × N N → N N , H : ⊆ N N → N N such that H ( id , GK ) ⊢ f , for every G ⊢ g . ✎ ☞ ✎ ☞ N N N N q q ✍ ✌ ✍ ✲ ✌ id p p ✎ ☞ ✎ ☞ ✎ ☞ ✎ ☞ N N N N N N N N q q q q K G H ✍ ✌ ✲ ✍ ✌ ✍ ✲ ✌ ✍ ✲ ✌ q p ✎ ρ Z ☞ ✎ ρ W ☞ ✞ ☎ q q Z ✝ ✆ W ❄ ❄ ✍ ✌ g ✍ ✌ ✲ ✲ ρ Y ρ X ✎ ☞ ✎ ☞ ✞ ☎ q q X Y ✝ ✆ ❄ ✍ ✌ ✍ ❄ ✌ ✲ ✲ f x y f ( x )

< W , ≡ W , | W are defined in the obvious way. Via ≡ W a lattice of equivalence classes is obtained, called the Weihrauch lattice.

< W , ≡ W , | W are defined in the obvious way. Via ≡ W a lattice of equivalence classes is obtained, called the Weihrauch lattice.

Definition (Positive Closed Choice) Given a computable metric space X with a Borel measure µ , PC X : ⊆ A − ( X ) ⇒ X, A �→ A, is the positive closed choice operator, which selects points from closed sets A ⊆ X of positive measure denoted by the negative representation ψ X − . ✬✩ ✗✔ ✎☞ ✖✕ ✍✌ ✬✩ ✫✪ ✛✘ ★ s ★ ✫✪ ★ ✚✙ ★★★ C X s ✰ ✛✘ ✗✔ ✓✏ ★✥ ✒✑ ✖✕ ✚✙ ✧✦

Theorem Let X and Y be represented spaces. The following are equivalent for f : ⊆ X ⇒ Y: ◮ f is Las Vegas computable, ◮ f ≤ W PC 2 N ≡ W PC [ 0 , 1 ] .

Theorem Let X and Y be represented spaces. The following are equivalent for f : ⊆ X ⇒ Y: ◮ f is Las Vegas computable, ◮ f ≤ W PC 2 N ≡ W PC [ 0 , 1 ] .

Theorem Let X and Y be represented spaces. The following are equivalent for f : ⊆ X ⇒ Y: ◮ f is Las Vegas computable, ◮ f ≤ W PC 2 N ≡ W PC [ 0 , 1 ] .

Definition (WWKL) Let T ∞ ⊆ 2 N be the set of (charachteristic functions of) infinite binary trees. We define then: WWKL : ⊆ T ∞ ⇒ 2 N , T �→ [ T ] := { p ∈ 2 N | p is an infinite path in T with µ ([ T ]) > 0 } . Theorem Let X and Y be represented spaces. The following are equivalent for f : ⊆ X ⇒ Y: ◮ f is Las Vegas computable, ◮ f ≤ W WWKL .

Definition (WWKL) Let T ∞ ⊆ 2 N be the set of (charachteristic functions of) infinite binary trees. We define then: WWKL : ⊆ T ∞ ⇒ 2 N , T �→ [ T ] := { p ∈ 2 N | p is an infinite path in T with µ ([ T ]) > 0 } . Theorem Let X and Y be represented spaces. The following are equivalent for f : ⊆ X ⇒ Y: ◮ f is Las Vegas computable, ◮ f ≤ W WWKL .

Basic properties of Las Vegas Computable Functions Theorem (Closure under composition) The class of Las Vegas computable multi-valued function is closed under composition. Hence: Las Vegas Computable multi-valued functions constitute a natural computational class.

Basic properties of Las Vegas Computable Functions Theorem (Closure under composition) The class of Las Vegas computable multi-valued function is closed under composition. Hence: Las Vegas Computable multi-valued functions constitute a natural computational class.