Realizing arithmetical formulæ

171 Pages·2013·4.1 MB·English

by  Étienne Miquey

Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Realizing arithmetical formulæ
E´tienne Miquey
JointworkwithMauricioGuillermo
February 4, 2013
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Curry-Howard
Proof/programm correspondence
Proof theory Functional programming
Proposition Data types
Deduction rule Typing rule
A⇒ B A→ B
Γ⊢ A ⇒ B Γ⊢ A Γ⊢ t :A → B Γ⊢ u : A
Γ ⊢B Γ⊢ (t)u : B
Correct (for the execution) program might be untypable :
let stupid n =
. if n=n+1 then 27 else true
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Realizability
Introduced by Kleene, relaxation of the previous
correspondence
Formula A realizers |A|
|A| of programs sharing the same computational behavior
... but intrinsically limited to intuitionnistic logic
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Realizability
Introduced by Kleene, relaxation of the previous
correspondence
Formula A realizers |A|
|A| of programs sharing the same computational behavior
... but intrinsically limited to intuitionnistic logic
E´tienneMiquey Realizingarithmeticalformulæ
Classicalrealizability
Realizabilitygame
Introduction
Zoology
Genderequality
Realizability
Introduced by Kleene, relaxation of the previous
correspondence
Formula A realizers |A|
|A| of programs sharing the same computational behavior
... but intrinsically limited to intuitionnistic logic
E´tienneMiquey Realizingarithmeticalformulæ