On the Indenspensability of Bar Recursion Interpreting the 2 - PowerPoint PPT Presentation

On the Indenspensability of Bar Recursion Interpreting the Σ 2 -fragment of classical Analysis in System T Danko Ilik (INRIA, France & ERC Advanced Grant ProofCert)

Contents 1. Background 2. Conservative extension of System T with control operators 3. A modified realizability interpretation Soundness Theorem Weak Church’s Rule 4. Conclusion Danko Ilik – On the Indenspensability of Bar Recursion 2

1 Background Danko Ilik – On the Indenspensability of Bar Recursion 3

Arithmetic Computational interpretations Heyting Arithmetic (HA) G¨ odel (1941/1958) Dialectica interpretation using System T (higher-type primitive recursion) Kleene (1945) Relizability using general recursion Kreisel (1962) Modified realizability via System T Danko Ilik – On the Indenspensability of Bar Recursion 4

Arithmetic Computational interpretations Heyting Arithmetic (HA) G¨ odel (1941/1958) Dialectica interpretation using System T (higher-type primitive recursion) Kleene (1945) Relizability using general recursion Kreisel (1962) Modified realizability via System T Peano Arithmetic (PA) • Works for formulas implied by their own double negation translations • Thanks to the fact that the induction axiom is one such formula Danko Ilik – On the Indenspensability of Bar Recursion 5

Analysis Computational interpretation What happens when the Axiom of Choice ∀ x ∃ yA ( x , y ) → ∃ f ∀ xA ( x , f ( x )) , (AC) is added to Arithmetic? Danko Ilik – On the Indenspensability of Bar Recursion 6

Analysis Computational interpretation What happens when the Axiom of Choice ∀ x ∃ yA ( x , y ) → ∃ f ∀ xA ( x , f ( x )) , (AC) is added to Arithmetic? Intuitionistic “Analysis” Computational interpretations still apply to HA+AC. Danko Ilik – On the Indenspensability of Bar Recursion 7

Analysis Computational interpretation What happens when the Axiom of Choice ∀ x ∃ yA ( x , y ) → ∃ f ∀ xA ( x , f ( x )) , (AC) is added to Arithmetic? Intuitionistic “Analysis” Computational interpretations still apply to HA+AC. Classical Analysis But double-negation translation of AC is not provable from AC+HA, so interpretations not directly applicable to classical Analysis. Danko Ilik – On the Indenspensability of Bar Recursion 8

Analysis Computational interpretation What happens when the Axiom of Choice ∀ x ∃ yA ( x , y ) → ∃ f ∀ xA ( x , f ( x )) , (AC) is added to Arithmetic? Intuitionistic “Analysis” Computational interpretations still apply to HA+AC. Classical Analysis But double-negation translation of AC is not provable from AC+HA, so interpretations not directly applicable to classical Analysis. Digression There are forms of AC that are resistant to double-negation translations: Raoult’s Open Induction Principle: ∀ α ( ∀ β < α U ( β ) → U ( α )) → ∀ α U ( α ) , where α ∈ N → { 0 , 1 } or α ∈ N → N and U is open (i.e. Σ 0 1 ). Danko Ilik – On the Indenspensability of Bar Recursion 9

Kuroda’s Principle (1951) If we add ¬¬∀ x ( A ( x ) ∨ ¬ A ( x )) (KC) to HA+AC, then the D-N translation of AC becomes provable! Danko Ilik – On the Indenspensability of Bar Recursion 10

Kuroda’s Principle (1951) If we add ¬¬∀ x ( A ( x ) ∨ ¬ A ( x )) (KC) to HA+AC, then the D-N translation of AC becomes provable! This was known to G¨ odel. Kreisel gives credit in § 2.43 of Spector’s (1962) paper. Double Negation Shift – intuitionistic equivalent of KC ∀ x ¬¬ B ( x ) → ¬¬∀ xB ( x ) . (DNS) Danko Ilik – On the Indenspensability of Bar Recursion 11

Double Negation Shift Computational interpretation? Double Negation Shift ¬¬∀ x ( A ( x ) ∨ ¬ A ( x )) (KC) Can we interpret it computationally? Danko Ilik – On the Indenspensability of Bar Recursion 12

Double Negation Shift Computational interpretation? Double Negation Shift ¬¬∀ x ( A ( x ) ∨ ¬ A ( x )) (KC) Can we interpret it computationally? Formal/False Church’s Thesis Already G¨ odel (1941) considers the special case of KC for A ( x ) := ∃ y T ( x , x , y ) . That directly refutes: ∀ x N ∃ y N A ( x , y ) → ∃ e N ∀ x N ∃ u N ( T ( e , x , u ) ∧ A ( x , U ( u ))) . (CT 0 ) Ex. A form of CT 0 is used to prove soundness of Kleene’s realizability. Danko Ilik – On the Indenspensability of Bar Recursion 13

Double Negation Shift Computational interpretation Bar Recursion Kreisel and Spector gave a computational interpretation of DNS by extending the primitive recursive System T with a general recursive schema: BR ( G , Y , H , s ) = � G ( s ) if Y ( λ k . if k < | s | then s k else 0 ) < | s | = H ( s , λ x . BR ( G , Y , H , s ∗ x )) otherwise • Soundness of BR is proven by an additional axiom like Bar Induction • Improved in works of Coquand, Kohlenbach, Berger, Oliva, ... Danko Ilik – On the Indenspensability of Bar Recursion 14

Analysis Alternative computational interpretation Interpretations based on computational side-effects Krivine 2003 Ex. “Dependent choice, ‘quote’ and the clock” Questions • Can one simplify the approach of side-effect and abstract machines? - Ex. Do call/cc and quote go beyond primitive recursion? • Is full classical logic necessary to prove soundness? - Ex. DNS does not brake the Disjunction Property of intuitionistic predicate calculus Danko Ilik – On the Indenspensability of Bar Recursion 15

Do we need more than System T? Schwichtenberg (1979) System T is closed over bar recursion at types N and N → N . Danko Ilik – On the Indenspensability of Bar Recursion 16

Do we need more than System T? Schwichtenberg (1979) System T is closed over bar recursion at types N and N → N . Kreisel ( § 12.2 of Spector (1962)) Those low types are sufficient for interpreting the classical AC for formulas of the form ∃ α N → N ∀ x N A 0 ( α, x ) , where A 0 is quantifier-free. Danko Ilik – On the Indenspensability of Bar Recursion 17

2 Conservative extension of System T with control operators Danko Ilik – On the Indenspensability of Bar Recursion 18

Goal: System T + and its properties Theorem (Normalization) There is a normalization function ↓ � − � s.t. for every term p of System T + of type γ ⊢ τ , the term ↓ � p � is a normal form of System T of the same type ( γ ⊢ r τ ). Proposition (Equations) ↓ � wkn p � α,ρ = ↓ � p � ρ ↓ � hyp � α,ρ = ↓ α ↓ � fst pair ( p , q ) � ρ = ↓ � p � ρ ↓ � snd pair ( p , q ) � ρ = ↓ � q � ρ ↓ � app ( lam p , q ) � ρ = ↓ � p � � q � ρ ,ρ ↓ � rec ( zero , p , q ) � ρ = ↓ � p � ρ ↓ � rec ( succ r , p , q ) � ρ = · · · ↓ N � shift p � ρ = ↓ N � p � φ,ρ ↓ N � app ( app ( hyp , x ) , y ) � φ,ρ = ↓ N � y � φ,ρ φ := η ( ≥ 2 ν �→ η ( ≥ 3 α �→ η ( µα ))) Danko Ilik – On the Indenspensability of Bar Recursion 19

T + = T + composable continuations Danvy-Filinski’s shift in call-by-name Types: T ∋ σ, τ ::= N | σ → τ | σ ∗ τ Terms: γ ⊢ σ lam ( σ ; γ ) ⊢ τ hyp ( σ ; γ ) ⊢ σ wkn ( τ ; γ ) ⊢ σ γ ⊢ σ → τ app γ ⊢ σ → τ γ ⊢ σ pair γ ⊢ σ γ ⊢ τ fst γ ⊢ σ ∗ τ γ ⊢ τ γ ⊢ σ ∗ τ γ ⊢ σ snd γ ⊢ σ ∗ τ succ γ ⊢ N zero γ ⊢ N γ ⊢ τ γ ⊢ N rec γ ⊢ N γ ⊢ σ γ ⊢ N → σ → σ shift ( N → σ → N ; γ ) ⊢ N γ ⊢ σ γ ⊢ σ Danko Ilik – On the Indenspensability of Bar Recursion 20

System T + Ackermann’s function (Example) A := λ m . R m ( λ n . n + 1 )( λ m ′ .λ u .λ n . R n ( u 1 )( λ n ′ .λ w . uw )) , is represented by lam ( rec hyp ( lam ( succ hyp )) ( lam ( lam ( lam ( rec hyp ( app ( wkn hyp )( succ zero )) ( lam ( lam ( app ( wkn ( wkn ( wkn hyp ))) hyp )))))))) i.e. a 1 st -order representation with de Bruijn indices 0 := hyp, 1 := wkn hyp, ... Danko Ilik – On the Indenspensability of Bar Recursion 21

System T + Ackermann’s function (Example) The Agda formalization really computes ex. A(3,2) to be succ · · · succ zero. � �� � 29 times (If one has enough RAM available) Danko Ilik – On the Indenspensability of Bar Recursion 22

Equations holding of the normalization function Proposition The following definitional equalities hold, ↓ � wkn p � α,ρ = ↓ � p � ρ (1) ↓ � hyp � α,ρ = ↓ α (2) ↓ � fst pair ( p , q ) � ρ = ↓ � p � ρ (3) ↓ � snd pair ( p , q ) � ρ = ↓ � q � ρ (4) ↓ � app ( lam p , q ) � ρ = ↓ � p � � q � ρ ,ρ (5) ↓ � rec ( zero , p , q ) � ρ = ↓ � p � ρ (6) ↓ � rec ( succ r , p , q ) � ρ = ↓ � app ( app ( q , r ) , rec ( r , p , q )) � ρ (7) ↓ N � shift p � ρ = ↓ N � p � φ,ρ (8) ↓ N � app ( app ( hyp , x ) , y ) � φ,ρ = ↓ N � y � φ,ρ (9) where for the last two equations, φ := η ( ≥ 2 ν �→ η ( ≥ 3 α �→ η ( µα ))) . Danko Ilik – On the Indenspensability of Bar Recursion 23

3 A modified realizability interpretation Danko Ilik – On the Indenspensability of Bar Recursion 24