Computational interpretation of classical forcing Lionel R ieg - PowerPoint PPT Presentation

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Computational interpretation of classical forcing Lionel R ieg Collège de France July 22 nd , 2016 July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 1 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing The question Logic Programs ¬¬ -translation CPS translation � formula ⊥ � return type Forcing � forcing conditions ??? � forcing transformation July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 2 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing in one drawing construction (model theory) g July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 3 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing in one drawing construction (model theory) g translation (proof theory) July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 3 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing in one drawing construction (model theory) g t : A translation t* : p F A (proof theory) July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 3 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Outline Formal proof system: PA ω + 1 Forcing in PA ω + 2 An example of computation by forcing 3 July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 4 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing PA ω + : syntax Sorts τ, σ := ι o τ → σ | | Expressions x τ λ x τ . M M , N , A , B := M N λ -calculus | | | 0 | S | rec τ arithmetic ∀ x τ . A | A ⇒ B | minimal logic Proof-terms t , u := λ x . t x | | t u | callcc July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 5 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing PA ω + : Logical connectives Second-order encodings: := ∀ Z . Z ⊥ ¬ A := A ⇒ ⊥ := ∀ Z . ( A ⇒ B ⇒ Z ) ⇒ Z A ∧ B A ∨ B := ∀ Z . ( A ⇒ Z ) ⇒ ( B ⇒ Z ) ⇒ Z ∃ x . A := ∀ Z . ( ∀ x . A ⇒ Z ) ⇒ Z e 1 = e 2 := ∀ Z . Z e 1 ⇒ Z e 2 Notations: x ∈ P := P ( x ) ∀ x ∈ P . A := ∀ x . x ∈ P ⇒ A ∃ x ∈ P . A := ∃ x . x ∈ P ∧ A July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 6 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing PA ω + : syntax Sorts τ, σ := ι o τ → σ | | Expressions x τ λ x τ . M M , N , A , B := M N | | | 0 | S | rec τ ∀ x τ . A M � τ N ֒ → A | A ⇒ B | | Proof-terms t , u := λ x . t x | | t u | callcc M � τ N ֒ → A ⇐⇒ M = N ⇒ A + some congruence on formulas July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 7 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing PA ω + : proof system Axiom E ; Γ , x : A ⊢ x : A Peirce E ; Γ ⊢ callcc : (( A ⇒ B ) ⇒ A ) ⇒ A E ; Γ ⊢ t : A A ≈ E A ′ Congruence E ; Γ ⊢ t : A ′ E ; Γ , x : A ⊢ t : B E ; Γ ⊢ t : A ⇒ B E ; Γ ⊢ u : A ⇒ i E ; Γ ⇒ e ⊢ λ x . t : A ⇒ B E ; Γ ⊢ t u : B E ; Γ ⊢ t : ∀ x τ . A E ; Γ ⊢ t : A ∀ i x � FV (Γ , E ) ∀ e E ; Γ ⊢ t : ∀ x τ . A E ; Γ ⊢ t : A [ N τ / x τ ] E , M = N ; Γ ⊢ t : A E ; Γ ⊢ t : M � τ M ֒ → A ֒ → i ֒ → e E ; Γ ⊢ t : M � τ N ֒ → A E ; Γ ⊢ t : A July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 8 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Classical realizability semantics Different from intuitionistic realizability intuitionistic: limits proofs, full extraction classical: full proofs, limits extraction July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 9 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Classical realizability semantics Different from intuitionistic realizability intuitionistic: limits proofs, full extraction classical: full proofs, limits extraction The KAM (Krivine’s Abstract Machine) Stack machine for λ -calculus + callcc July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 9 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Classical realizability semantics Different from intuitionistic realizability intuitionistic: limits proofs, full extraction classical: full proofs, limits extraction The KAM (Krivine’s Abstract Machine) Stack machine for λ -calculus + callcc Realizability interpretation Based on a pole � (set of processes of the KAM) Propositions interpreted by stacks (refutations) Realizers defined by orthogonality: | A | := � A � � July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 9 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Classical realizability semantics Different from intuitionistic realizability intuitionistic: limits proofs, full extraction classical: full proofs, limits extraction The KAM (Krivine’s Abstract Machine) Stack machine for λ -calculus + callcc Realizability interpretation Based on a pole � (set of processes of the KAM) Propositions interpreted by stacks (refutations) Realizers defined by orthogonality: | A | := � A � � Results: Adequacy: ⊢ t : A implies t � A Logical consistency: when � = ∅ , Tarski model Simple methods to extract witnesses for Σ 0 1 formulas July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 9 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Outline Formal proof system: PA ω + 1 Forcing in PA ω + 2 An example of computation by forcing 3 July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 10 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: overall idea PAω⁺+G g PAω⁺ t : A translation t* : p F A (proof theory) July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 11 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: input Definition (Forcing structure) A forcing structure is given by a sort κ of forcing conditions a predicate C κ → o of well-formed conditions ( p ∈ C written C [ p ] ) a product operation · on forcing conditions a maximal condition 1 a bunch of proof terms α 0 , . . . , α 8 G = generic filter on the set of forcing conditions = “approximations of g ” g = � G July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 12 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: input (example) Example (Forcing structure) The forcing structure to add a single Cohen real κ := ι (finite relations between N and Bool) C [ p ] := “ p is functional” ( p : N ⇀ Bool ) p · q := p ∪ q 1 := ∅ α 0 , . . . , α 8 G := pair-wise compatible finite functions from N to Bool = “approximations of g ” g = � G (a full function from N to Bool) July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 13 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: output 3 translations ( _ ) ∗ : July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 14 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: output 3 translations ( _ ) ∗ : on kinds: ι ∗ := ι o ∗ := κ → o ( σ → τ ) ∗ := σ ∗ → τ ∗ July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 14 / 29

Formal proof system: PA ω + Forcing in PA ω + An example of computation by forcing Forcing: output 3 translations ( _ ) ∗ : on kinds: ι ∗ := ι o ∗ := κ → o ( σ → τ ) ∗ := σ ∗ → τ ∗ on expressions: ( A ⇒ B ) ∗ p := ∀ q ∀ r . p � q · r ֒ → ( ∀ s . C [ q · s ] ⇒ A ∗ s ) ⇒ B ∗ r merely propagates through other constructions July 22 nd , 2016 Lionel R ieg (Collège de France) Computational interpretation of classical forcing 14 / 29