Realizability games for the specification problem Mauricio Guillermo - PowerPoint PPT Presentation

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Semantics Intuition falsity value � A � : stacks, opponent to A truth value | A | : terms, player of A pole ⊥ ⊥ : processes, referee t ⋆ π ≻ p 0 ≻ · · · ≻ p n ∈ ⊥ ⊥ ? � ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction Truth value defined by orthogonality : ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} ´ Etienne Miquey Realizability games for the specification problem 8/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Semantics Intuition falsity value � A � : stacks, opponent to A truth value | A | : terms, player of A pole ⊥ ⊥ : processes, referee t ⋆ π ≻ p 0 ≻ · · · ≻ p n ∈ ⊥ ⊥ ? � ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction Truth value defined by orthogonality : ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} ´ Etienne Miquey Realizability games for the specification problem 8/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Semantics Intuition falsity value � A � : stacks, opponent to A truth value | A | : terms, player of A pole ⊥ ⊥ : processes, referee t ⋆ π ≻ p 0 ≻ · · · ≻ p n ∈ ⊥ ⊥ ? � ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction Truth value defined by orthogonality : ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} ´ Etienne Miquey Realizability games for the specification problem 8/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Pole ⊥ ⊥ ⊂ Λ c ⋆ Π closed by anti-reduction : ∀ p , p ′ ∈ Λ c ⋆ Π : ( p ≻ p ′ ) ∧ ( p ′ ∈ ⊥ ⊥ ) ⇒ p ∈ ⊥ ⊥ Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Models ( M , ⊥ ⊥ ) Ground model M Truth value (player): ⊥ = { t ∈ Λ c : ∀ π ∈ � A � , t ⋆ π ∈ ⊥ | A | = � A � ⊥ ⊥} Falsity value (opponent): � A ⇒ B � = { t · π : t ∈ | A | ∧ π ∈ � B �} �∀ xA � = � n ∈ N � A { x := n }� F : N k →P (Π) � A { X := ˙ �∀ XA � = � F }� � ˙ F ( e 1 , ..., e k ) � = F ( � e 1 � , . . . , � e k � ) Notation t ∈ | A | = � A � ⊥ ⊥ t � A iff t � A iff t � A for all ⊥ ⊥ ´ Etienne Miquey Realizability games for the specification problem 9/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Remarks Case ⊥ ⊥ = ∅ (degenerated model) Truth as in the standard model: � Λ if � A � = 1 | A | = ∅ if � A � = 0 Realizable ⇔ True in the standard model Case ⊥ ⊥ � = ∅ t ⋆ π ∈ ⊥ ⊥ ⇒ forall A , k π t � A Restriction to proof-like ´ Etienne Miquey Realizability games for the specification problem 10/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Remarks Case ⊥ ⊥ = ∅ (degenerated model) Truth as in the standard model: � Λ if � A � = 1 | A | = ∅ if � A � = 0 Realizable ⇔ True in the standard model Case ⊥ ⊥ � = ∅ t ⋆ π ∈ ⊥ ⊥ ⇒ forall A , k π t � A Restriction to proof-like ´ Etienne Miquey Realizability games for the specification problem 10/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Properties Adequacy � x 1 : A 1 , . . . , x k : A k ⊢ t : A ⇒ t [ t 1 / x 1 , . . . , t k / x k ] � A ∀ i ∈ [1 , k ]( t i � A i ) Realizing Peano axioms If PA 2 − ⊢ A , then there is a closed proof-like term t s.t. t � A . Witness extraction If t � ∃ N x A ( x ) and A ( x ) is atomic or decidable, then we can build a term u s.t. that ∀ π ∈ Π: t ⋆ u · π ≻ stop ⋆ n · π ∧ A ( n ) holds ´ Etienne Miquey Realizability games for the specification problem 11/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Relativization Nat( x ) ≡ ∀ Z ( Z (0) ⇒ ∀ y ( Z ( y ) ⇒ Z ( s ( y ))) ⇒ Z ( x )) Proposition There is no t ∈ Λ c such that t � ∀ n . Nat ( n ) ´ Etienne Miquey Realizability games for the specification problem 12/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Relativization Nat( x ) ≡ ∀ Z ( Z (0) ⇒ ∀ y ( Z ( y ) ⇒ Z ( s ( y ))) ⇒ Z ( x )) Proposition There is no t ∈ Λ c such that t � ∀ n . Nat ( n ) Fix: ∀ nat x A := ∀ x (Nat( x ) ⇒ A ) Obviously, λ x . x � ∀ nat x Nat( x ) ´ Etienne Miquey Realizability games for the specification problem 12/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Relativization Nat( x ) ≡ ∀ Z ( Z (0) ⇒ ∀ y ( Z ( y ) ⇒ Z ( s ( y ))) ⇒ Z ( x )) Proposition There is no t ∈ Λ c such that t � ∀ n . Nat ( n ) Better : A , B ::= . . . | { e } ⇒ A �{ e } ⇒ A � = { ¯ n · π : � e � = n ∧ π ∈ � A �} ∀ N x A ( x ) ≡ ∀ x ( { x } ⇒ A ( x )) Let T be a storage operator. The following holds for any formula A ( x ): 1 λ x . x � ∀ nat x A ( x ) ⇒ ∀ N x A ( x ) 2 λ x . Tx � ∀ N x A ( x ) ⇒ ∀ nat x A ( x ) ´ Etienne Miquey Realizability games for the specification problem 12/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our problem Specification of A Can we give a characterization of { t ∈ Λ c : t � A } ? Absoluteness Are arithmetical formulæ absolute for realizability models ( M , ⊥ ⊥ )? ´ Etienne Miquey Realizability games for the specification problem 13/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion The specification problem ´ Etienne Miquey Realizability games for the specification problem 14/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole Two ways of building poles from any set P of processes. 1 goal-oriented : ⊥ := { p ∈ Λ c ⋆ Π : ∃ p ′ ∈ P , p ≻ p ′ } ⊥ 2 thread-oriented : th p = { p ′ ∈ Λ c ⋆ Π : p ≻ p ′ } Thread of p : th p ) c ≡ � � th c ⊥ ⊥ := ( p p ∈ P p ∈ P ´ Etienne Miquey Realizability games for the specification problem 15/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole Two ways of building poles from any set P of processes. 1 goal-oriented : ⊥ := { p ∈ Λ c ⋆ Π : ∃ p ′ ∈ P , p ≻ p ′ } ⊥ Proof: Let p 0 ∈ Λ ⋆ Π and ⊥ ⊥ 0 := { p ∈ Λ c ⋆ Π : p ≻ p 0 } . Let p 1 , p 2 ∈ Λ ⋆ Π be such that: p 1 ≻ p 2 and p 2 ∈ ⊥ ⊥ 0 ≡ p 2 ≻ p 0 . Then p 1 ≻ p 2 ≻ p 0 , thus p 1 ∈ ⊥ ⊥ 0 . � ´ Etienne Miquey Realizability games for the specification problem 15/ 39 2 thread-oriented :

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole Two ways of building poles from any set P of processes. 1 goal-oriented : ⊥ := { p ∈ Λ c ⋆ Π : ∃ p ′ ∈ P , p ≻ p ′ } ⊥ Proof: Let p 0 ∈ Λ ⋆ Π and ⊥ ⊥ 0 := { p ∈ Λ c ⋆ Π : p ≻ p 0 } . Let p 1 , p 2 ∈ Λ ⋆ Π be such that: p 1 ≻ p 2 and p 2 ∈ ⊥ ⊥ 0 ≡ p 2 ≻ p 0 . Then p 1 ≻ p 2 ≻ p 0 , thus p 1 ∈ ⊥ ⊥ 0 . � ´ Etienne Miquey Realizability games for the specification problem 15/ 39 2 thread-oriented :

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole 2 thread-oriented : th p = { p ′ ∈ Λ c ⋆ Π : p ≻ p ′ } Thread of p : th p ) c ≡ � � th c ⊥ ⊥ := ( p p ∈ P p ∈ P ´ Etienne Miquey Realizability games for the specification problem 15/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole 2 thread-oriented : th p = { p ′ ∈ Λ c ⋆ Π : p ≻ p ′ } Thread of p : th p ) c ≡ � � th c ⊥ ⊥ := ( p p ∈ P p ∈ P Proof: Let p 0 ∈ Λ ⋆ Π and ⊥ 0 := th c ⊥ p 0 ≡ { p ∈ Λ c ⋆ Π : p 0 �≻ p } . Let p 1 , p 2 ∈ Λ ⋆ Π be such that: p 1 ≻ p 2 and p 2 ∈ ⊥ ⊥ 0 ≡ p 0 �≻ p 2 . Assume p 0 ≻ p 1 , then p 0 ≻ p 1 ≻ p 2 / ∈ ⊥ ⊥ 0 Absurd ! � ´ Etienne Miquey Realizability games for the specification problem 15/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Our leverage: the pole 2 thread-oriented : th p = { p ′ ∈ Λ c ⋆ Π : p ≻ p ′ } Thread of p : th p ) c ≡ � � th c ⊥ ⊥ := ( p p ∈ P p ∈ P Proof: Let p 0 ∈ Λ ⋆ Π and ⊥ 0 := th c ⊥ p 0 ≡ { p ∈ Λ c ⋆ Π : p 0 �≻ p } . Let p 1 , p 2 ∈ Λ ⋆ Π be such that: p 1 ≻ p 2 and p 2 ∈ ⊥ ⊥ 0 ≡ p 0 �≻ p 2 . Assume p 0 ≻ p 1 , then p 0 ≻ p 1 ≻ p 2 / ∈ ⊥ ⊥ 0 Absurd ! � ´ Etienne Miquey Realizability games for the specification problem 15/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) iff ? By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) iff ? By definition: S ∈P (Π) � ˙ S ⇒ ˙ �∀ X . ( X ⇒ X ) � = � S � u ∈ | ˙ π ∈ � ˙ = � S ∈P (Π) { u · π : S | ∧ S �} u � ˙ = � S ∈P (Π) { u · π : ∧ π ∈ S } S Proof: ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: S ∈P (Π) � ˙ S ⇒ ˙ �∀ X . ( X ⇒ X ) � = � S � u ∈ | ˙ π ∈ � ˙ = � S ∈P (Π) { u · π : S | ∧ S �} u � ˙ = � S ∈P (Π) { u · π : ∧ π ∈ S } S Proof: ( ⇐ ) Obvious. ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. goal-oriented : 1 ⊥ ⊥ := { p : p ≻ u ⋆ π } Amounts to: u · π ∈ �∀ X . ( X ⇒ X ) � ? Define S := { π } . We check that: π ∈ S u � ˙ S ⇔ ∀ ρ ∈ S , u ⋆ ρ ∈ ⊥ ⊥ ⇔ u ⋆ π ∈ ⊥ ⊥ Thus t ⋆ u · π ∈ ⊥ ⊥ and t ⋆ u · π ≻ u ⋆ π . � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. goal-oriented : 1 ⊥ ⊥ := { p : p ≻ u ⋆ π } Amounts to: u · π ∈ �∀ X . ( X ⇒ X ) � ? Define S := { π } . We check that: π ∈ S u � ˙ S ⇔ ∀ ρ ∈ S , u ⋆ ρ ∈ ⊥ ⊥ ⇔ u ⋆ π ∈ ⊥ ⊥ Thus t ⋆ u · π ∈ ⊥ ⊥ and t ⋆ u · π ≻ u ⋆ π . � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. goal-oriented : 1 ⊥ ⊥ := { p : p ≻ u ⋆ π } Amounts to: u · π ∈ �∀ X . ( X ⇒ X ) � ? Define S := { π } . We check that: π ∈ S u � ˙ S ⇔ ∀ ρ ∈ S , u ⋆ ρ ∈ ⊥ ⊥ ⇔ u ⋆ π ∈ ⊥ ⊥ Thus t ⋆ u · π ∈ ⊥ ⊥ and t ⋆ u · π ≻ u ⋆ π . � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. thread-oriented : 2 ⊥ := th c ⊥ t ⋆ u · π ≡ { p ∈ Λ c ⋆ Π : t ⋆ u · π �≻ p } . Obviously, t ⋆ u · π / ∈ ⊥ ⊥ . Thus u · π / ∈ �∀ X . ( X ⇒ X ) � . Defining S := { π } , we deduce that : u � ˙ S ⇔ ∃ ρ ∈ S , u ⋆ ρ / ∈ ⊥ ⊥ ⇔ u ⋆ π / ∈ ⊥ ⊥ ⇔ t ⋆ u · π ≻ u ⋆ π � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. thread-oriented : 2 ⊥ := th c ⊥ t ⋆ u · π ≡ { p ∈ Λ c ⋆ Π : t ⋆ u · π �≻ p } . Obviously, t ⋆ u · π / ∈ ⊥ ⊥ . Thus u · π / ∈ �∀ X . ( X ⇒ X ) � . Defining S := { π } , we deduce that : u � ˙ S ⇔ ∃ ρ ∈ S , u ⋆ ρ / ∈ ⊥ ⊥ ⇔ u ⋆ π / ∈ ⊥ ⊥ ⇔ t ⋆ u · π ≻ u ⋆ π � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example: Identity-like t � ∀ X . ( X ⇒ X ) t ⋆ u · π ≻ u ⋆ π iff By definition: u � ˙ �∀ X . ( X ⇒ X ) � = � S ∈P (Π) { u · π : S ∧ π ∈ S } Proof: ( ⇒ ) Assume t � ∀ X ( X ⇐ X ) and fix u ∈ Λ and π ∈ Π. thread-oriented : 2 ⊥ := th c ⊥ t ⋆ u · π ≡ { p ∈ Λ c ⋆ Π : t ⋆ u · π �≻ p } . Obviously, t ⋆ u · π / ∈ ⊥ ⊥ . Thus u · π / ∈ �∀ X . ( X ⇒ X ) � . Defining S := { π } , we deduce that : u � ˙ S ⇔ ∃ ρ ∈ S , u ⋆ ρ / ∈ ⊥ ⊥ ⇔ u ⋆ π / ∈ ⊥ ⊥ ⇔ t ⋆ u · π ≻ u ⋆ π � ´ Etienne Miquey Realizability games for the specification problem 16/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π ⊥ 0 := ( th ( p 0 )) c and � X � = { π } : 0 Define p 0 := t 0 ⋆ κ s · κ z · π , ⊥ � κ z � 0 X implies p 0 ≻ κ z ⋆ π � κ z � 0 X implies κ s � 0 X ⇒ X and p 0 ≻ κ s ⋆ t 1 · π ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π ⊥ 0 := ( th ( p 0 )) c and � X � = { π } : 0 Define p 0 := t 0 ⋆ κ s · κ z · π , ⊥ � κ z � 0 X implies p 0 ≻ κ z ⋆ π � κ z � 0 X implies κ s � 0 X ⇒ X and p 0 ≻ κ s ⋆ t 1 · π ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π j ∈ [0 , i ] ( th ( p j )) c and � X � = { π } : Define p i := t i ⋆ π , ⊥ ⊥ i := � i � κ z � i X implies p i ≻ κ z ⋆ π � κ z � i X implies κ s � i X ⇒ X and p i ≻ κ s ⋆ t i +1 · π ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ X . ( X ⇒ X ) ⇒ X ⇒ X iff ??? t 0 ⋆ κ s · κ z · π ≻ κ s ⋆ t 1 · π t 1 ⋆ π ≻ κ s ⋆ t 2 · π . . . t i ⋆ π ≻ κ s ⋆ t i +1 · π . . . t s ⋆ π ≻ κ z ⋆ π j ∈ [0 , i ] ( th ( p j )) c and � X � = { π } : Define p i := t i ⋆ π , ⊥ ⊥ i := � i � κ z � i X implies p i ≻ κ z ⋆ π � κ z � i X implies κ s � i X ⇒ X and p i ≻ κ s ⋆ t i +1 · π Termination: i ∈ N ( th ( p i )) c , get a If ∀ i ∈ N ( κ z � i X ), define ⊥ ⊥ ∞ := � contradiction. ´ Etienne Miquey Realizability games for the specification problem 17/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ XY . ( X ⇒ Y ) ⇒ X ⇒ Y iff ??? t 0 ⋆ κ f · κ x · π ≻ κ f ⋆ t 1 · π t 1 ⋆ π ′ ≻ κ f ⋆ t 2 · π . . . t i ⋆ π ′ ≻ κ f ⋆ t i +1 · π . . . t s ⋆ π ′ κ x ⋆ π ′ ≻ (Same proof) ´ Etienne Miquey Realizability games for the specification problem 18/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion t 0 � ∀ XY . ( X ⇒ Y ) ⇒ X ⇒ Y iff ??? t 0 ⋆ κ f · κ x · π ≻ κ f ⋆ t 1 · π t 1 ⋆ π ′ ≻ κ f ⋆ t 2 · π . . . t i ⋆ π ′ ≻ κ f ⋆ t i +1 · π . . . t s ⋆ π ′ κ x ⋆ π ′ ≻ (Same proof) ´ Etienne Miquey Realizability games for the specification problem 18/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Arithmetical formulæ by hand Depth Player Game θ ⋆ k �·� · π �·� (1) j �·� ∃ 2 1 1 2 · · · 2 k �·� ⋆ m � 1 � · ξ � 1 � · π �·� k �·� ⋆ m � 2 � · ξ � 2 � · π �·� k �·� ⋆ m k �·� · ξ j �·� · π �·� (2) · · · ∀ ξ k �·� ⋆ p k �·� · k j �·� · π j �·� (3) ξ � 1 � ⋆ p � 1 � · k � 1 � · π � 1 � ξ � 2 � ⋆ p � 2 � · k � 2 � · π � 2 � ∃ 1 3 j � 21 � 2 (4) k � 2 � ⋆ m � 21 � · ξ � 21 � · π � 2 � k � 2 � ⋆ m j � 2 � · ξ j � 2 � · π � 2 � k � 2 � ⋆ m � 22 � · ξ � 22 � · π � 2 � . . . . ∀ . . . . . . . . . . . . ξ ˚ σ ⋆ p ˚ σ · k ˚ σ · π ˚ . . σ · · · · · · 1 j ˚ i σ ∃ k ˚ σ ⋆ m ˚ σ · 1 · ξ ˚ σ · 1 · π ˚ k ˚ σ ⋆ m ˚ σ · ξ ˚ σ · π ˚ σ · j ˚ σ · j ˚ σ σ k ˚ σ ⋆ m ˚ σ · i · ξ ˚ σ · i · π ˚ σ k ∀ σ ⋆ p ˚ σ · k ˚ σ · π ˚ (2n-1) ξ ˚ σ · 1 ⋆ p ˚ σ · 1 · k ˚ σ · 1 · π ˚ ξ ˚ σ · j ˚ σ · j ˚ σ · j ˚ σ · 1 σ · j ˚ σ · i ⋆ p ˚ σ · i · k ˚ σ · i · π ˚ σ ξ ˚ σ · i k ∃ (2n) k ˚ σ · i ⋆ π ˚ σ · i ∀ X ´ Etienne Miquey Realizability games for the specification problem 19/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Advertisement Problem You want to specify A . Methodology: � requirement: some intuition... 1 direct-style : define the good poles, 2 indirect-style : try the thread method, 3 induction-style : define a game ´ Etienne Miquey Realizability games for the specification problem 20/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Advertisement Problem You want to specify A . Methodology: � requirement: some intuition... 1 direct-style : define the good poles, 2 indirect-style : try the thread method, 3 induction-style : define a game ´ Etienne Miquey Realizability games for the specification problem 20/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion A first notion of game ´ Etienne Miquey Realizability games for the specification problem 21/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Coquand’s game Arithmetical formula Φ 2 h : ∃ x 1 ∀ y 1 . . . ∃ x h ∀ y h f ( � x h , � y h ) = 0 Rules: Players : Eloise ∃ and Abelard ∀ . Moves : - at his turn, each player instantiates his variable - Eloise allowed to backtrack Final position : evaluation of f ( � m h , � n h ) = 0 : true : Eloise wins false : game continues Abelard wins if the game never ends Winning strategy Way of playing that ensures the victory, independently of the opponent moves. ´ Etienne Miquey Realizability games for the specification problem 22/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Coquand’s game Arithmetical formula Φ 2 h : ∃ x 1 ∀ y 1 . . . ∃ x h ∀ y h f ( � x h , � y h ) = 0 Rules: Players : Eloise ∃ and Abelard ∀ . Moves : - at his turn, each player instantiates his variable - Eloise allowed to backtrack Final position : evaluation of f ( � m h , � n h ) = 0 : true : Eloise wins false : game continues Abelard wins if the game never ends Winning strategy Way of playing that ensures the victory, independently of the opponent moves. ´ Etienne Miquey Realizability games for the specification problem 22/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion The need of backtrack f ( m , n , p ) = 0 := ( n > 0 ∧ Halt ( m , n )) ∨ ( n = 0 ∧ ¬ Halt ( m , p )) the m th Turing machine stops before n steps Halt ( m , n ) := Formula ∀ m ∃ n ∀ p ( f ( m , n , p ) = 0) Winning strategy ? Abelard plays the code m of a Turing machine M . Eloise chooses to play n = 0 (” M never stops ”) Abelard answers a given number of steps p Eloise checks whether M stops before p steps: either M is still running after p steps : � Eloise wins. either M stops before p steps : � Eloise backtracks and plays p instead of 0 (“ M stops before p steps ”) ´ Etienne Miquey Realizability games for the specification problem 23/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion The need of backtrack f ( m , n , p ) = 0 := ( n > 0 ∧ Halt ( m , n )) ∨ ( n = 0 ∧ ¬ Halt ( m , p )) the m th Turing machine stops before n steps Halt ( m , n ) := Formula ∀ m ∃ n ∀ p ( f ( m , n , p ) = 0) Winning strategy : Abelard plays the code m of a Turing machine M . Eloise chooses to play n = 0 (” M never stops ”) Abelard answers a given number of steps p Eloise checks whether M stops before p steps: either M is still running after p steps : � Eloise wins. either M stops before p steps : � Eloise backtracks and plays p instead of 0 (“ M stops before p steps ”) ´ Etienne Miquey Realizability games for the specification problem 23/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula ∃ x ∀ y ∃ z ( x · y = 2 · z ) Player Action Position Start P 0 = ( · , · , · ) ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula ∀ y ∃ z (1 · y = 2 · z ) Player Action Position Start P 0 = ( · , · , · ) ∃ x := 1 P 1 = (1 , · , · ) ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula ∃ z (1 = 2 · z ) Player Action Position Start P 0 = ( · , · , · ) ∃ x := 1 P 1 = (1 , · , · ) ∀ y := 1 P 2 = (1 , 1 , · ) ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula ∀ y ∃ z (2 · y = 2 · z ) Player Action Position Start P 0 = ( · , · , · ) ∃ x := 1 P 1 = (1 , · , · ) ∀ y := 1 P 2 = (1 , 1 , · ) ∃ P 3 = (2 , · , · ) backtrack to P 0 + x := 2 ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula ∃ z (2 = 2 · z ) Player Action Position Start P 0 = ( · , · , · ) ∃ x := 1 P 1 = (1 , · , · ) ∀ y := 1 P 2 = (1 , 1 , · ) ∃ P 3 = (2 , · , · ) backtrack to P 0 + x := 2 P 4 = (2 , 1 , · ) ∀ y := 1 ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula 2 = 2 Player Action Position Start P 0 = ( · , · , · ) P 1 = (1 , · , · ) ∃ x := 1 P 2 = (1 , 1 , · ) ∀ y := 1 ∃ backtrack to P 0 + x := 2 P 3 = (2 , · , · ) ∀ y := 1 P 4 = (2 , 1 , · ) ∃ z := 1 P 5 = (2 , 1 , 1) ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula 2 = 2 Player Action Position Start P 0 = ( · , · , · ) P 1 = (1 , · , · ) ∃ x := 1 P 2 = (1 , 1 , · ) ∀ y := 1 ∃ backtrack to P 0 + x := 2 P 3 = (2 , · , · ) ∀ y := 1 P 4 = (2 , 1 , · ) ∃ z := 1 P 5 = (2 , 1 , 1) evaluation ∃ wins ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Example Formula 2 = 2 Player Action Position Start P 0 = ( · , · , · ) P 1 = (1 , · , · ) ∃ x := 1 P 2 = (1 , 1 , · ) ∀ y := 1 ∃ backtrack to P 0 + x := 2 P 3 = (2 , · , · ) ∀ y := 1 P 4 = (2 , 1 , · ) ∃ z := 1 P 5 = (2 , 1 , 1) evaluation ∃ wins History H := � n P n ´ Etienne Miquey Realizability games for the specification problem 24/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 0 : deductive system Rules: If there exists ( � m h , � n h ) ∈ H such that M � f ( � m h , � n h ) = 0: Win H ∈ W 0 Φ For all i < h , ( � m i , � n i ) ∈ H and m ∈ N : n i · n ) } ∈ W 0 H ∪ { ( � m i · m , � ∀ n ∈ N Φ Play H ∈ W 0 Φ ´ Etienne Miquey Realizability games for the specification problem 25/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ ≡ ∃ N x 1 ∀ N y 1 . . . ∃ N x h ∀ y h ( f ( � x h , � y h ) = 0) ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ ≡ ∃ N x 1 ∀ N y 1 . . . ∃ N x h ∀ y h ( f ( � x h , � y h ) = 0) Φ 0 ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) Φ i − 1 ≡ ∀ X i ( ∀ N x i ( ∀ N y i Φ i ⇒ X i ) ⇒ X i ) Φ h ≡ ∀ W ( W ( f ( � x h , � y h )) ⇒ W (0)) ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ 0 ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) Φ i − 1 ≡ ∀ X i ( ∀ N x i ( ∀ N y i Φ i ⇒ X i ) ⇒ X i ) Φ h ≡ ∀ W ( W ( f ( � x h , � y h )) ⇒ W (0)) Realizability � A ⇒ B � = { u · π : u ∈ | A | ∧ π ∈ � B �} �∀ N xA ( x ) � = � n ∈ N { n · π : π ∈ � A ( n ) �} ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ 0 ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) Start : Eloise proposes t 0 to defend Φ 0 Abelard proposes u 0 · π 0 to attack Φ 0 move p i ( ∃ -position) history 0 t 0 ⋆ u 0 · π 0 H 0 := { ( ∅ , ∅ , u 0 , π 0 ) } ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ 0 ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) move p i ( ∃ -position) history 0 t 0 ⋆ u 0 · π 0 H 0 := { ( ∅ , ∅ , u 0 , π 0 ) } Eloise reduces p 0 until p 0 ≻ u 0 ⋆ m 1 · t 1 · π 0 � she can decide to play ( m 1 , t 1 ) and ask for Abelard’s answer � Abelard must give n 1 · u ′ · π ′ . ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ 0 ≡ ∀ X 1 ( ∀ N x 1 ( ∀ N y 1 Φ 1 ⇒ X 1 ) ⇒ X 1 ) move p i ( ∃ -position) history 0 t 0 ⋆ u 0 · π 0 H 0 := { ( ∅ , ∅ , u 0 , π 0 ) } 1 t 1 ⋆ n 1 · u 1 · π 1 H 1 := { ( m 1 , n 1 , u 1 , π 1 ) } ∪ H 0 Eloise reduces p 0 until p 0 ≻ u 0 ⋆ m 1 · t 1 · π 0 � she can decide to play ( m 1 , t 1 ) and ask for Abelard’s answer � Abelard must give n 1 · u ′ · π ′ . ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ i − 1 ≡ ∀ X i ( ∀ N x i ( ∀ N y i Φ i ⇒ X i ) ⇒ X i ) move p i ( ∃ -position) history 1 t 1 ⋆ n 1 · u 1 · π 1 H 1 := { ( m 1 , n 1 , u 1 , π 1 ) } ∪ H 0 . . . . . . . . . t i ⋆ n i · u i · π i H i := { ( m i , n i , u i , π i ) } ∪ H i − 1 i Eloise reduces p i until p i ≻ u ⋆ m · t · π with ( � m j , � n j , u , π ) ∈ H j where j < h . � she can decide to play ( m i +1 , t i +1 ) � Abelard must give n i · u ′ · π ′ . ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : playing with realizability Formulæ structure Φ h ≡ ∀ W ( W ( f ( � x h , � y h )) ⇒ W (0)) move p i ( ∃ -position) history 1 t 1 ⋆ n 1 · u 1 · π 1 H 1 := { ( m 1 , n 1 , u 1 , π 1 ) } ∪ H 0 . . . . . . . . . t i ⋆ n i · u i · π i H i := { ( m i , n i , u i , π i ) } ∪ H i − 1 i Eloise reduces p i until p i ≻ u ⋆ m · t · π with ( � m j , � n j , u , π ) ∈ H j where j < h . � she can decide to play ( m i +1 , t i +1 ) � Abelard must give n i · u ′ · π ′ . p i ≻ u ⋆ π with ( � m h , � n h , u , π ) ∈ H j � she wins iff M | = f ( � m h , � n h ) = 0. ´ Etienne Miquey Realizability games for the specification problem 26/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : formal definition if ∃ ( � m h , � n h , u , π ) ∈ H s.t. p ≻ u ⋆ π and M � f ( � m h , � n h ) = 0 : Win � p , H � ∈ W 1 Φ for every ( � m i , � n i , u , π ) ∈ H , m ∈ N s.t. p ≻ u ⋆ m · t · π : � t ⋆ n · u ′ · π ′ , H ∪ { ( � n i · n , u ′ , π ′ ) }� ∈ W 1 ∀ ( n ′ , u ′ , π ′ ) m i · m , � Φ Play � p , H � ∈ W 1 Φ Winning strategy t ∈ Λ c s.t. for any handle ( u , π ) ∈ Λ × Π : � t ⋆ u · π, { ( ∅ , ∅ , u , π ) }� ∈ W 1 Φ ´ Etienne Miquey Realizability games for the specification problem 27/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 1 : formal definition if ∃ ( � m h , � n h , u , π ) ∈ H s.t. p ≻ u ⋆ π and M � f ( � m h , � n h ) = 0 : Win � p , H � ∈ W 1 Φ for every ( � m i , � n i , u , π ) ∈ H , m ∈ N s.t. p ≻ u ⋆ m · t · π : � t ⋆ n · u ′ · π ′ , H ∪ { ( � n i · n , u ′ , π ′ ) }� ∈ W 1 ∀ ( n ′ , u ′ , π ′ ) m i · m , � Φ Play � p , H � ∈ W 1 Φ Winning strategy t ∈ Λ c s.t. for any handle ( u , π ) ∈ Λ × Π : � t ⋆ u · π, { ( ∅ , ∅ , u , π ) }� ∈ W 1 Φ ´ Etienne Miquey Realizability games for the specification problem 27/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Specification result Adequacy If t is a winning strategy for G 1 Φ , then t � Φ Proof (sketch): - play a match with stacks from falsity value, - conclude by anti-reduction. ´ Etienne Miquey Realizability games for the specification problem 28/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Specification result Adequacy If t is a winning strategy for G 1 Φ , then t � Φ Proof (sketch): - play a match with stacks from falsity value, - conclude by anti-reduction. Completeness of G 1 If the calculus is deterministic and substitutive, then if t � Φ then t is a winning strategy for the game G 1 Φ Proof (sketch): by contradiction - substitute Abelard’s winning answers along the thread scheme, - reach a winning position anyway. ´ Etienne Miquey Realizability games for the specification problem 28/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Specification result Adequacy If t is a winning strategy for G 1 Φ , then t � Φ Proof (sketch): - play a match with stacks from falsity value, - conclude by anti-reduction. Completeness of G 1 If the calculus is deterministic and substitutive , then if t � Φ then t is a winning strategy for the game G 1 Φ Proof (sketch): by contradiction - substitute Abelard’s winning answers along the thread scheme, - reach a winning position anyway. ´ Etienne Miquey Realizability games for the specification problem 28/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion The general case ´ Etienne Miquey Realizability games for the specification problem 29/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Loosing the substition quote quote ⋆ ϕ · t · π ≻ t ⋆ n ϕ · π the calculus is no longer substitutive there are some wild realizers which are not winning strategies! Consider Φ ≤ ≡ ∃ N x ∀ N y ( x ≤ y ) and t ≤ s.t. : t ≤ ⋆ u · π ≻ T 0 ⋆ π ≻ u ⋆ 0 · T 1 · π and : � � ⋆ π ′ if u ′ ≡ T 0 and π ′ ≡ π T 1 ⋆ n · u ′ · π ′ ≻ u ′ ⋆ π ′ otherwise ´ Etienne Miquey Realizability games for the specification problem 30/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Loosing the substition quote quote ⋆ ϕ · t · π ≻ t ⋆ n ϕ · π the calculus is no longer substitutive there are some wild realizers which are not winning strategies! Consider Φ ≤ ≡ ∃ N x ∀ N y ( x ≤ y ) and t ≤ s.t. : t ≤ ⋆ u · π ≻ T 0 ⋆ π ≻ u ⋆ 0 · T 1 · π and : � � ⋆ π ′ if u ′ ≡ T 0 and π ′ ≡ π T 1 ⋆ n · u ′ · π ′ ≻ u ′ ⋆ π ′ otherwise ´ Etienne Miquey Realizability games for the specification problem 30/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Loosing the substition quote quote ⋆ ϕ · t · π ≻ t ⋆ n ϕ · π the calculus is no longer substitutive there are some wild realizers which are not winning strategies! Consider Φ ≤ ≡ ∃ N x ∀ N y ( x ≤ y ) and t ≤ s.t. : t ≤ ⋆ u · π ≻ T 0 ⋆ π ≻ u ⋆ 0 · T 1 · π and : � � ⋆ π ′ if u ′ ≡ T 0 and π ′ ≡ π T 1 ⋆ n · u ′ · π ′ ≻ u ′ ⋆ π ′ otherwise � Idea : I’ve already been there ... ´ Etienne Miquey Realizability games for the specification problem 30/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 2 : non-substitutive case � Idea : I’ve already been there... if ∃ ( � m h , � n h , u , π ) ∈ H s.t. p ≻ u ⋆ π and M � f ( � m h , � n h ) = 0 : Win � p , H � ∈ W 1 Φ for every ( � m i , � n i , u , π ) ∈ H , m ∈ N s.t. p ≻ u ⋆ m · t · π : � t ⋆ n · u ′ · π ′ , H ∪ { ( � n i · n , u ′ , π ′ ) }� ∈ W 1 ∀ ( n ′ , u ′ , π ′ ) m i · m , � Φ Play � p , H � ∈ W 1 Φ ´ Etienne Miquey Realizability games for the specification problem 31/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion G 2 : non-substitutive case � Idea : I’ve already been there... if ∃ ( � m h , � n h , u , π ) ∈ H , ∃ p ∈ P s.t. p ≻ u ⋆ π and M � f ( � m h , � n h ) = 0 : Win � P , H � ∈ W 2 Φ for every ( � m i , � n i , u , π ) ∈ H , m ∈ N s.t. ∃ p ∈ P , p ≻ u ⋆ m · t · π : �{ t ⋆ n · u ′ · π ′ } ∪ P � , H ∪ { ( � n i · n , u ′ , π ′ ) } ∈ W 2 ∀ ( n ′ , u ′ , π ′ ) m i · m , � Φ P � P , H � ∈ W 2 Φ ´ Etienne Miquey Realizability games for the specification problem 31/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Specification result Adequacy If t is a winning strategy for G 2 Φ , then t � Φ Proof (sketch): - play a match with stacks from falsity value, - conclude by anti-reduction. ´ Etienne Miquey Realizability games for the specification problem 32/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Specification result Adequacy If t is a winning strategy for G 2 Φ , then t � Φ Proof (sketch): - play a match with stacks from falsity value, - conclude by anti-reduction. Completeness of G 2 If t � Φ then t is a winning strategy for the game G 2 Φ Proof (sketch): by contradiction, - build an increasing sequence � P i , H i � using ∀ winning answers, p ∈ P ∞ th ( p )) c , - define ⊥ ⊥ := ( � - reach a contradiction. ´ Etienne Miquey Realizability games for the specification problem 32/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Consequences Proposition : Uniform winning strategy There exists a term T such that if: - M � ∃ x 1 ∀ y 1 ... f ( � x , � y ) = 0 - θ f computes f then T θ f is a winning strategy for ∃ x 1 ∀ y 1 ... f ( � x , � y ) = 0. Proof: Enumeration of N k , using θ f to check whether we reached a winning position. Theorem : Absoluteness If Φ is an arithmetical formula, then ∃ t ∈ Λ c , t � Φ M � Φ iff ´ Etienne Miquey Realizability games for the specification problem 33/ 39

G 1 : a first game G 2 : general case Introduction Classical realizability Specification Conclusion Consequences Proposition : Uniform winning strategy There exists a term T such that if: - M � ∃ x 1 ∀ y 1 ... f ( � x , � y ) = 0 - θ f computes f then T θ f is a winning strategy for ∃ x 1 ∀ y 1 ... f ( � x , � y ) = 0. Proof: Enumeration of N k , using θ f to check whether we reached a winning position. Theorem : Absoluteness If Φ is an arithmetical formula, then ∃ t ∈ Λ c , t � Φ M � Φ iff ´ Etienne Miquey Realizability games for the specification problem 33/ 39