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Introduction A constructive proof of DC Algebraic models of classical realizability Classical realizability and side-effects Etienne Miquey 1,2 1 Equipe r 2 (INRIA), IRIF, Universit e Paris-Diderot 2 IMERL, Fac. de Ingenier a,

  1. Introduction A constructive proof of DC Algebraic models of classical realizability Theory vs Model What is the status of axioms ( e.g. A ∨ ¬ A )? � neither true nor false in the ambient theory (here, true means provable ) There is another point of view: Theory : provability in an axiomatic representation (syntax) Model : validity in a particular structure (semantic) Example: A ∧ B A ∨ B A ¬ A A ∨ ¬ A B B ✓ ✗ ✓ ✗ A A ✓ ✗ ✓ ✓ ✓ ✗ ✓ ✓ ✓ ✗ ✓ ✓ ✗ ✗ ✗ ✗ ✓ ✗ Valid formula ´ Etienne Miquey Classical realizability and side-effects 11/ 47

  2. Introduction A constructive proof of DC Algebraic models of classical realizability Teaser #2 Classical realizability: A �→ { t : t � A } (intuition: programs that share a common computational behavior given by A) Great news Classical realizability semantics gives surprisingly new models! In Part III, we will study the algebraic structure of these models. ´ Etienne Miquey Classical realizability and side-effects 12/ 47

  3. Introduction A constructive proof of DC Algebraic models of classical realizability Part II - A constructive proof of dependent choice compatible with classical logic ´ Etienne Miquey Classical realizability and side-effects 13/ 47

  4. Introduction A constructive proof of DC Algebraic models of classical realizability The axiom of choice Chapter 5 Axiom of Choice : ∀ x A . ∃ y B . P ( x , y ) → ∃ f A → B . ∀ x A . P ( x , f ( x )) AC : ´ Etienne Miquey Classical realizability and side-effects 14/ 47

  5. Introduction A constructive proof of DC Algebraic models of classical realizability The axiom of choice Chapter 5 Axiom of Choice : ∀ x A . ∃ y B . P ( x , y ) → ∃ f A → B . ∀ x A . P ( x , f ( x )) AC : : = λ H . ( λ x . wit ( H x ) , λ x . prf ( H x )) Computational content through dependent types: Γ ⊢ p : A [ t / x ] Γ ⊢ t : T Γ , x : T ⊢ t : A ( ∀ I ) ( ∃ I ) Γ ⊢ λ x . t : ∀ x T . A Γ ⊢ ( t , p ) : ∃ x T . A Γ ⊢ p : ∃ x T . A ( x ) Γ ⊢ p : ∃ x T . A ( x ) ( wit ) ( prf ) Γ ⊢ wit p : T Γ ⊢ prf p : A ( wit p ) ´ Etienne Miquey Classical realizability and side-effects 14/ 47

  6. Introduction A constructive proof of DC Algebraic models of classical realizability Incompatibility with classical logic Bad news dependent sum + classical logic = ❆ One can define: H 0 : = call/cc α ( 1 , throw α ( 0 , p )) : ∃ x . P ( x ) and reach a contradiction: P ( 0 ) ���� ( wit H 0 , prf H 0 ) → ( 1 , p ) � ����� �� ����� � ✘✘✘ ∃ x . P ( x ) We need to: � share � restrict dependent types ´ Etienne Miquey Classical realizability and side-effects 15/ 47

  7. Introduction A constructive proof of DC Algebraic models of classical realizability Incompatibility with classical logic Bad news dependent sum + classical logic = ❆ One can define: H 0 : = call/cc α ( 1 , throw α ( 0 , p )) : ∃ x . P ( x ) and reach a contradiction: P ( 0 ) ���� ( wit H 0 , prf H 0 ) → ( 1 , p ) � ����� �� ����� � ✘✘✘ ∃ x . P ( x ) We need to: � share � restrict dependent types ´ Etienne Miquey Classical realizability and side-effects 15/ 47

  8. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a solution ? Restriction to countable choice: AC � : ∀ x � . ∃ y B . P ( x , y ) → ∃ f � → B . ∀ x � . P ( x , f ( x )) Proof: AC : = λ H . ( λ n . if n = 0 then wit ( H 0 ) else λ n . if n = 1 then wit ( H 1 ) else ... , λ n . if n = 0 then prf ( H 0 ) else λ n . if n = 1 then prf ( H 1 ) else ... ) ´ Etienne Miquey Classical realizability and side-effects 16/ 47

  9. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a solution ? Restriction to countable choice: AC � : ∀ x � . ∃ y B . P ( x , y ) → ∃ f � → B . ∀ x � . P ( x , f ( x )) Proof: AC � : = λ H . let H 0 = H 0 in let H 1 = H 1 in ... ( λ n . if n = 0 then wit H 0 else λ n . if n = 1 then wit H 1 else ... , λ n . if n = 0 then prf H 0 else λ n . if n = 1 then prf H 1 else ... ) ´ Etienne Miquey Classical realizability and side-effects 16/ 47

  10. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a solution ? Restriction to countable choice: AC � : ∀ x � . ∃ y B . P ( x , y ) → ∃ f � → B . ∀ x � . P ( x , f ( x )) Proof: AC � : = λ H . let H ∞ = ( H 0 , H 1 ,. . . , H n ,. . . ) in ( λ n . wit ( nth n H ∞ ) , λ n . prf ( nth n H ∞ )) ´ Etienne Miquey Classical realizability and side-effects 16/ 47

  11. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a solution ? Restriction to countable choice: AC � : ∀ x � . ∃ y B . P ( x , y ) → ∃ f � → B . ∀ x � . P ( x , f ( x )) Proof: AC � : = λ H . let H ∞ = cofix 0 bn ( H n , b ( S ( n ))) in ( λ n . wit ( nth n H ∞ ) , λ n . prf ( nth n H ∞ )) ´ Etienne Miquey Classical realizability and side-effects 16/ 47

  12. Introduction A constructive proof of DC Algebraic models of classical realizability dPA ω (Herbelin’s recipe) A proof system: classical : p , q :: = ... | catch α p | throw α p with stratified dependent types : t , u :: = ... | wit p terms: A , B :: = ... | ∀ x T . A | ∃ x T . A | Π a : A . B | t = u formulas: p , q :: = ... | λ x . p | ( t , p ) | λ a . p proofs: a syntactical restriction of dependencies to nef proofs call-by-value and sharing : p , q :: = ... | let a = q in p with inductive and coinductive constructions: p , q :: = ... | ind t bn [ p | p ] | cofix t bn p lazy evaluation for the cofix ´ Etienne Miquey Classical realizability and side-effects 17/ 47

  13. Introduction A constructive proof of DC Algebraic models of classical realizability dPA ω (Herbelin’s recipe) A proof system: classical : p , q :: = ... | catch α p | throw α p with stratified dependent types : t , u :: = ... | wit p terms: A , B :: = ... | ∀ x T . A | ∃ x T . A | Π a : A . B | t = u formulas: p , q :: = ... | λ x . p | ( t , p ) | λ a . p proofs: a syntactical restriction of dependencies to nef proofs call-by-value and sharing : p , q :: = ... | let a = q in p with inductive and coinductive constructions: p , q :: = ... | ind t bn [ p | p ] | cofix t bn p lazy evaluation for the cofix ´ Etienne Miquey Classical realizability and side-effects 17/ 47

  14. Introduction A constructive proof of DC Algebraic models of classical realizability State of the art Subject reduction If Γ ⊢ p : A and p → q , then Γ ⊢ q : A . Normalization If Γ ⊢ p : A then p is normalizable. requires Consistency � dPA ω ⊥ ´ Etienne Miquey Classical realizability and side-effects 18/ 47

  15. Introduction A constructive proof of DC Algebraic models of classical realizability Roadmap dPA ω [Herbelin’12] : + control operators Subject reduction + dependent types + co-fixpoints + sharing & laziness typing/reduction preservation dLPA ω ? + sequent calculus Subject reduction + dependent types + co-fixpoints + sharing & laziness CPS-translation? ?-calculus Normalization ´ Etienne Miquey Classical realizability and side-effects 19/ 47

  16. Introduction A constructive proof of DC Algebraic models of classical realizability Roadmap dPA ω [Herbelin’12] : + control operators Subject reduction + dependent types + co-fixpoints + sharing & laziness typing/reduction preservation dLPA ω ? + sequent calculus Subject reduction + dependent types λ [ lv τ ⋆ ] + co-fixpoints + sharing & laziness CPS-translation? ?-calculus Normalization ´ Etienne Miquey Classical realizability and side-effects 19/ 47

  17. Introduction A constructive proof of DC Algebraic models of classical realizability Roadmap dPA ω [Herbelin’12] : + control operators Subject reduction + dependent types + co-fixpoints + sharing & laziness typing/reduction preservation dLPA ω ? + sequent calculus dL Subject reduction + dependent types + co-fixpoints + sharing & laziness CPS-translation? ?-calculus Normalization ´ Etienne Miquey Classical realizability and side-effects 19/ 47

  18. Introduction A constructive proof of DC Algebraic models of classical realizability Classical call-by-need Chapter 6 The λ [ lv τ ⋆ ] -calculus (Ariola et al. ): a sequent calculus with explicit stores Danvy’s method of semantics artifact: derive a small-step reduction system 1 derive context-free small-step reduction rules 2 derive an (untyped) CPS 3 Qestions: � Does it normalize? � Can the CPS be typed? � Can we define a realizability interpretation? ´ Etienne Miquey Classical realizability and side-effects 20/ 47

  19. Introduction A constructive proof of DC Algebraic models of classical realizability Danvy’s semantics artifacts Syntax: p :: = V | µα . c (Proofs) V :: = v | a (Weak values) v :: = λ a . p (Strong values) e :: = E | ˜ µ a . c (Contexts) E :: = α | F | ˜ µ [ a ] . � a || F � τ (Catchable contexts) F :: = p · E (Forcing contexts) c :: = � p || e � (Commands) l :: = c τ (Closures) τ :: = ϵ | τ [ a : = p ] (Store) Reduction rules: � p || ˜ µ a . c � τ c τ [ a : = p ] (Lazy storage) → � µα . c || E � τ ( c [ E / α ] ) τ → � a || F � τ [ a : = p ] τ ′ µ [ a ] . � a || F � τ ′ � τ � p || ˜ (Lookup) → µ [ a ] . � a || F � τ ′ � τ � V || F � τ [ a : = V ] τ ′ � V || ˜ (Forced eval.) → � λ a . p || q · E � τ � q || ˜ µ a . � p || E �� τ → ´ Etienne Miquey Classical realizability and side-effects 21/ 47

  20. Introduction A constructive proof of DC Algebraic models of classical realizability Danvy’s semantics artifacts Small steps: e � p || ˜ µ a . c � e τ c e τ [ a : = p ] → � p || E � e τ → � p || E � p τ p � µα . c || E � p τ → ( c [ E / α ] ) τ � V || E � p τ � V || E � E τ → µ [ a ] . � a || F � τ ′ � E τ � V || F � V τ [ a : = V ] τ ′ � V || ˜ E → � V || F � E τ → � V || F � V τ � a || F � V τ [ a : = p ] τ ′ µ [ a ] . � a || F � τ ′ � p τ → � p || ˜ V � λ a . p || F � V τ → � λ a . p || F � F τ F � λ a . p || q · E � F τ → � q || ˜ µ a . � p || E �� e τ ´ Etienne Miquey Classical realizability and side-effects 21/ 47

  21. Introduction A constructive proof of DC Algebraic models of classical realizability Danvy’s semantics artifacts CPS : � � p || e � τ � : = [ [ e ] ] e [ [ τ ] ] τ [ [ p ] ] p [ [˜ µ a . c ] ] e : = λτ p . � c � τ [ a : = p ] e [ [ E ] ] e : = λτ p . p τ [ [ E ] ] E [ [ µα . c ] ] p : = λτ E . ( [ [ c ] ] c τ ) [ E / α ] p [ [ V ] ] p : = λτ E . E τ [ [ V ] ] v λτ V . V τ [ a : = V ] τ ′ [ µ [ a ] . � a || F � τ ′ ] [ [˜ ] E : = [ F ] ] F E [ [ F ] ] E : = λτ V . V τ [ [ F ] ] F λτ F . τ ( a ) τ ( λτ V . V τ [ a : = V ] τ ′ [ [ [ a ] ] v : = [ F ] ] F ) V [ [ λ a . p ] ] v : = λτ F . F τ ( λ q τ E . [ [ p ] ] p τ [ a : = q ] E ) [ [ q · E ] ] F : = λτ v . v [ [ q ] ] p τ [ [ E ] ] E F ´ Etienne Miquey Classical realizability and side-effects 21/ 47

  22. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Store extension: τ ⊳ τ ′ Term-in-store ( t | τ ) : closed store τ s.t. FV ( t ) ⊆ dom ( τ ) . ( � generalizes closed terms) Pole : set of closures ⊥ ⊥ which is: saturated : c ′ τ ′ ∈ ⊥ c τ → c ′ τ ′ ⊥ and implies c τ ∈ ⊥ ⊥ closed by store extension : c τ ′ ∈ ⊥ τ ⊳ τ ′ c τ ∈ ⊥ ⊥ and implies ⊥ Compatible stores: τ ⋄ τ ′ ⊥ ( e | τ ′ ) : τ ⋄ τ ′ and � t || e � ττ ′ ∈ ⊥ Orthogonality ( t | τ ) ⊥ ⊥ . Realizers : definitions derived from the small-step rules! ´ Etienne Miquey Classical realizability and side-effects 22/ 47

  23. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Store extension: τ ⊳ τ ′ Term-in-store ( t | τ ) : closed store τ s.t. FV ( t ) ⊆ dom ( τ ) . ( � generalizes closed terms) Pole : set of closures ⊥ ⊥ which is: saturated : c ′ τ ′ ∈ ⊥ c τ → c ′ τ ′ ⊥ and implies c τ ∈ ⊥ ⊥ closed by store extension : c τ ′ ∈ ⊥ τ ⊳ τ ′ c τ ∈ ⊥ ⊥ and implies ⊥ Compatible stores: τ ⋄ τ ′ ⊥ ( e | τ ′ ) : τ ⋄ τ ′ and � t || e � ττ ′ ∈ ⊥ Orthogonality ( t | τ ) ⊥ ⊥ . Realizers : definitions derived from the small-step rules! ´ Etienne Miquey Classical realizability and side-effects 22/ 47

  24. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Adequacy ⊥ , if τ � Γ and Γ ⊢ c c , then c τ ∈ ⊥ For all ⊥ ⊥ . Normalization If ⊢ l c τ then c τ normalizes. Proof: The set ⊥ ⊥ ⇓ = { c τ ∈ C 0 : c τ normalizes } is a pole. (+ Bonus: typed CPS translation unveiling Kripke’s forcing.) ´ Etienne Miquey Classical realizability and side-effects 22/ 47

  25. Introduction A constructive proof of DC Algebraic models of classical realizability A classical sequent calculus with dependent types Chapter 7 Can this work? Π p Π q Π e . . . . . . . . . . . . Γ , a : A ⊢ p : B [ a ] | ∆ Γ ⊢ q : A | ∆ Γ | e : B [ q ] ⊢ ∆ q ∈ V ( → r ) ( → l ) Γ ⊢ λ a . p : Π a : A . B | ∆ Γ | q · e : Π a : A . B ⊢ ∆ ( Cut ) � λ a . p || q · e � : (Γ ⊢ ∆) −→ ´ Etienne Miquey Classical realizability and side-effects 23/ 47

  26. Introduction A constructive proof of DC Algebraic models of classical realizability A classical sequent calculus with dependent types Chapter 7 Can this work? Π p Π q Π e . . . . . . . . . . . . Γ , a : A ⊢ p : B [ a ] | ∆ Γ ⊢ q : A | ∆ Γ | e : B [ q ] ⊢ ∆ q ∈ V ( → r ) ( → l ) Γ ⊢ λ a . p : Π a : A . B | ∆ Γ | q · e : Π a : A . B ⊢ ∆ ( Cut ) � λ a . p || q · e � : (Γ ⊢ ∆) −→ ✚ ✟ Γ , a : A ⊢ p : ✚✚ Γ , a : A | e : ✟✟ B [ a ] | ∆ B [ q ] ⊢ ∆ Π q . Mismatch . � p || e � : (Γ , a : A ⊢ ∆) . . ( ˜ µ ) Γ ⊢ q : A | ∆ Γ | ˜ µ a . � p || e � : A ⊢ ∆ ( Cut ) � q || ˜ µ a . � p || e �� : (Γ ⊢ ∆) ´ Etienne Miquey Classical realizability and side-effects 23/ 47

  27. Introduction A constructive proof of DC Algebraic models of classical realizability A classical sequent calculus with dependent types Chapter 7 Can this work? � Π p Π q Π e . . . . . . . . . . . . Γ , a : A ⊢ p : B [ a ] | ∆ Γ ⊢ q : A | ∆ Γ | e : B [ q ] ⊢ ∆ q ∈ V ( → r ) ( → l ) Γ ⊢ λ a . p : Π a : A . B | ∆ Γ | q · e : Π a : A . B ⊢ ∆ ( Cut ) � λ a . p || q · e � : (Γ ⊢ ∆) −→ Γ , a : A ⊢ p : B [ a ] | ∆ Γ , a : A | e : B [ q ] ⊢ ∆ ; {·| p }{ a | q } Π q . ( Cut ) . � p || e � : Γ , a : A ⊢ ∆ ; { a | q } . . ( ˜ µ ) Γ ⊢ q : A | ∆ Γ | ˜ µ a . � p || e � : A ⊢ ∆ ; { . | q } ( Cut ) � q || ˜ µ a . � p || e �� : (Γ ⊢ ∆) ; {·|·} ´ Etienne Miquey Classical realizability and side-effects 23/ 47

  28. Introduction A constructive proof of DC Algebraic models of classical realizability dL A type system with: a list of dependencies : Γ | e : A ′ ⊢ ∆ ; σ {·| p } A ′ ∈ A σ Γ ⊢ p : A | ∆ ; σ ( Cut ) � p || e � : (Γ ⊢ ∆ ; σ ) a value restriction Is it enough? subject reduction normalization consistency as a logic suitable for CPS translation ´ Etienne Miquey Classical realizability and side-effects 24/ 47

  29. Introduction A constructive proof of DC Algebraic models of classical realizability dL A type system with: a list of dependencies : Γ | e : A ′ ⊢ ∆ ; σ {·| p } A ′ ∈ A σ Γ ⊢ p : A | ∆ ; σ ( Cut ) � p || e � : (Γ ⊢ ∆ ; σ ) a value restriction Is it enough? subject reduction � normalization � consistency as a logic � suitable for CPS translation ✗ ´ Etienne Miquey Classical realizability and side-effects 24/ 47

  30. Introduction A constructive proof of DC Algebraic models of classical realizability dL A type system with: a list of dependencies : Γ | e : A ′ ⊢ ∆ ; σ {·| p } A ′ ∈ A σ Γ ⊢ p : A | ∆ ; σ ( Cut ) � p || e � : (Γ ⊢ ∆ ; σ ) a value restriction Is it enough? subject reduction � normalization � consistency as a logic � suitable for CPS translation ✗ � q � � ˜ µ a . � p || e � � = � q � ( λ a . � p � � e � ) ���� ���� ���� ¬¬ A ¬¬ B ( a ) ¬ B ( q ) ´ Etienne Miquey Classical realizability and side-effects 24/ 47

  31. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (1/2) This is quite normal: we observed a desynchronization we compensated only within the type system � we need to do this already in the calculus! Who’s guilty ? � � q || ˜ µ a . � p || e �� � = � q � ( λ a . � p �� e � ) Moto: � p � shouldn’t be applied to � e � before � q � has reduced ( � q � ( λ a . � p � )) � e � So, we’re looking for: ´ Etienne Miquey Classical realizability and side-effects 25/ 47

  32. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (1/2) This is quite normal: we observed a desynchronization we compensated only within the type system � we need to do this already in the calculus! Who’s guilty ? � � q || ˜ µ a . � p || e �� � = � q � ( λ a . � p �� e � ) Moto: � p � shouldn’t be applied to � e � before � q � has reduced ( � q � ( λ a . � p � )) � e � So, we’re looking for: ´ Etienne Miquey Classical realizability and side-effects 25/ 47

  33. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (1/2) This is quite normal: we observed a desynchronization we compensated only within the type system � we need to do this already in the calculus! Who’s guilty ? � � q || ˜ µ a . � p || e �� � = � q � ( λ a . � p �� e � ) Moto: � p � shouldn’t be applied to � e � before � q � has reduced ( � q � ( λ a . � p � )) � e � So, we’re looking for: ´ Etienne Miquey Classical realizability and side-effects 25/ 47

  34. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (1/2) This is quite normal: we observed a desynchronization we compensated only within the type system � we need to do this already in the calculus! Who’s guilty ? � � q || ˜ µ a . � p || e �� � = � q � ( λ a . � p �� e � ) Moto: � p � shouldn’t be applied to � e � before � q � has reduced ( � q � ( λ a . � p � )) � e � So, we’re looking for: � λ a . p || q · e � → � µ ˆ µ a . � p || ˆ tp . � q || ˜ tp �� || e � ´ Etienne Miquey Classical realizability and side-effects 25/ 47

  35. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (2/2) ? � � λ a . p || q · e � � −→ ( � q � ( λ a . � p � )) � e � Qestions: Is any q compatible with such a reduction ? 1 Is this typable ? 2 ´ Etienne Miquey Classical realizability and side-effects 26/ 47

  36. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (2/2) ? � � λ a . p || q · e � � −→ ( � q � ( λ a . � p � )) � e � Qestions: Is any q compatible with such a reduction ? 1 � � p [ V / a ] �� e � If q eventually gives a value V : ✓ � t � e � If � q � → λ . t and drops its continuation: ✗ ´ Etienne Miquey Classical realizability and side-effects 26/ 47

  37. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (2/2) ? � � λ a . p || q · e � � −→ ( � q � ( λ a . � p � )) � e � Qestions: � q ∈ nef Is any q compatible with such a reduction ? 1 � � p [ V / a ] �� e � If q eventually gives a value V : ✓ � t � e � If � q � → λ . t and drops its continuation: ✗ Negative-elimination free (Herbelin’12) Values + one continuation variable + no application ´ Etienne Miquey Classical realizability and side-effects 26/ 47

  38. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (2/2) ? � � λ a . p || q · e � � −→ ( � q � ( λ a . � p � )) � e � Qestions: � q ∈ nef Is any q compatible with such a reduction ? 1 Is this typable ? 2 Naive atempt: ( � q � ( λ a . � p � )) � e � ���� � � �� � � ���� ( A →⊥ ) →⊥ Π ( a : A ) ¬¬ B ( a ) ¬ B [ q ] ´ Etienne Miquey Classical realizability and side-effects 26/ 47

  39. Introduction A constructive proof of DC Algebraic models of classical realizability Toward a CPS translation (2/2) ? � � λ a . p || q · e � � −→ ( � q � ( λ a . � p � )) � e � Qestions: � q ∈ nef Is any q compatible with such a reduction ? 1 Is this typable ? 2 Beter: ( � q � ( λ a . � p � )) � e � ���� � � �� � � ���� ∀ R . (Π ( a : A ) R ( a )) → R ( q ) Π ( a : A ) ¬¬ B ( a ) ¬ B [ q ] � ������������������������������������� �� ������������������������������������� � ¬¬ B ( q ) ( Remark: not possible without q ∈ nef ) ´ Etienne Miquey Classical realizability and side-effects 26/ 47

  40. Introduction A constructive proof of DC Algebraic models of classical realizability dL ˆ tp An extension of dL with: delimited continuations dependent types restricted to the nef fragment Regular mode Dependent mode Γ | e : A ⊢ d ∆ , ˆ Γ ⊢ p : A | ∆ tp : B ; σ {·| p } Γ ⊢ p : A | ∆ Γ | e : A ⊢ ∆ � p || e � : Γ ⊢ d ∆ , ˆ � p || e � : Γ ⊢ ∆ tp : B ; σ delimited scope of dependencies: c : (Γ ⊢ d ∆ , ˆ tp : A ; {·|·} ) B ∈ A σ ˆ ˆ tp I tp E Γ ⊢ µ ˆ Γ | ˆ tp : A ⊢ d ∆ , ˆ tp . c : A | ∆ tp : B ; σ {·| p } Mission accomplished? subject reduction (Bonus) embedding into Lepigre’s calculus � normalization � realizability consistency as a logic interpretation CPS translation ´ Etienne Miquey Classical realizability and side-effects 27/ 47

  41. Introduction A constructive proof of DC Algebraic models of classical realizability dL ˆ tp An extension of dL with: delimited continuations dependent types restricted to the nef fragment Regular mode Dependent mode Γ | e : A ⊢ d ∆ , ˆ Γ ⊢ p : A | ∆ tp : B ; σ {·| p } Γ ⊢ p : A | ∆ Γ | e : A ⊢ ∆ � p || e � : Γ ⊢ d ∆ , ˆ � p || e � : Γ ⊢ ∆ tp : B ; σ delimited scope of dependencies: c : (Γ ⊢ d ∆ , ˆ tp : A ; {·|·} ) B ∈ A σ ˆ ˆ tp I tp E Γ ⊢ µ ˆ Γ | ˆ tp : A ⊢ d ∆ , ˆ tp . c : A | ∆ tp : B ; σ {·| p } Mission accomplished � subject reduction � (Bonus) embedding into Lepigre’s calculus � normalization � consistency as a logic � � realizability CPS translation � interpretation ´ Etienne Miquey Classical realizability and side-effects 27/ 47

  42. Introduction A constructive proof of DC Algebraic models of classical realizability dLPA ω Chapter 8 A classical sequent calculus with: stratified dependent types : t , u :: = ... | wit p terms: formulas: A , B :: = ... | ∀ x T . A | ∃ x T . A | Π a : A . B | t = u p , q :: = ... | λ x . p | ( t , p ) | λ a . p proofs: a restriction to the nef fragment arithmetical terms: t , u :: = ... | 0 | S ( t ) | rec t xy [ t 0 | t S ] | λ x . t | t u stores : τ :: = ε | τ [ a : = p τ ] | τ [ α : = e ] inductive and coinductive constructions: p , q :: = ... | ind t bn [ p | p ] | cofix t bn p a call-by-value reduction and lazy evaluation of cofix ´ Etienne Miquey Classical realizability and side-effects 28/ 47

  43. Introduction A constructive proof of DC Algebraic models of classical realizability End of the road dPA ω + control operators Subject reduction + dependent types + co-fixpoints + sharing & laziness ? dLPA ω ? + sequent calculus Subject reduction + dependent types + co-fixpoints + sharing & laziness ? ?-calculus Normalization ´ Etienne Miquey Classical realizability and side-effects 29/ 47

  44. Introduction A constructive proof of DC Algebraic models of classical realizability End of the road dPA ω + control operators Subject reduction + dependent types + co-fixpoints + sharing & laziness macros � dLPA ω � + sequent calculus Subject reduction + dependent types + co-fixpoints + sharing & laziness Normalization realizability ´ Etienne Miquey Classical realizability and side-effects 29/ 47

  45. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Same methodology: small-step reductions 1 derive the realizability interpretation 2 Resembles λ [ lv τ ⋆ ] -interpretation, plus: dependent types from Lepigre’s calculus co-inductive formulas ´ Etienne Miquey Classical realizability and side-effects 30/ 47

  46. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Same methodology: small-step reductions 1 derive the realizability interpretation 2 Resembles λ [ lv τ ⋆ ] -interpretation, plus: dependent types from Lepigre’s calculus: � Π a : A . B ∀ a . ( a ∈ A → B ) co-inductive formulas ´ Etienne Miquey Classical realizability and side-effects 30/ 47

  47. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Same methodology: small-step reductions 1 derive the realizability interpretation 2 Resembles λ [ lv τ ⋆ ] -interpretation, plus: dependent types from Lepigre’s calculus co-inductive formulas: by finite approximations Xx A � f � � � ν t n ∈ � � F n A , t � f ´ Etienne Miquey Classical realizability and side-effects 30/ 47

  48. Introduction A constructive proof of DC Algebraic models of classical realizability Realizability interpretation Same methodology: small-step reductions 1 derive the realizability interpretation 2 Resembles λ [ lv τ ⋆ ] -interpretation, plus: dependent types from Lepigre’s calculus co-inductive formulas: by finite approximations Consequences of adequacy : Normalization If Γ ⊢ σ c , then c is normalizable. Consistency � dLPA ω p : ⊥ ´ Etienne Miquey Classical realizability and side-effects 30/ 47

  49. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion and further work Contributions of this part : classical call-by-need: realizability interpretation typed continuation-and-store-passing style translation dependent classical sequent calculus: list of dependencies use of delimited continuations for soundness dependently-typed continuation-passing style translation dLPA ω : soundness and normalization, realizability interpretation of co-fixpoints ´ Etienne Miquey Classical realizability and side-effects 31/ 47

  50. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion and further work Contributions of this part : classical call-by-need dependent classical sequent calculus dLPA ω Further work : Can dL tp be related to: 1 P´ edrot-Tabareau’s Baclofen Type Theory ? V´ ak´ ar’s categorical presentation ? Bowman et. al. CPS for CC ? Relation to Krivine’s realizability semantics of DC: 2 Compatible with quote ? Approximation of the limits required for bar recursion? Algebraic counterpart of side-effects in realizability structures? 3 ´ Etienne Miquey Classical realizability and side-effects 31/ 47

  51. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion and further work Contributions of this part : classical call-by-need dependent classical sequent calculus dLPA ω Further work : Can dL tp be related to: 1 P´ edrot-Tabareau’s Baclofen Type Theory ? V´ ak´ ar’s categorical presentation ? Bowman et. al. CPS for CC ? Relation to Krivine’s realizability semantics of DC: 2 Compatible with quote ? Approximation of the limits required for bar recursion? Algebraic counterpart of side-effects in realizability structures? 3 ´ Etienne Miquey Classical realizability and side-effects 31/ 47

  52. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion and further work Contributions of this part : classical call-by-need dependent classical sequent calculus dLPA ω Further work : Can dL tp be related to: 1 P´ edrot-Tabareau’s Baclofen Type Theory ? V´ ak´ ar’s categorical presentation ? Bowman et. al. CPS for CC ? Relation to Krivine’s realizability semantics of DC: 2 Compatible with quote ? Approximation of the limits required for bar recursion? Algebraic counterpart of side-effects in realizability structures? 3 ´ Etienne Miquey Classical realizability and side-effects 31/ 47

  53. Introduction A constructive proof of DC Algebraic models of classical realizability Part III - Algebraic models of classical realizability ´ Etienne Miquey Classical realizability and side-effects 32/ 47

  54. Introduction A constructive proof of DC Algebraic models of classical realizability Algebraization of classical realizability Chapter 9 Realizers: t � A defined as t ∈ � A � ⊥ ⊥ Key elements: the pole ⊥ ⊥ the latice ( P (Π) , ⊇ ) Observations: this induces a semantic subtyping: ⊥ B � A ≤ ⊥ � B � ⊆ � A � connectives/quantifiers: � ∀ = ∧ = × Realizability This is to be compared with: � ∀ = = ∧ Forcing ´ Etienne Miquey Classical realizability and side-effects 33/ 47

  55. Introduction A constructive proof of DC Algebraic models of classical realizability Algebraization of classical realizability Chapter 9 Realizers: t � A defined as t ∈ � A � ⊥ ⊥ Key elements: the pole ⊥ ⊥ the latice ( P (Π) , ⊇ ) Observations: this induces a semantic subtyping: ⊥ B � A ≤ ⊥ � B � ⊆ � A � connectives/quantifiers: � ∀ = ∧ = × Realizability This is to be compared with: � ∀ = = ∧ Forcing ´ Etienne Miquey Classical realizability and side-effects 33/ 47

  56. Introduction A constructive proof of DC Algebraic models of classical realizability Algebraization of classical realizability Chapter 9 Realizers: t � A defined as t ∈ � A � ⊥ ⊥ Key elements: the pole ⊥ ⊥ the latice ( P (Π) , ⊇ ) Observations: this induces a semantic subtyping: ⊥ B � A ≤ ⊥ � B � ⊆ � A � connectives/quantifiers: � ∀ = ∧ = × Realizability This is to be compared with: � ∀ = = ∧ Forcing ´ Etienne Miquey Classical realizability and side-effects 33/ 47

  57. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative structures Chapter 10 Types Formulas λ -terms Proofs In particular, a � b reads: a is a subtype of b a is a realizer of b the realizer a is more defined than b ´ Etienne Miquey Classical realizability and side-effects 34/ 47

  58. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative structures Chapter 10 Types Formulas ⊆ λ -terms Proofs In particular, a � b reads: a is a subtype of b a is a realizer of b the realizer a is more defined than b ´ Etienne Miquey Classical realizability and side-effects 34/ 47

  59. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative structures Chapter 10 Types Formulas λ -terms Proofs In particular, a � b reads: a is a subtype of b a is a realizer of b the realizer a is more defined than b ´ Etienne Miquey Classical realizability and side-effects 34/ 47

  60. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative structures Chapter 10 Definition: Complete meet-semilatice ( A , � , → ) s.t.: if a 0 � a and b � b 0 then ( a → b ) � ( a 0 → b 0 ) � b ∈ B ( a → b ) = a → � b ∈ B b Examples: complete Heyting/Boolean algebras classical realizability: A � P (Π) ; a � b � a ⊇ b ⊥ · b = { t · π : t ∈ a ⊥ a → b � a ⊥ ⊥ , π ∈ b } ´ Etienne Miquey Classical realizability and side-effects 34/ 47

  61. Introduction A constructive proof of DC Algebraic models of classical realizability Interpretating of λ -terms λ -terms: a @ b � � { c ∈ A : a � ( b → c ) } λ f � � a ∈A ( a → f ( a )) call/cc : cc � � a , b ∈A ((( a → b ) → a ) → a ) a � b → c a @ b � c Adjunction: ⇔ If ⊢ t : A then t A � A A Adequacy: In particular: = � k A a , b ∈A ( a → b → a ) = � s A a , b , c ∈A (( a → b → c ) → ( a → b ) → a → c ) ´ Etienne Miquey Classical realizability and side-effects 35/ 47

  62. Introduction A constructive proof of DC Algebraic models of classical realizability Interpretating of λ -terms λ -terms: a @ b � � { c ∈ A : a � ( b → c ) } λ f � � a ∈A ( a → f ( a )) call/cc : cc � � a , b ∈A ((( a → b ) → a ) → a ) a � b → c a @ b � c Adjunction: ⇔ If ⊢ t : A then t A � A A Adequacy: In particular: = � k A a , b ∈A ( a → b → a ) = � s A a , b , c ∈A (( a → b → c ) → ( a → b ) → a → c ) ´ Etienne Miquey Classical realizability and side-effects 35/ 47

  63. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative algebras Separator S : k A ∈ S , and s A ∈ S (Combinators) 1 If a ∈ S and a � b , then b ∈ S . 2 (Upwards closure) If ( a → b ) ∈ S and a ∈ S , then b ∈ S . (Modus ponens) 3 Implicative algebras: ( A , � , → ) + separator S Entailment: a ⊢ S b � a → b ∈ S . Adjunction a ⊢ S b → c if and only if a × b ⊢ S c ´ Etienne Miquey Classical realizability and side-effects 36/ 47

  64. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative algebras Separator S : k A ∈ S , and s A ∈ S (Combinators) 1 If a ∈ S and a � b , then b ∈ S . 2 (Upwards closure) If ( a → b ) ∈ S and a ∈ S , then b ∈ S . (Modus ponens) 3 Implicative algebras: ( A , � , → ) + separator S Entailment: a ⊢ S b � a → b ∈ S . Adjunction a ⊢ S b → c if and only if a × b ⊢ S c ´ Etienne Miquey Classical realizability and side-effects 36/ 47

  65. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative tripos Adjunction a ⊢ S b → c if and only if a × b ⊢ S c ( � ( A / S , ⊢ S , × , + , → ) is a Heyting algebra ) Tripos: � Set op → HA T : A I / S [ I ] I �→ Collapse criteria The following are equivalent: T is isomorphic to a forcing tripos 1 S ⊆ A is a principal filter of A . 2 S ⊆ A is finitely generated and ⋔ ∈ S . 3 ´ Etienne Miquey Classical realizability and side-effects 37/ 47

  66. Introduction A constructive proof of DC Algebraic models of classical realizability Implicative tripos Adjunction a ⊢ S b → c if and only if a × b ⊢ S c ( � ( A / S , ⊢ S , × , + , → ) is a Heyting algebra ) Tripos: � Set op → HA T : A I / S [ I ] I �→ Collapse criteria The following are equivalent: T is isomorphic to a forcing tripos 1 S ⊆ A is a principal filter of A . 2 S ⊆ A is finitely generated and ⋔ ∈ S . 3 ´ Etienne Miquey Classical realizability and side-effects 37/ 47

  67. Introduction A constructive proof of DC Algebraic models of classical realizability Decomposing the arrow Chapter 11 Logic: � A → B ¬ A ∨ B Different axiomatic: S 1 : ( A ∨ A ) → A S 3 : ( A ∨ B ) → ( B ∨ A ) : : S 2 A → ( A ∨ B ) S 4 ( A → B ) → (( C ∨ A ) → ( C ∨ B )) λ -calculus: λ x . t � ˜ µ ( [ x ] , β ) . � t || β � : ¬ A ` B L ` fragment of Munch-Maccagnoni’s system L embedding of the call-by-name λ -calculus ´ Etienne Miquey Classical realizability and side-effects 38/ 47

  68. Introduction A constructive proof of DC Algebraic models of classical realizability Disjunctive structures Complete meet-semilatice ( A , � , ` , ¬ ) : ¬ is anti-monotonic 1 ` is monotonic 2 � b ∈ B ( a ` b ) = a ` ( � � b ∈ B ( b ` a ) = ( � b ∈ B b ) b ∈ B b ) ` a and 3 ¬ � a ∈ A a = � a ∈ A ¬ a 4 Examples: complete Boolean algebras classical realizability in L ` : A � P (Π) a ` b � ( a , b ) a � b � a ⊇ b ¬ a � [ a ⊥ ⊥ ] Induced implication ( A , � , ` → b � ¬ a ` b is an implicative structure ` → ) with a ´ Etienne Miquey Classical realizability and side-effects 39/ 47

  69. Introduction A constructive proof of DC Algebraic models of classical realizability Disjunctive structures Complete meet-semilatice ( A , � , ` , ¬ ) : ¬ is anti-monotonic 1 ` is monotonic 2 � b ∈ B ( a ` b ) = a ` ( � � b ∈ B ( b ` a ) = ( � b ∈ B b ) b ∈ B b ) ` a and 3 ¬ � a ∈ A a = � a ∈ A ¬ a 4 Examples: complete Boolean algebras classical realizability in L ` : A � P (Π) a ` b � ( a , b ) a � b � a ⊇ b ¬ a � [ a ⊥ ⊥ ] Induced implication ( A , � , ` → b � ¬ a ` b is an implicative structure ` → ) with a ´ Etienne Miquey Classical realizability and side-effects 39/ 47

  70. Introduction A constructive proof of DC Algebraic models of classical realizability Disjunctive structures Complete meet-semilatice ( A , � , ` , ¬ ) : ¬ is anti-monotonic 1 ` is monotonic 2 � b ∈ B ( a ` b ) = a ` ( � � b ∈ B ( b ` a ) = ( � b ∈ B b ) b ∈ B b ) ` a and 3 ¬ � a ∈ A a = � a ∈ A ¬ a 4 Examples: complete Boolean algebras classical realizability in L ` : A � P (Π) a ` b � ( a , b ) a � b � a ⊇ b ¬ a � [ a ⊥ ⊥ ] Induced implication ( A , � , ` → b � ¬ a ` b is an implicative structure ` → ) with a ´ Etienne Miquey Classical realizability and side-effects 39/ 47

  71. Introduction A constructive proof of DC Algebraic models of classical realizability Interpreting L ` Terms: Contexts: µ − . c � � a ∈A { a : c ( a ) ∈ ⊥ ( a , b ) � a ` b ⊥} µ () . c � � [ a ] � ¬ a a , b ∈A { a ` b : c ( a , b ) ∈ ⊥ ⊥} µ + . c � � µ [] . c � � a ∈A { a : c ( a ) ∈ ⊥ ⊥} a ∈A {¬ a : c ( a ) ∈ ⊥ ⊥} Adequacy for any term t , if Γ ⊢ t : A | ∆ , then ( t [ σ ] ) A � A [ σ ] A ; 1 for any context e , if Γ | e : A ⊢ ∆ , then ( e [ σ ] ) A � A [ σ ] A ; 2 Besides: � � L ` λ -calculus [ ] A → [ ] A ` ι ( A → , � , → ) ( A ` , � , ` , ¬ ) ´ Etienne Miquey Classical realizability and side-effects 40/ 47

  72. Introduction A constructive proof of DC Algebraic models of classical realizability Interpreting L ` Terms: Contexts: µ − . c � � a ∈A { a : c ( a ) ∈ ⊥ ( a , b ) � a ` b ⊥} µ () . c � � [ a ] � ¬ a a , b ∈A { a ` b : c ( a , b ) ∈ ⊥ ⊥} µ + . c � � µ [] . c � � a ∈A { a : c ( a ) ∈ ⊥ ⊥} a ∈A {¬ a : c ( a ) ∈ ⊥ ⊥} Adequacy for any term t , if Γ ⊢ t : A | ∆ , then ( t [ σ ] ) A � A [ σ ] A ; 1 for any context e , if Γ | e : A ⊢ ∆ , then ( e [ σ ] ) A � A [ σ ] A ; 2 Besides: � � L ` λ -calculus [ ] A → [ ] A ` ι ( A → , � , → ) ( A ` , � , ` , ¬ ) ´ Etienne Miquey Classical realizability and side-effects 40/ 47

  73. Introduction A constructive proof of DC Algebraic models of classical realizability Disjunctive algebras Bourbaki’s axioms: � � s ` a ∈A [ ( a ` a ) → a ] � 1 � s ` a , b ∈A [ a → ( a ` b ) ] � 2 � s ` a , b ∈A [ ( a ` b ) → ( b ` a ) ] � 3 � s ` a , b , c ∈A [ ( a → b ) → ( c ` a ) → ( c ` b ) ] � 4 � s ` a , b , c ∈A [ ( a ` ( b ` c )) → (( a ` b ) ` c ) ] 5 Separator S : (1) If a ∈ S and a � b then b ∈ S (upward closure) (2) s 1 , s 2 , s 3 , s 4 and s 5 are in S (combinators) ` (3) If a → b ∈ S and a ∈ S then b ∈ S (closure under modus ponens) ´ Etienne Miquey Classical realizability and side-effects 41/ 47

  74. Introduction A constructive proof of DC Algebraic models of classical realizability Internal logic Recall: a ⊢ S b � a ` → b ∈ S Sum type: 1. a ` b ⊢ S a + b 2. a + b ⊢ S a ` b Negation: ` ` 1. ¬ a ⊢ S a → ⊥ 2. a → ⊥ ⊢ S ¬ a Double-negation elimination: 1. a ⊢ S ¬¬ a 2. ¬¬ a ⊢ S a Theorem S is an implicative separator ´ Etienne Miquey Classical realizability and side-effects 42/ 47

  75. Introduction A constructive proof of DC Algebraic models of classical realizability Internal logic Recall: a ⊢ S b � a ` → b ∈ S Sum type: 1. a ` b ⊢ S a + b 2. a + b ⊢ S a ` b Negation: ` ` 1. ¬ a ⊢ S a → ⊥ 2. a → ⊥ ⊢ S ¬ a Double-negation elimination: 1. a ⊢ S ¬¬ a 2. ¬¬ a ⊢ S a Theorem S is an implicative separator ´ Etienne Miquey Classical realizability and side-effects 42/ 47

  76. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion Disjunctive structures: induced by classical realizability allow to adequately embed L ` are implicative structures Disjunctive algebras: are intrinsically classical are implicative algebras do not necessarily collapse to a forcing situation Conclusion Implicative algebras are more general. ´ Etienne Miquey Classical realizability and side-effects 43/ 47

  77. Introduction A constructive proof of DC Algebraic models of classical realizability Conclusion Disjunctive structures: induced by classical realizability allow to adequately embed L ` are implicative structures Disjunctive algebras: are intrinsically classical are implicative algebras do not necessarily collapse to a forcing situation Conclusion Implicative algebras are more general. ´ Etienne Miquey Classical realizability and side-effects 43/ 47

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