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A System F accounting for scalars

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The linear-algebraic lambda-calculus (arXiv:quant-ph/0612199) extends the lambda-calculus with the possibility of making arbitrary linear combinations of lambda-calculus terms a.t+b.u. In this paper we provide a System F -like type system for the linear-algebraic lambda-calculus, which keeps track of \'the amount of a type\' that is present in a term. We show that this scalar type system enjoys both the subject-reduction property and the strong-normalisation property, which constitute our main technical results. The latter yields a significant simplification of the linear-algebraic lambda-calculus itself, by removing the need for some restrictions in its reduction rules - and thus leaving it more intuitive. More importantly we show that our type system can readily be modified into a probabilistic type system, which guarantees that terms define correct probabilistic functions. Thus we are able to specialize the linear-algebraic lambda-calculus into a higher-order, probabilistic lambda-calculus. Finally we discuss the more long-term aims of this reseach in terms of establishing connections with linear logic, and building up towards a quantum physical logic through the Curry-Howard isomorphism. Thus we begin to investigate the logic induced by the scalar type system, and prove a no-cloning theorem expressed solely in terms of the possible proof methods in this logic.

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A System F accounting for scalars

  1. 1. A System F accounting for scalars arXiv:0903.3741 Pablo Arrighi and Alejandro Díaz-Caro Université de Grenoble Laboratoire d'Informatique de Grenoble November 19th, 2009. PPS (Paris)
  2. 2. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Global context 1 Oddity of Quantum theory =⇒ Quantum Logic? (developed ad hoc before quantum computing, no clear relation with quantum programs). Birkho, G. and J. von Neumann, The logic of quantum mechanics, 1 Annals of Mathematics 37 (1936), pp. 823843. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 2/31
  3. 3. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Global context 1 Oddity of Quantum theory =⇒ Quantum Logic? (developed ad hoc before quantum computing, no clear relation with quantum programs). Models of Linear Logics =⇒ Quantum Theory? (Coherent spaces, Micromechanics loses duplicability.) Birkho, G. and J. von Neumann, The logic of quantum mechanics, 1 Annals of Mathematics 37 (1936), pp. 823843. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 2/31
  4. 4. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Global context 1 Oddity of Quantum theory =⇒ Quantum Logic? (developed ad hoc before quantum computing, no clear relation with quantum programs). Models of Linear Logics =⇒ Quantum Theory? (Coherent spaces, Micromechanics loses duplicability.) Curry-Howard : (programs,types)=⇒(proofs,logics). Quantum Computation : (quantum programs, quantum types). CH+QC : (quantum th. proofs, quantum th. logics)? Quantum logics : isolating the reasoning behind quantum algorithms? Birkho, G. and J. von Neumann, The logic of quantum mechanics, 1 Annals of Mathematics 37 (1936), pp. 823843. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 2/31
  5. 5. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Global context 1 Oddity of Quantum theory =⇒ Quantum Logic? (developed ad hoc before quantum computing, no clear relation with quantum programs). Models of Linear Logics =⇒ Quantum Theory? (Coherent spaces, Micromechanics loses duplicability.) Curry-Howard : (programs,types)=⇒(proofs,logics). Quantum Computation : (quantum programs, quantum types). CH+QC : (quantum th. proofs, quantum th. logics)? Quantum logics : isolating the reasoning behind quantum algorithms? What are quantum types? Birkho, G. and J. von Neumann, The logic of quantum mechanics, 1 Annals of Mathematics 37 (1936), pp. 823843. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 2/31
  6. 6. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Quantum theory States are (normalized) vectors v. Vector space of o.n.b. (bi ). Then v= i αi bi . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 3/31
  7. 7. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Quantum theory States are (normalized) vectors v. Vector space of o.n.b. (bi ). Then v= i αi bi . Evolutions are (unitary) linear operators U . v = U v. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 3/31
  8. 8. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Quantum theory States are (normalized) vectors v. Vector space of o.n.b. (bi ). Then v= i αi bi . Evolutions are (unitary) linear operators U . v = U v. Systems are put next to one another with ⊗. Bilinear just like application : u + v ⊗ w = u ⊗ w + v ⊗ w, u ⊗ v + w = u ⊗ v + u ⊗ w, . . . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 3/31
  9. 9. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Quantum theory States are (normalized) vectors v. Vector space of o.n.b. (bi ). Then v= i αi bi . Evolutions are (unitary) linear operators U . v = U v. Systems are put next to one another with ⊗. Bilinear just like application : u + v ⊗ w = u ⊗ w + v ⊗ w, u ⊗ v + w = u ⊗ v + u ⊗ w, . . . No-cloning theorem! Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 3/31
  10. 10. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work No-cloning theorem Statement: ∃U / ∀v : U v = v ⊗ v. Proof: Vector space of o.n.b. (bi ), so v= αi bi . We can have i U bi = bi ⊗ bi (=copying, OK) But then U v = αi bi = αi U bi U i i = αi bi ⊗ bi = αi αj bi ⊗ bj i ij = ( αi bi ) ⊗ ( αj bj ) i j = v⊗v (=cloning, Not OK) Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 4/31
  11. 11. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work No-cloning theorem Statement: ∃U / ∀v : U v = v ⊗ v. Proof: Vector space of o.n.b. (bi ), so v= αi bi . We can have i U bi = bi ⊗ bi (=copying, OK) But then U v = αi bi = αi U bi U i i = αi bi ⊗ bi = αi αj bi ⊗ bj i ij = ( αi bi ) ⊗ ( αj bj ) i j = v⊗v (=cloning, Not OK) Conicts with β -reduction? Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 4/31
  12. 12. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus2 The language Higher-order computation t ::= x | λx .t | (t t) | Arrighi, P. and G. Dowek. Linear-algebraic λ-calculus: higher-order, 2 encodings and conuence. Lecture Notes in Computer Science (RTA'08), 5117 (2008), pp. 1731. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 5/31
  13. 13. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus2 The language Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 Arrighi, P. and G. Dowek. Linear-algebraic λ-calculus: higher-order, 2 encodings and conuence. Lecture Notes in Computer Science (RTA'08), 5117 (2008), pp. 1731. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 5/31
  14. 14. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus2 The language Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) (*) b an abstraction or a variable. Arrighi, P. and G. Dowek. Linear-algebraic λ-calculus: higher-order, 2 encodings and conuence. Lecture Notes in Computer Science (RTA'08), 5117 (2008), pp. 1731. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 5/31
  15. 15. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus2 The language Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) Elementary rules such as (*) b an abstraction or a u + 0 → u and α.(u + v) → α.u + α.v. variable. Factorisation rules such as (**) u closed normal. α.u + β.u → (α + β).u. (**) (***) u and u + v closed normal. Application rules such as u (v + w) → (u v) + (u w). (***) Arrighi, P. and G. Dowek. Linear-algebraic λ-calculus: higher-order, 2 encodings and conuence. Lecture Notes in Computer Science (RTA'08), 5117 (2008), pp. 1731. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 5/31
  16. 16. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Copying vs Cloning Untyped λ-calculus + linear algebra ⇒ Cloning? Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 6/31
  17. 17. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Copying vs Cloning Untyped λ-calculus + linear algebra ⇒ Cloning? λx .(x ⊗ x ) αi bi →∗ αi bi ⊗ bi i i ↓ ( αi bi ) ⊗ ( αi bi ) i i Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 6/31
  18. 18. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Copying vs Cloning Untyped λ-calculus + linear algebra ⇒ Cloning? λx .(x ⊗ x ) αi bi →∗ αi bi ⊗ bi i i ↓ ( αi bi ) ⊗ ( αi bi ) i i No-cloning says bottom reduction forbidden. We must delay beta reduction till after linearity. So restrict beta reduction to base vectors → i.e. abstractions or variables. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 6/31
  19. 19. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  20. 20. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Yb ≡ λx .(b + (x x )) λx .(b + (x x )) Yb → b + Yb Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  21. 21. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Yb ≡ λx .(b + (x x )) λx .(b + (x x )) Yb → b + Yb But whoever says innity says trouble says. . . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  22. 22. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Yb ≡ λx .(b + (x x )) λx .(b + (x x )) Yb → b + Yb But whoever says innity says trouble says. . . indenite forms. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  23. 23. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Yb ≡ λx .(b + (x x )) λx .(b + (x x )) Yb → b + Yb But whoever says innity says trouble says. . . indenite forms. Yb − Yb → b + Yb − Yb → b ↓∗ 0 Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  24. 24. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work Linear-Algebraic λ-Calculus Why the restrictions : Innities Untyped λ-calculus + linear algebra ⇒∞ Yb ≡ λx .(b + (x x )) λx .(b + (x x )) Yb → b + Yb But whoever says innity says trouble says. . . indenite forms. Yb − Yb → b + Yb − Yb → b ↓∗ 0 High school teacher says we must restrict factorization rules to nite vectors → i.e. closed-normal forms. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 7/31
  25. 25. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Straightforward extension of System F (λ2la ) System F rules plus simple rules to type algebraic terms ax0 Γ u: A Γ v: A Γ t: A +I αI Γ 0: A Γ u + v: A Γ α.t : A Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 8/31
  26. 26. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Straightforward extension of System F (λ2la ) System F rules plus simple rules to type algebraic terms ax0 Γ u: A Γ v: A Γ t: A +I αI Γ 0: A Γ u + v: A Γ α.t : A Theorem (Strong normalization) Γ t: T ⇒t is strongly normalising. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 8/31
  27. 27. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Straightforward extension of System F (λ2la ) System F rules plus simple rules to type algebraic terms ax0 Γ u: A Γ v: A Γ t: A +I αI Γ 0: A Γ u + v: A Γ α.t : A Theorem (Strong normalization) Γ t: T ⇒t is strongly normalising. Proof. Sketch: We extend the notion of saturated sets. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 8/31
  28. 28. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la SN : Set of strongly normalising terms Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 9/31
  29. 29. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la SN : Set of strongly normalising terms A subset X ∈ SN is saturated if 1 ∀n ≥ 0, (((x t1 ) . . . ) tn ) ∈ X where ti ∈ SN ; → − → − 2 v[b/x ] t ∈ X ⇒ (λx v) b t ∈ X ; 3 t, u ∈ X ⇒ t + u ∈ X ; 4 ∀α, t ∈ X ⇒ α.t ∈ X ; 5 ∀i ∈ I , ((ui w1 ) . . . wn ) ∈ X ⇒ u i w1 . . . w ∈ X ; n i ∈I 6 ∀i ∈ I , (u wi ) ∈ X ⇒ u w i ∈ X; i ∈I 7 α.((t1 t2 ) . . . tn ) ∈ X ⇔ ((t1 t2 ) . . . α.tk ) . . . tn ∈ X (1 ≤ k ≤ n); 8 0 ∈ X; → − → − 9 ∀ t ∈ SN , (0 t ) ∈ X ; 10 → ∈ SN , (t 0) → ∈ X . − ∀t, u −u X stable by construction and anti-reduction Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 9/31
  30. 30. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la SN : Set of strongly normalising terms A subset X ∈ SN is saturated if 1 ∀n ≥ 0, (((x t1 ) . . . ) tn ) ∈ X where ti ∈ SN ; → − → − 2 v[b/x ] t ∈ X ⇒ (λx v) b t ∈ X ; 3 t, u ∈ X ⇒ t + u ∈ X ; 4 ∀α, t ∈ X ⇒ α.t ∈ X ; 5 ∀i ∈ I , ((ui w1 ) . . . wn ) ∈ X ⇒ u i w1 . . . w ∈ X ; n i ∈I 6 ∀i ∈ I , (u wi ) ∈ X ⇒ u w i ∈ X; i ∈I 7 α.((t1 t2 ) . . . tn ) ∈ X ⇔ ((t1 t2 ) . . . α.tk ) . . . tn ∈ X (1 ≤ k ≤ n); 8 0 ∈ X; → − → − 9 ∀ t ∈ SN , (0 t ) ∈ X ; 10 → ∈ SN , (t 0) → ∈ X . − ∀t, u −u X stable by construction and anti-reduction SAT is the set of all saturated sets Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 9/31
  31. 31. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (II) Rening the sketch: The idea is that types correspond to saturated sets. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 10/31
  32. 32. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (II) Rening the sketch: The idea is that types correspond to saturated sets. This correspondance is achived by a maping from types to SAT . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 10/31
  33. 33. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (III) Lemma 1 SN ∈ SAT , 2 A B , ∈ SAT ⇒ A → B ∈ SAT , 3 For all collection A i of members of SAT , i Ai ∈ SAT , Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 11/31
  34. 34. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (III) Lemma 1 SN ∈ SAT , 2 A B , ∈ SAT ⇒ A → B ∈ SAT , 3 For all collection A i of members of SAT , i Ai ∈ SAT , Denition (Mapping) [[X ]]ξ = ξ(X ) (where ξ(·) : TVar → SAT ) [[A → B ]]ξ = [[A]]ξ → [[B ]]ξ [[∀X .T ]]ξ = Y ∈SAT [[T ]]ξ(X :=Y ) Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 11/31
  35. 35. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (III) Lemma 1 SN ∈ SAT , 2 A B , ∈ SAT ⇒ A → B ∈ SAT , 3 For all collection A i of members of SAT , i Ai ∈ SAT , Denition (Mapping) [[X ]]ξ = ξ(X ) (where ξ(·) : TVar → SAT ) [[A → B ]]ξ = [[A]]ξ → [[B ]]ξ [[∀X .T ]]ξ = Y ∈SAT [[T ]]ξ(X :=Y ) Lemma Given a valuation ξ , [[T ]]ξ ∈ SAT Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 11/31
  36. 36. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (IV) Denition ( ) For Γ = x1 : A1 , . . . , xn : An , Γ t: T means that ∀ξ , x1 ∈ [[A1 ]]ξ , . . . xn ∈ [[An ]]ξ ⇒ t ∈ [[T ]]ξ Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 12/31
  37. 37. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (IV) Denition ( ) For Γ = x1 : A1 , . . . , xn : An , Γ t: T means that ∀ξ , x1 ∈ [[A1 ]]ξ , . . . xn ∈ [[An ]]ξ ⇒ t ∈ [[T ]]ξ Rening the sketch: Prove that Γ t: T ⇒ Γ t: T We prove this by induction on the derivation of Γ t: T (In fact the denition of is slightly dierent to strengthen the induction hypothesis) Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 12/31
  38. 38. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (V) Then, the proof of the strong normalisation theorem is: Let Γ t: T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 13/31
  39. 39. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (V) Then, the proof of the strong normalisation theorem is: Let Γ t: T ⇒ Γ t: T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 13/31
  40. 40. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (V) Then, the proof of the strong normalisation theorem is: Let Γ t: T ⇒ Γ t: T ⇒ If ∀(xi : Ai ) ∈ Γ, xi ∈ [[Ai ]]ξ then t ∈ [[T ]]ξ Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 13/31
  41. 41. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (V) Then, the proof of the strong normalisation theorem is: Let Γ t: T ⇒ Γ t: T ⇒ If ∀(xi : Ai ) ∈ Γ, xi ∈ [[Ai ]]ξ then t ∈ [[T ]]ξ Note that x i ∈ [[Ai ]]ξ because [[Ai ]]ξ is saturated, Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 13/31
  42. 42. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Strong normalization of λ2la (V) Then, the proof of the strong normalisation theorem is: Let Γ t: T ⇒ Γ t: T ⇒ If ∀(xi : Ai ) ∈ Γ, xi ∈ [[Ai ]]ξ then t ∈ [[T ]]ξ Note that x i ∈ [[Ai ]]ξ because [[Ai ]]ξ is saturated, then t is strong normalising because [[T ]]ξ ⊆ SN Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 13/31
  43. 43. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Linear-Algebraic λ-Calculus with λ2la Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) Elementary rules such as (*) b an abstraction or a variable. u + 0 → u and α.(u + v) → α.u + α.v. Factorisation rules such as α.u + β.u → (α + β).u. Application rules such as u (v + w) → (u v) + (u w). Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 14/31
  44. 44. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Linear-Algebraic λ-Calculus with λ2la Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) Elementary rules such as (*) b an abstraction or a variable. u + 0 → u and α.(u + v) → α.u + α.v. Every typable term is strong normalizing Factorisation rules such as α.u + β.u → (α + β).u. Application rules such as u (v + w) → (u v) + (u w). Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 14/31
  45. 45. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Linear-Algebraic λ-Calculus with λ2la Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) Elementary rules such as (*) b an abstraction or a variable. u + 0 → u and α.(u + v) → α.u + α.v. Every typable term is strong normalizing Factorisation rules such as Hence Yb is no typable! α.u + β.u → (α + β).u. Application rules such as u (v + w) → (u v) + (u w). Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 14/31
  46. 46. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work System F Linear-Algebraic λ-Calculus with λ2la Higher-order computation Linear algebra t ::= x | λx .t | (t t) | t + t | α.t | 0 λx .t b → t[b/x ] (∗) Elementary rules such as (*) b an abstraction or a variable. u + 0 → u and α.(u + v) → α.u + α.v. Every typable term is strong normalizing Factorisation rules such as Hence Yb is no typable! α.u + β.u → (α + β).u. t − t → 0 always, so it is not Application rules such as necesary to reduce t rst. we can u (v + w) → (u v) + (u w). remove the closed-normal restrictions! Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 14/31
  47. 47. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Grammar Types grammar: T = U | ∀X .T | α.T | 0, U = X | U → T | ∀X .U where α∈S and (S, +, ×) is a conmutative ring. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 15/31
  48. 48. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Type inference rules ax U [ ] Γ, x : U x :U Γ u: U → T Γ v: U Γ, x : U t: T →E → I [U ] Γ (u v) : T Γ λx t : U → T Γ u : ∀X .T Γ u: T ∀E [X := U ] ∀I [X ] with X ∈ FV (Γ) / Γ u : T [U /X ] Γ u : ∀X .T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 16/31
  49. 49. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Type inference rules ax U [ ] Γ, x : U x :U Γ u: U → T Γ v: U Γ, x : U t: T →E → I [U ] Γ (u v) : T Γ λx t : U → T Γ u : ∀X .T Γ u: T ∀E [X := U ] ∀I [X ] with X ∈ FV (Γ) / Γ u : T [U /X ] Γ u : ∀X .T Γ u: T Γ v: T Γ u: T ax0 +I αI Γ 0: T Γ u + v: T Γ α.u : T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 16/31
  50. 50. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Type inference rules ax U [ ] Γ, x : U x :U Γ u : α.(U → T ) Γ v : β.U Γ, x : U t: T →E → I [U ] Γ (u v) : (α × β).T Γ λx t : U → T Γ u : ∀X .T Γ u: T ∀E [X := U ] ∀I [X ] with X ∈ FV (Γ) / Γ u : T [U /X ] Γ u : ∀X .T Γ u : α.T Γ v : β.T Γ u: T ax +I 0 sI [α] Γ 0: 0 Γ u + v : (α + β).T Γ α.u : α.T Where U ∈U and types in contexts are are in U. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 17/31
  51. 51. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Strong normalisation (·) : map that take types and remove all the scalars on it. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 18/31
  52. 52. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Strong normalisation (·) : map that take types and remove all the scalars on it. Example: (U → α.X ) = U → X Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 18/31
  53. 53. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Strong normalisation (·) : map that take types and remove all the scalars on it. Example: (U → α.X ) = U → X We also dene 0 =T for some T without scalars. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 18/31
  54. 54. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Strong normalisation (·) : map that take types and remove all the scalars on it. Example: (U → α.X ) = U → X We also dene 0 =T for some T without scalars. Lemma (Correspondence with λ2la ) Γ t: T ⇒Γ λ2la t : T . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 18/31
  55. 55. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Strong normalisation (·) : map that take types and remove all the scalars on it. Example: (U → α.X ) = U → X We also dene 0 =T for some T without scalars. Lemma (Correspondence with λ2la ) Γ t: T ⇒Γ λ2la t : T . Theorem (Strong normalisation) Γ t: T ⇒t is strongly normalising. Proof. By previous lemma Γ λ2la t: T , then t is strong normalising. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 18/31
  56. 56. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction Theorem (Subject Reduction) Let t →∗ t . Then Γ t :T ⇒ Γ t :T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 19/31
  57. 57. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction Theorem (Subject Reduction) Let t →∗ t . Then Γ t :T ⇒ Γ t :T Proof. (sketch) We proof rule by rule that if t→t using that rule and Γ t : T , then Γ t : T . In general, the method is to take the term t, decompose it into its small parts and recompose to t. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 19/31
  58. 58. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (I) To show a rule as example, we need some auxiliary lemmas and denitions: Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 20/31
  59. 59. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (I) To show a rule as example, we need some auxiliary lemmas and denitions: Order: Write A B if either B ≡ ∀X .A or A ≡ ∀X .C and B ≡ C [U /X ] for some U ∈ U. ≥ is the reexive and transitive closure of . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 20/31
  60. 60. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (I) To show a rule as example, we need some auxiliary lemmas and denitions: Order: Write A B if either B ≡ ∀X .A or A ≡ ∀X .C and B ≡ C [U /X ] for some U ∈ U. ≥ is the reexive and transitive closure of . Instuition of this denition: Types in the numerator of Γ t: A Γ t : ∀X .C ∀I with X ∈ FV (Γ) / or ∀E Γ t : ∀X .A Γ t : C [U /X ] are greater than the types in the denominator. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 20/31
  61. 61. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (II) Some lemmas needed: Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 21/31
  62. 62. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (II) Some lemmas needed: 1 A ≥B and Γ t: A ⇒ Γ t: B Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 21/31
  63. 63. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (II) Some lemmas needed: 1 A ≥ B and Γ t : A ⇒ Γ t: B 2 A ≥ B ⇒ αA ≥ αB Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 21/31
  64. 64. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (II) Some lemmas needed: 1 A ≥ B and Γ t : A ⇒ Γ t: B 2 A ≥ B ⇒ αA ≥ αB 3 Generation lemma (app): Let Γ (u v) : γ.T , then  Γ u : β.U → T  Γ v : α.U    T ≥ T  γ =α×β  Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 21/31
  65. 65. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (II) Some lemmas needed: 1 A ≥ B and Γ t : A ⇒ Γ t: B 2 A ≥ B ⇒ αA ≥ αB 3 Generation lemma (app): Let Γ (u v) : γ.T , then  Γ u : β.U → T  Γ v : α.U    T ≥ T  γ =α×β  4 Generation lemma (sum): Let Γ u + v : γ.T , then  Γ u : α.T  Γ v : β.T γ =α+β  Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 21/31
  66. 66. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (III) Example: Rule ( u + v) w → ( u w ) + ( v w ) . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 22/31
  67. 67. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (III) Example: Rule ( u + v) w → ( u w ) + ( v w ) . Let Γ (u + v) w : T . Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 22/31
  68. 68. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (III) Example: Rule ( u + v) w → ( u w ) + ( v w ) . Let Γ (u + v) w : T . Using the previous lemmas we can prove that u : (δ × β).U → T   Γ Γ v : ((1 − δ) × β).U → T Γ w : α.U  where α × β = 1, T ≥T and δ is some scalar. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 22/31
  69. 69. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (IV) Then Γ u : (δ × β).U → T Γ w : α.U →E Γ (u w) : (δ × β × α).T ≥ δ.T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 23/31
  70. 70. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (IV) Then Γ u : (δ × β).U → T Γ w : α.U →E Γ (u w) : (δ × β × α).T ≥ δ.T Also, Γ v : ((1 − δ) × β).(U → T ) Γ w : α.U →E Γ (v w) : ((1 − δ) × β × α).T ≥ (1 − δ).T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 23/31
  71. 71. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Subject reduction proof: example (IV) Then Γ u : (δ × β).U → T Γ w : α.U →E Γ (u w) : (δ × β × α).T ≥ δ.T Also, Γ v : ((1 − δ) × β).(U → T ) Γ w : α.U →E Γ (v w) : ((1 − δ) × β × α).T ≥ (1 − δ).T So Γ (u w) : δ.T Γ (v w) : (1 − δ).T +I and ≡ Γ (u w) + (v w) : T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 23/31
  72. 72. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Intuition Conditional functions → same type on each branch. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 24/31
  73. 73. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Intuition Conditional functions → same type on each branch. By restricting the scalars to positive reals → probabilistic type system. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 24/31
  74. 74. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Intuition Conditional functions → same type on each branch. By restricting the scalars to positive reals → probabilistic type system. For example, one can type functions such as λx {x [ 1 .(true + false)] [ 4 .true + 3 .false]} : B → B 2 1 4 with the type system serving as a guarantee that the function conserves probabilities summing to one. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 24/31
  75. 75. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Formalisation We dene the probabilistic type system to be the scalar type system with the following restrictions: Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 25/31
  76. 76. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Formalisation We dene the probabilistic type system to be the scalar type system with the following restrictions: S = R+ , Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 25/31
  77. 77. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Formalisation We dene the probabilistic type system to be the scalar type system with the following restrictions: S = R+ , Contexts: Types in contexts are classic types (C ), i.e. types in U exempt of any scalar, Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 25/31
  78. 78. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Formalisation We dene the probabilistic type system to be the scalar type system with the following restrictions: S = R+ , Contexts: Types in contexts are classic types (C ), i.e. types in U exempt of any scalar, We change the rule ∀E to Γ t : ∀X .T ∀E with C ∈C Γ t : T [C /X ] Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 25/31
  79. 79. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Formalisation We dene the probabilistic type system to be the scalar type system with the following restrictions: S = R+ , Contexts: Types in contexts are classic types (C ), i.e. types in U exempt of any scalar, We change the rule ∀E to Γ t : ∀X .T ∀E with C ∈C Γ t : T [C /X ] The nal conclusion must be classic. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 25/31
  80. 80. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Proof Denition (Weight function to check probability distributions) Let ω : Λ → R+ be a function dened inductively by: ω(0) = 0 ω(t1 + t2 ) = ω(t1 ) + ω(t2 ) ω(b) = 1 ω(α.t) = α × ω(t) ω(t1 t2 ) = ω(t1 ) × ω(t2 ) Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 26/31
  81. 81. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Proof Denition (Weight function to check probability distributions) Let ω : Λ → R+ be a function dened inductively by: ω(0) = 0 ω(t1 + t2 ) = ω(t1 ) + ω(t2 ) ω(b) = 1 ω(α.t) = α × ω(t) ω(t1 t2 ) = ω(t1 ) × ω(t2 ) Theorem (Normal terms in probabilistic have weight 1) ΓC t: C ⇒ ω(t ↓) = 1. Proof. We prove that Γ t : α.C ⇒ ω(t ↓) = α by structural induction over t ↓. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 26/31
  82. 82. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Proof Denition (Weight function to check probability distributions) Let ω : Λ → R+ be a function dened inductively by: ω(0) = 0 ω(t1 + t2 ) = ω(t1 ) + ω(t2 ) ω(b) = 1 ω(α.t) = α × ω(t) ω(t1 t2 ) = ω(t1 ) × ω(t2 ) Theorem (Normal terms in probabilistic have weight 1) ΓC t: C ⇒ ω(t ↓) = 1. Proof. We prove that Γ t : α.C ⇒ ω(t ↓) = α by structural induction over t ↓. 1 Example: 2.(λx .x ) y 2 Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 26/31
  83. 83. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Proof Denition (Weight function to check probability distributions) Let ω : Λ → R+ be a function dened inductively by: ω(0) = 0 ω(t1 + t2 ) = ω(t1 ) + ω(t2 ) ω(b) = 1 ω(α.t) = α × ω(t) ω(t1 t2 ) = ω(t1 ) × ω(t2 ) Theorem (Normal terms in probabilistic have weight 1) ΓC t: C ⇒ ω(t ↓) = 1. Proof. We prove that Γ t : α.C ⇒ ω(t ↓) = α by structural induction over t ↓. 1 Example: 2.(λx .x ) y 2 1 ω(2.(λx 2 .x ) y ) = 2 Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 26/31
  84. 84. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Probabilistic type system: Proof Denition (Weight function to check probability distributions) Let ω : Λ → R+ be a function dened inductively by: ω(0) = 0 ω(t1 + t2 ) = ω(t1 ) + ω(t2 ) ω(b) = 1 ω(α.t) = α × ω(t) ω(t1 t2 ) = ω(t1 ) × ω(t2 ) Theorem (Normal terms in probabilistic have weight 1) ΓC t: C ⇒ ω(t ↓) = 1. Proof. We prove that Γ t : α.C ⇒ ω(t ↓) = α by structural induction over t ↓. 1 Example: 2.(λx 2 .x ) y →y 1 ω(2.(λx 2 .x ) y ) = 2 ω(y ) = 1 Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 26/31
  85. 85. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (I) Denition (Proof method of depth n) Π0 (S ) = S Πn−1 (S ) Πk (S ) πh πk Πh ( S ) Πn (S ) = R or R or R PS P S S P where S is a sequent, πn is a constant derivation tree of size n, max{k , h} = n − 1, R is a typing rule, and P S is a sequent such that the resulting derivation tree is well-formed. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 27/31
  86. 86. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (I) Denition (Proof method of depth n) Π0 (S ) = S Πn−1 (S ) Πk (S ) πh πk Πh ( S ) Πn (S ) = R or R or R PS P S P S where S is a sequent, πn is a constant derivation tree of size n, max{k , h} = n − 1, R is a typing rule, and P S is a sequent such that the resulting derivation tree is well-formed. C (Πn (S )) denote the conclusion (root) of the tree Πn (S ). Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 27/31
  87. 87. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (II) Examples: S Π1 ( S ) = ∀I P S Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 28/31
  88. 88. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (II) Examples: S Γ t: T Π1 ( S ) = ∀I Π1 (Γ t: T) = ∀I P S Γ t : ∀X .T Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 28/31
  89. 89. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (II) Examples: S Γ t: T Π1 ( S ) = ∀I Π1 (Γ t: T) = ∀I P S Γ t : ∀X .T π n −2 Πn (Γ u : α.A) = Γ u : α.A Γ v : 2.A +I Γ u + v : (2 + α).A (Partial function or pattern matching) Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 28/31
  90. 90. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (II) Examples: S Γ t: T Π1 ( S ) = ∀I Π1 (Γ t: T) = ∀I P S Γ t : ∀X .T π n −2 Πn (Γ u : α.A) = Γ u : α.A Γ v : 2.A +I Γ u + v : (2 + α).A (Partial function or pattern matching) Basically C (Πn (S )) = PS means that PS can be derived from S by using S once, with the xed proof method Π. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 28/31
  91. 91. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (III) Theorem (No-cloning of scalars) Πn such that ∀α, C (Πn (Γ α.U )) = ∆ (δ × αs + γ).V with δ = 0 and γ constants in S , s ∈ N1 and U , V constants in U . Proof. Induction over n. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 29/31
  92. 92. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (III) Theorem (No-cloning of scalars) Πn such that ∀α, C (Πn (Γ α.U )) = ∆ (δ × αs + γ).V with δ = 0 and γ constants in S , s ∈ N1 and U , V constants in U . Proof. Induction over n. Corollary (No-cloning Theorem) Πn such that ∀T , C (Πn (Γ T )) = ∆ T ⊗ T. where A ⊗B is the classical encoding for the type of tuples. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 29/31
  93. 93. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (III) Theorem (No-cloning of scalars) Πn such that ∀α, C (Πn (Γ α.U )) = ∆ (δ × αs + γ).V with δ = 0 and γ constants in S , s ∈ N1 and U , V constants in U . Proof. Induction over n. Corollary (No-cloning Theorem) Πn such that ∀T , C (Πn (Γ T )) = ∆ T ⊗ T. where A ⊗B is the classical encoding for the type of tuples. Proof. It is easy to show that ∀T , T ≡ α.U with U ∈U Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 29/31
  94. 94. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (III) Theorem (No-cloning of scalars) Πn such that ∀α, C (Πn (Γ α.U )) = ∆ (δ × αs + γ).V with δ = 0 and γ constants in S , s ∈ N1 and U , V constants in U . Proof. Induction over n. Corollary (No-cloning Theorem) Πn such that ∀T , C (Πn (Γ T )) = ∆ T ⊗ T. where A ⊗B is the classical encoding for the type of tuples. Proof. It is easy to show that ∀T , T ≡ α.U with U ∈U T ⊗ T ≡ α.U ⊗ α.U ≡ α .(U ⊗ U ) = (1 × α2 + 0).(U ⊗ U ). 2 Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 29/31
  95. 95. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Logical content: No-cloning theorem (III) Theorem (No-cloning of scalars) Πn such that ∀α, C (Πn (Γ α.U )) = ∆ (δ × αs + γ).V with δ = 0 and γ constants in S , s ∈ N1 and U , V constants in U . Proof. Induction over n. Corollary (No-cloning Theorem) Πn such that ∀T , C (Πn (Γ T )) = ∆ T ⊗ T. where A ⊗B is the classical encoding for the type of tuples. Proof. It is easy to show that ∀T , T ≡ α.U with U ∈U T ⊗ T ≡ α.U ⊗ α.U ≡ α .(U ⊗ U ) = (1 × α2 + 0).(U ⊗ U ). 2 By the previous theorem, the corollary holds. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 29/31
  96. 96. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Summary of contributions S.N. : Simplied the Linear-algebraic λ-calculus by lifting most restrictions. S.R. OK. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 30/31
  97. 97. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Summary of contributions S.N. : Simplied the Linear-algebraic λ-calculus by lifting most restrictions. S.R. OK. Scalar type system : types keep track of the `amount of a type' by holding sum of amplitudes of terms of that types. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 30/31
  98. 98. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Summary of contributions S.N. : Simplied the Linear-algebraic λ-calculus by lifting most restrictions. S.R. OK. Scalar type system : types keep track of the `amount of a type' by holding sum of amplitudes of terms of that types. =⇒ probabilistic type system, yielding a higher-order probabilistic λ-calculus. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 30/31
  99. 99. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Polemics, future work Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 31/31
  100. 100. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Polemics, future work Captured no-cloning theorem is a way that is faithful to quantum theory and linear algebra, unlike LL: For all one can nd a copying proof method, but there is no A proof method for cloning all A. Algebraic linearity is about taking α.U to something in αγ + δ . . . not just for γ = 1 = δ + 1. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 31/31
  101. 101. Motivation Linear-Algebraic λ-Calculus System F The scalar type system Conclusions and future work The scalar type system Polemics, future work Captured no-cloning theorem is a way that is faithful to quantum theory and linear algebra, unlike LL: For all one can nd a copying proof method, but there is no A proof method for cloning all A. Algebraic linearity is about taking α.U to something in αγ + δ . . . not just for γ = 1 = δ + 1. C.-H.+Q.C.=(quantum th. proofs, quantum th. logics)? Need a Vectorial type system: Scalar type system → magnitude and signs for type vectors. Future system → direction, (i.e. addition and orthogonality of types). Then it would be possible to check norm on amplitudes rather than probabilities. Pablo Arrighi and Alejandro Díaz-Caro, LIG A System F accounting for scalars, arXiv:0903.3741 31/31

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