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Church call-by-name (x.e) e x e{x:=e x - PowerPoint PPT Presentation

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T HE C ALL-BY-NEED L AMBDA C ALCULUS, R EVISITED Stephen Chang and Matthias Felleisen Northeastern University 26/3/2012 Church call-by-name (x.e) e x e{x:=e x } ()

  1. T HE C ALL-BY-NEED L AMBDA C ALCULUS, R EVISITED Stephen Chang and Matthias Felleisen Northeastern University 26/3/2012 �

  2. Church ��

  3. ��

  4. ��

  5. ��

  6. ��

  7. ��

  8. λ call-by-name (λx.e) e x → e{x:=e x } (β) �����

  9. λ call-by-name E[(λx.e) e x ] → E[e{x:=e x }] (β) "leftmost-outermost" E @ λx e x e ��

  10. λ v call-by-value E[(λx.e) v x ] → E[e{x:=v x }] (β v ) "leftmost-outermost" E @ λx v x e ��

  11. call-by-need ��

  12. call-by-need 1) Evaluate argument only when needed. ��

  13. call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. ��

  14. λ need ? ? call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. ��

  15. λ need λ need-af λ need-mow ? ? = = Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. ��

  16. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. ��

  17. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need ��

  18. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need x ��

  19. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need λx x ��

  20. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need λx e x x ��

  21. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need → → @ λx e x λx e x x x (reshuffle) ��

  22. λ need λ need-af λ need-mow � � Ariola/Felleisen Maraist/Odersky/Wadler '94,'95,'97 '94,'95,'98 call-by-need @ @ → λx v x λx v x x v x (like β) ���

  23. our λ need call-by-need λx e x x ���

  24. our λ need call-by-need λx e x x ���

  25. our λ need call-by-need λx "demand path" e x x ���

  26. our λ need call-by-need λx "demand path" e x x (β need ) ���

  27. O LD λ need : O PERATIONAL O VERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. ���

  28. O LD λ need : O PERATIONAL O VERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. ���

  29. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e ���

  30. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e ���

  31. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e | (λx.D) e ���

  32. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e | (λx.D) e ���

  33. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e | (λx.D) e ���

  34. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e | (λx.D) e binding structure ���

  35. O LD λ need : D EMAND C ONTEXTS D = [ ] | D e | B[D] binding structure B = [ ] | (λx.B) e ���

  36. O LD λ need : B INDING S TRUCTURE B = [ ] | (λx.B) e (λx.(λy.(λz. ...) e z ) e y ) e x ���

  37. O LD λ need : B INDING S TRUCTURE B = [ ] | (λx.B) e (λx.(λy.(λz. ...) e z ) e y ) e x ���

  38. O LD λ need : B INDING S TRUCTURE B = [ ] | (λx.B) e (λx.(λy.(λz. ...) e z ) e y ) e x ���

  39. O LD λ need : B INDING S TRUCTURE B = [ ] | (λx.B) e (λx.(λy.(λz. ...) e z ) e y ) e x @ λx e x @ λy e y @ λz e z ... ���

  40. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (((λx.λy.λz. ...) e x ) e y ) e z @ @ e z @ e y λx e x λy λz ... ���

  41. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (((λx.λy.λz. ...) e x ) e y ) e z @ @ e z @ e y λx e x λy λz ... ���

  42. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (((λx.λy.λz. ...) e x ) e y ) e z @ @ e z @ e y λx e x λy λz ... ���

  43. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e ((λx.(λy.λz. ...) e y ) e x ) e z @ @ e z λx e x @ λy e y λz ... ���

  44. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e ((λx.(λy.λz. ...) e y ) e x ) e z @ @ e z λx e x @ λy e y λz ... ���

  45. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (λx.((λy.λz. ...) e y ) e z ) e x @ λx e x @ @ e z λy e y λz ... ���

  46. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (λx.((λy.λz. ...) e y ) e z ) e x @ λx e x @ @ e z λy e y λz ... ���

  47. O LD λ need : R ESHUFFLING OF B INDINGS B = [ ] | (λx.B) e (λx.(λy.(λz. ...) e z ) e y ) e x @ λx e x @ λy e y @ λz e z ... ���

  48. P ROBLEMS WITH O LD C ALL-BY-NEED C ALCULUS 1) Reshuffling rules. ���

  49. O LD λ need : O PERATIONAL O VERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. ���

  50. O LD λ need : O PERATIONAL O VERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. ���

  51. O LD λ need : O PERATIONAL O VERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. ���

  52. O LD λ need : D EREFERENCING (λx.(λy.(λz. ...) e z ) e y ) e x @ λx e x @ λy e y @ λz e z ... ���

  53. O LD λ need : D EREFERENCING (λx.(λy.(λz. y ) e z ) e y ) e x @ λx e x @ λy e y @ λz e z y ���

  54. O LD λ need : D EREFERENCING (λx.(λy.(λz. y ) e z ) v y ) e x @ λx e x @ λy v y @ λz e z y ���

  55. O LD λ need : D EREFERENCING (λx.(λy.(λz. y ) e z ) v y ) e x @ λx e x @ λy v y @ λz e z y (λy.D[y]) v → (λy.D[v]) v (deref) ���

  56. O LD λ need : D EREFERENCING (λx.(λy.(λz. v y ) e z ) v y ) e x @ λx e x @ λy v y @ λz e z v y (λy.D[y]) v → (λy.D[v]) v (deref) ���

  57. P ROBLEMS WITH O LD C ALL-BY-NEED C ALCULUS 1) Reshuffling rules. 2) Arguments and applications never go away. ���

  58. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ λx e x @ λy e y @ λz e z ... �������

  59. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ λx e x @ λy e y @ λz e z ... B = [ ] | (λx.B) e ���

  60. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z @ e y λx e x λy λz ... A = [ ] | ??? ���

  61. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ λx e x @ @ e z λy e y λz ... A = [ ] | ??? ���

  62. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | ??? ���

  63. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | (λx.A) e ���

  64. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | (λx.A) e ���

  65. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | (λx.A) e ���

  66. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | A[(λx.A)] e ���

  67. N EW λ need : H ANDLING A RBITRARY B INDING S TRUCTURE @ @ e z λx e x @ λy e y λz ... A = [ ] | A[(λx.A)] e D = [ ] | D e | A[D] ���

  68. P ROBLEMS WITH O LD C ALL-BY-NEED C ALCULUS 1) Reshuffling rules. 2) Arguments and applications never go away. ���

  69. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz ... ���

  70. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz ... A[ ...] ���

  71. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] A[D[y]] ���

  72. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] A[D[y]] ���

  73. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] ... (λy...D[y])...e y ... ���

  74. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] ... (λy...D[y])...e y ... ���

  75. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] ...A[λy...D[y] ] e y ... ���

  76. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] ...A[λy...D[y] ] e y ... ���

  77. N EW λ need : S PLITTING C ONTEXTS @ @ e z @ e y λx e x λy λz D[y] ...A[λy. [D[y]]] e y ... = [ ] | A[λx. ] ���

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