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Reduction strategies for CBN and CBV Structural operational style - PowerPoint PPT Presentation

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Reduction strategies for CBN and CBV Structural operational style and Felleisen style Bas Broere April 24th 2018 Bas Broere Reduction strategies for CBN and CBV 1/22 Introduction Disclaimer I have (almost) no experience in functional

  1. Features/limitations of SOS in CBV We cannot reduce under λ -abstractions (weak reduction). So this is NOT a thing: M → M ′ λ x . M → λ x . M ′ Bas Broere — Reduction strategies for CBN and CBV 9/22

  2. Features/limitations of SOS in CBV We cannot reduce under λ -abstractions (weak reduction). So this is NOT a thing: M → M ′ λ x . M → λ x . M ′ We need to reduce N to a value if we want to evaluate ( λ x . M ) N (call-by-value). So also this is NOT a thing: ( λ x . M )( N P ) → M [ x := N P ] Bas Broere — Reduction strategies for CBN and CBV 9/22

  3. Features/limitations of SOS in CBV We cannot reduce under λ -abstractions (weak reduction). So this is NOT a thing: M → M ′ λ x . M → λ x . M ′ We need to reduce N to a value if we want to evaluate ( λ x . M ) N (call-by-value). So also this is NOT a thing: ( λ x . M )( N P ) → M [ x := N P ] In the application M N , we need to reduce M to a value first (left-to-right), because: N → N ′ v N → v N ′ (app-r) . Bas Broere — Reduction strategies for CBN and CBV 9/22

  4. Features/limitations of SOS in CBV We cannot reduce under λ -abstractions (weak reduction). So this is NOT a thing: M → M ′ λ x . M → λ x . M ′ We need to reduce N to a value if we want to evaluate ( λ x . M ) N (call-by-value). So also this is NOT a thing: ( λ x . M )( N P ) → M [ x := N P ] In the application M N , we need to reduce M to a value first (left-to-right), because: N → N ′ v N → v N ′ (app-r) . It is deterministic, i.e. for every M there is at most one M ′ s.t. M → M ′ . Bas Broere — Reduction strategies for CBN and CBV 9/22

  5. Multiple reductions When making a reduction sequence we have 3 possibilities: Bas Broere — Reduction strategies for CBN and CBV 10/22

  6. Multiple reductions When making a reduction sequence we have 3 possibilities: 1 The reductions end in a value (termination): M → M 1 → M 2 → . . . → v . Example: ( λ x .λ y . y x ) (( λ x . x )1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) → ( λ y . y 1) ( λ x . x ) → ( λ x . x ) 1 → 1 Bas Broere — Reduction strategies for CBN and CBV 10/22

  7. Multiple reductions When making a reduction sequence we have 3 possibilities: 1 The reductions end in a value (termination): M → M 1 → M 2 → . . . → v . Example: ( λ x .λ y . y x ) (( λ x . x )1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) → ( λ y . y 1) ( λ x . x ) → ( λ x . x ) 1 → 1 2 The reduction never ends (divergence): M → M 1 → M 2 → . . . → M n → . . . . Example: Bas Broere — Reduction strategies for CBN and CBV 10/22

  8. Multiple reductions When making a reduction sequence we have 3 possibilities: 1 The reductions end in a value (termination): M → M 1 → M 2 → . . . → v . Example: ( λ x .λ y . y x ) (( λ x . x )1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) → ( λ y . y 1) ( λ x . x ) → ( λ x . x ) 1 → 1 2 The reduction never ends (divergence): M → M 1 → M 2 → . . . → M n → . . . . Example: ω ω , where ω = λ x . x x . Bas Broere — Reduction strategies for CBN and CBV 10/22

  9. Multiple reductions When making a reduction sequence we have 3 possibilities: 1 The reductions end in a value (termination): M → M 1 → M 2 → . . . → v . Example: ( λ x .λ y . y x ) (( λ x . x )1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) → ( λ y . y 1) ( λ x . x ) → ( λ x . x ) 1 → 1 2 The reduction never ends (divergence): M → M 1 → M 2 → . . . → M n → . . . . Example: ω ω , where ω = λ x . x x . 3 Error: M → M 1 → M 2 → . . . → M n �→ , where M n is not a value, but also does not reduce. Example: Bas Broere — Reduction strategies for CBN and CBV 10/22

  10. Multiple reductions When making a reduction sequence we have 3 possibilities: 1 The reductions end in a value (termination): M → M 1 → M 2 → . . . → v . Example: ( λ x .λ y . y x ) (( λ x . x )1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) → ( λ y . y 1) ( λ x . x ) → ( λ x . x ) 1 → 1 2 The reduction never ends (divergence): M → M 1 → M 2 → . . . → M n → . . . . Example: ω ω , where ω = λ x . x x . 3 Error: M → M 1 → M 2 → . . . → M n �→ , where M n is not a value, but also does not reduce. Example: ω 2 → 2 2 �→ . Bas Broere — Reduction strategies for CBN and CBV 10/22

  11. Reduction strategies for CBV Reduction contexts (felleisen style) Head reductions: ( λ x . M ) v → ǫ M [ x := v ] ( β v ) Bas Broere — Reduction strategies for CBN and CBV 11/22

  12. Reduction strategies for CBV Reduction contexts (felleisen style) Head reductions: ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , with E ::= [ ] | E b | v E terms with a hole [ ] in it. Bas Broere — Reduction strategies for CBN and CBV 11/22

  13. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  14. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  15. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  16. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  17. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  18. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  19. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  20. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 12/22

  21. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) E = [ ] Bas Broere — Reduction strategies for CBN and CBV 12/22

  22. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) E = [ ] → ( λ x . x ) 1 Bas Broere — Reduction strategies for CBN and CBV 12/22

  23. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) E = [ ] → ( λ x . x ) 1 E = [ ] Bas Broere — Reduction strategies for CBN and CBV 12/22

  24. Example Reduction contexts (Felleisen style) ( λ x . M ) v → ǫ M [ x := v ] ( β v ) M → ǫ M ′ E [ M ] → E [ M ′ ] (context) , E ::= [ ] | E b | v E Same example as before: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) E = ( λ x .λ y . x y ) [ ] ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) E = [ ] ( λ x . x ) → ( λ y . y 1) ( λ x . x ) E = [ ] → ( λ x . x ) 1 E = [ ] → 1 Bas Broere — Reduction strategies for CBN and CBV 12/22

  25. Equivalence SOS and Felleisen Remember the earlier example: ( λ x . x ) 1 → 1 ( λ x .λ y . y x ) (( λ x . x ) 1) → ( λ x .λ y . y x ) 1 ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) This corresponds to the context: Bas Broere — Reduction strategies for CBN and CBV 13/22

  26. Equivalence SOS and Felleisen Remember the earlier example: ( λ x . x ) 1 → 1 ( λ x .λ y . y x ) (( λ x . x ) 1) → ( λ x .λ y . y x ) 1 ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) This corresponds to the context: E = Bas Broere — Reduction strategies for CBN and CBV 13/22

  27. Equivalence SOS and Felleisen Remember the earlier example: ( λ x . x ) 1 → 1 ( λ x .λ y . y x ) (( λ x . x ) 1) → ( λ x .λ y . y x ) 1 ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) This corresponds to the context: E = (( λ x .λ y . y x ) [ ]) ( λ x . x ) with head reduction ( λ x . x ) 1 → ǫ 1. Bas Broere — Reduction strategies for CBN and CBV 13/22

  28. Equivalence SOS and Felleisen Remember the earlier example: ( λ x . x ) 1 → 1 ( λ x .λ y . y x ) (( λ x . x ) 1) → ( λ x .λ y . y x ) 1 ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ x . x ) → ( λ x .λ y . y x ) 1 ( λ x . x ) This corresponds to the context: E = (( λ x .λ y . y x ) [ ]) ( λ x . x ) with head reduction ( λ x . x ) 1 → ǫ 1. Equivalence SOS and Felleisen approach are one-to-one! Bas Broere — Reduction strategies for CBN and CBV 13/22

  29. Result From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*. Bas Broere — Reduction strategies for CBN and CBV 14/22

  30. Result From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*. *: The determinism is also a consequence of the following theorem: Unique decomposition theorem For all terms M , there exists at most one reduction context E and one term N , s.t. M = E [ N ] and N can reduce by head-reduction. Bas Broere — Reduction strategies for CBN and CBV 14/22

  31. Result From this equivalence, it follow that Felleisen style also has the previously discussed properties: Left-to-right, Call-by-value, Weak reduction, Deterministic*. *: The determinism is also a consequence of the following theorem: Unique decomposition theorem For all terms M , there exists at most one reduction context E and one term N , s.t. M = E [ N ] and N can reduce by head-reduction. Proof: Induction on the terms. Bas Broere — Reduction strategies for CBN and CBV 14/22

  32. Reduction strategies for CBN Call-by-name The arguments to a function are not evaluated before the function is called (perform β -reduction asap). Bas Broere — Reduction strategies for CBN and CBV 15/22

  33. Reduction strategies for CBN Call-by-name The arguments to a function are not evaluated before the function is called (perform β -reduction asap). Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) Bas Broere — Reduction strategies for CBN and CBV 15/22

  34. Reduction strategies for CBN Call-by-name The arguments to a function are not evaluated before the function is called (perform β -reduction asap). Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , with E ::= [ ] | E M Bas Broere — Reduction strategies for CBN and CBV 15/22

  35. Reduction strategies for CBN Call-by-name The arguments to a function are not evaluated before the function is called (perform β -reduction asap). Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , with E ::= [ ] | E M , so E ::= [ ] M 1 . . . M n . Bas Broere — Reduction strategies for CBN and CBV 15/22

  36. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 16/22

  37. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 16/22

  38. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 16/22

  39. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 16/22

  40. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) → ( λ z . z ) (( λ x . x ) 1) Bas Broere — Reduction strategies for CBN and CBV 16/22

  41. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) → ( λ z . z ) (( λ x . x ) 1) → ( λ x . x ) 1 Bas Broere — Reduction strategies for CBN and CBV 16/22

  42. Examples Structural operation style (SOS) M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) M N → M ′ N (app-l) The same example once again, in SOS-style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) → ( λ z . z ) (( λ x . x ) 1) → ( λ x . x ) 1 → 1 . Bas Broere — Reduction strategies for CBN and CBV 16/22

  43. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 17/22

  44. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ x . x ) Bas Broere — Reduction strategies for CBN and CBV 17/22

  45. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 17/22

  46. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 17/22

  47. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) Bas Broere — Reduction strategies for CBN and CBV 17/22

  48. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] Bas Broere — Reduction strategies for CBN and CBV 17/22

  49. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] → ( λ z . z ) (( λ x . x ) 1) Bas Broere — Reduction strategies for CBN and CBV 17/22

  50. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] → ( λ z . z ) (( λ x . x ) 1) E = [ ] Bas Broere — Reduction strategies for CBN and CBV 17/22

  51. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] → ( λ z . z ) (( λ x . x ) 1) E = [ ] → ( λ x . x ) 1 Bas Broere — Reduction strategies for CBN and CBV 17/22

  52. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] → ( λ z . z ) (( λ x . x ) 1) E = [ ] → ( λ x . x ) 1 E = [ ] Bas Broere — Reduction strategies for CBN and CBV 17/22

  53. Examples Felleisen style M → M ′ ( λ x . M ) N → M [ x := N ] ( β n ) E [ M ] → E [ M ′ ](context) , E ::= [ ] | E M . And in Felleisen style: ( λ x .λ y . y x ) (( λ x . x ) 1) ( λ z . z ) E = [ ] ( λ z . z ) → ( λ y . y (( λ x . x ) 1)) ( λ z . z ) E = [ ] → ( λ z . z ) (( λ x . x ) 1) E = [ ] → ( λ x . x ) 1 E = [ ] → 1 . Bas Broere — Reduction strategies for CBN and CBV 17/22

  54. CBV vs CBN Some examples: Termination: Bas Broere — Reduction strategies for CBN and CBV 18/22

  55. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → Bas Broere — Reduction strategies for CBN and CBV 18/22

  56. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; Bas Broere — Reduction strategies for CBN and CBV 18/22

  57. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → Bas Broere — Reduction strategies for CBN and CBV 18/22

  58. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . Bas Broere — Reduction strategies for CBN and CBV 18/22

  59. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Bas Broere — Reduction strategies for CBN and CBV 18/22

  60. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: Bas Broere — Reduction strategies for CBN and CBV 18/22

  61. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: ( λ x . x + x ) M : Bas Broere — Reduction strategies for CBN and CBV 18/22

  62. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: ( λ x . x + x ) M : CBV evaluates M once, CBN twice; Bas Broere — Reduction strategies for CBN and CBV 18/22

  63. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: ( λ x . x + x ) M : CBV evaluates M once, CBN twice; ( λ x . 1) M : Bas Broere — Reduction strategies for CBN and CBV 18/22

  64. CBV vs CBN Some examples: Termination: CBN: ( λ x . 1) ( ω ω ) → 1; CBV: ( λ x . 1) ( ω ω ) → ( λ x . 1) ( ω ω ) → . . . In general: Termination in CBV implies termination in CBN, but not the other way around. Efficiency: ( λ x . x + x ) M : CBV evaluates M once, CBN twice; ( λ x . 1) M : CBV evaluates M once, CBN not at all. Bas Broere — Reduction strategies for CBN and CBV 18/22

  65. CBN in CBV language Bas Broere — Reduction strategies for CBN and CBV 19/22

  66. CBN in CBV language Introduce functions λ x . M to delay computation of M (thunks). Bas Broere — Reduction strategies for CBN and CBV 19/22

  67. CBN in CBV language Introduce functions λ x . M to delay computation of M (thunks). 1 � x � = x () (application); Bas Broere — Reduction strategies for CBN and CBV 19/22

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