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Evaluating Call-by-need on the Control Stack Stephen Chang, David Van Horn, Matthias Felleisen Northeastern University Lazy Abstract Machines Sharing implemented with: heap Lazy Abstract Machines Sharing implemented with: heap

  1. Evaluating Call-by-need on the Control Stack Stephen Chang, David Van Horn, Matthias Felleisen Northeastern University �

  2. Lazy Abstract Machines Sharing implemented with: heap �

  3. Lazy Abstract Machines Sharing implemented with: heap stack operations (alternative approach) �

  4. Lazy Abstract Machines Sharing implemented with: heap stack operations (alternative approach) [Garcia et al. 2009] �

  5. Our Paper • New way to resolve variable references in the stack �

  6. Our Paper • New way to resolve variable references in the stack • Reorganize stack structure to allow indexing �

  7. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] �

  8. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed �

  9. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once �

  10. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M ��

  11. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M | (λx.E[x]) E ��

  12. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M [ ] | E M [ ] | E M | (λx.E) M | (λx.E[x]) E [ ] | E M ��

  13. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M (λx.E) M (λx.E) M (λx.E) M | (λx.E[x]) E ��

  14. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M | (λx.E[x]) E (λx.E[x]) E (λx.E[x]) E (λx.E[x]) E ��

  15. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M | (λx.E[x]) E deref (β alternative): (λx.E[x]) V (λx.E[V V V V]) V ��

  16. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M | (λx.E[x]) E deref (β alternative): (λx.E[x]) V (λx.E[V V V V]) V • One-at-a-time substitution (only when needed) ��

  17. Call-by-need λ-Calculus [Ariola et al. 1995] [Ariola and Felleisen 1997] • Delay evaluation of argument until needed • Evaluate each argument only once M = x | M M | λx.M E = [ ] | E M | (λx.E) M | (λx.E[x]) E deref (β alternative): (λx.E[x]) V (λx.E[V V V V]) V • One-at-a-time substitution (only when needed) • Argument not removed (may need it again) ��

  18. An Initial Abstract Machine ��

  19. An Initial Abstract Machine Standard Reduction = abstract machine SR E[N] E[M] if M N ��

  20. An Initial Abstract Machine Standard Reduction = abstract machine SR E[N] E[M] if M N • Re-partition into E and M after every reduction ��

  21. CK Machine [Felleisen 1986] (For by-value λ calculus) • Separate program into two registers: C = Current subterm being evaluated C C C K = Continuation (equiv. to eval. context) K K K ��

  22. CK Machine [Felleisen 1986] (For by-value λ calculus) • Separate program into two registers: C = Current subterm being evaluated C C C K = Continuation (equiv. to eval. context) K K K Don't need to re-partition program after every reduction ��

  23. CK Machine [Felleisen 1986] (For by-value λ calculus) • Separate program into two registers: C = Current subterm being evaluated C C C K = Continuation (equiv. to eval. context) K K K Don't need to re-partition program after every reduction [Garcia et al. 2009] : lazy CK machine ��

  24. Evaluation Contexts (E) vs Continuations (K) [ ] ~ mt E[[ ] M] ~ (arg M K) E ~ K E[(λx.[ ]) M] ~ (bind x M K) E ~ K E[(λx.E'[x]) [ ]] ~ (op x K' K) K' ~ E', K ~ E ��

  25. Evaluation Contexts (E) vs Continuations (K) [ ] ~ mt [ ] ~ mt [ ] ~ mt [ ] ~ mt E[[ ] M] ~ (arg M K) E ~ K E[(λx.[ ]) M] ~ (bind x M K) E ~ K E[(λx.E'[x]) [ ]] ~ (op x K' K) K' ~ E', K ~ E ��

  26. Evaluation Contexts (E) vs Continuations (K) [ ] ~ mt E[[ ] M] ~ (arg M K) E[[ ] M] ~ (arg M K) E[[ ] M] ~ (arg M K) E[[ ] M] ~ (arg M K) E ~ K E[(λx.[ ]) M] ~ (bind x M K) E ~ K E[(λx.E'[x]) [ ]] ~ (op x K' K) K' ~ E', K ~ E ��

  27. Evaluation Contexts (E) vs Continuations (K) [ ] ~ mt E[[ ] M] ~ (arg M K) E ~ K E[(λx.[ ]) M] ~ (bind x M K) E[(λx.[ ]) M] ~ (bind x M K) E[(λx.[ ]) M] ~ (bind x M K) E[(λx.[ ]) M] ~ (bind x M K) E ~ K E[(λx.E'[x]) [ ]] ~ (op x K' K) K' ~ E', K ~ E ��

  28. Evaluation Contexts (E) vs Continuations (K) [ ] ~ mt E[[ ] M] ~ (arg M K) E ~ K E[(λx.[ ]) M] ~ (bind x M K) E ~ K E[(λx.E'[x]) [ ]] ~ (op x K' K) E[(λx.E'[x]) [ ]] ~ (op x K' K) E[(λx.E'[x]) [ ]] ~ (op x K' K) E[(λx.E'[x]) [ ]] ~ (op x K' K) K' ~ E', K ~ E ��

  29. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 ��

  30. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 ��

  31. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 K K K K = mt ��

  32. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) K K K K = (arg M5) mt ��

  33. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λy.(λz.(y M)) M0 M1 M2) M3 M4 K K K K = (bind x M5) mt ��

  34. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λy.(λz.(y M)) M0 M1 M2) M3 K K K K = (arg M4) (bind x M5) mt ��

  35. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λy.(λz.(y M)) M0 M1 M2) K K K K = (arg M3) (arg M4) (bind x M5) mt ��

  36. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λz.(y M)) M0 M1 M2 K K K K = (bind y M3) (arg M4) (bind x M5) mt ��

  37. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λz.(y M)) M0 M1 K K K K = (arg M2) (bind y M3) (arg M4) (bind x M5) mt ��

  38. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λz.(y M)) M0 K K K K = (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt ��

  39. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (λz.(y M)) K = (arg M0) K K K (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt ��

  40. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = (y M) K = (bind z M0) K K K (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt ��

  41. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K = (arg M) K K (bind z M0) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt ��

  42. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K = (arg M) K K (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt (arg M) (arg M) ��

  43. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K = (arg M) K (bind z M0) (bind z M0) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt (bind z M0) (bind z M0) ��

  44. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K K = (arg M) (bind z M0) (arg M1) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt (arg M1) (arg M1) ��

  45. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K K = (arg M) (bind z M0) (arg M1) (arg M2) (arg M2) (bind y M3) (arg M4) (bind x M5) mt (arg M2) (arg M2) ��

  46. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K = (arg M) K (bind z M0) (arg M1) (arg M2) (bind y M3) (bind y M3) (arg M4) (bind x M5) mt (bind y M3) (bind y M3) ��

  47. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K = (arg M) K (bind z M0) (arg M1) (arg M2) (bind y M3) (bind y M3) (arg M4) (bind x M5) mt (bind y M3) (bind y M3) ��

  48. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K = (arg M) K (bind z M0) (arg M1) (arg M2) (bind y M3) (bind y M3) (arg M4) (bind x M5) mt (bind y M3) (bind y M3) ��

  49. Example (Garcia Machine) (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5 C C C C = y K K K = (arg M) K (bind z M0) (arg M1) (arg M2) (bind y M3) (bind y M3) (arg M4) (bind x M5) mt (bind y M3) (bind y M3) ��

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