Evaluating Call-by-need on the Control Stack

Stephen Chang, David Van Horn, Matthias Felleisen

Northeastern University

1

Lazy Abstract Machines

Sharing implemented with:heap

2

Lazy Abstract Machines

Sharing implemented with:heap

stack operations(alternative approach)

3

Lazy Abstract Machines

Sharing implemented with:heap

stack operations(alternative approach)

[Garcia et al. 2009]

4

Our Paper

• New way to resolve variable references in the stack

5

Our Paper

• New way to resolve variable references in the stack

• Reorganize stack structure to allow indexing

6

Call-by-need λ-Calculus

[Ariola et al. 1995][Ariola and Felleisen 1997]

7

Call-by-need λ-Calculus

[Ariola et al. 1995][Ariola and Felleisen 1997]

• Delay evaluation of argument until needed

8

Call-by-need λ-Calculus

[Ariola et al. 1995][Ariola and Felleisen 1997]

• Delay evaluation of argument until needed

• Evaluate each argument only once

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

M = x | M M | λx.M

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

E = [ ] | E M | (λx.E) M | (λx.E[x]) E

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[ ] | E M[ ] | E M[ ] | 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 | (λx.E) M(λx.E) M(λx.E) M(λx.E) M | (λx.E[x]) E

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[x]) E(λx.E[x]) E(λx.E[x]) E(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

deref (β alternative):

(λx.E[x]) V (λx.E[VVVV]) V

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[VVVV]) V

• One-at-a-time substitution (only when needed)

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[VVVV]) V

• One-at-a-time substitution (only when needed)

• Argument not removed (may need it again)

17

An Initial Abstract Machine

18

An Initial Abstract Machine

Standard Reduction = abstract machine

E[M] SR E[N]

if M N

19

An Initial Abstract Machine

Standard Reduction = abstract machine

E[M] SR E[N]

if M N

• Re-partition into E and M after every reduction

20

CK Machine

[Felleisen 1986]

(For by-value λ calculus)

• Separate program into two registers:

CCCC = Current subterm being evaluated

KKKK = Continuation (equiv. to eval. context)

21

CK Machine

[Felleisen 1986]

(For by-value λ calculus)

• Separate program into two registers:

CCCC = Current subterm being evaluated

KKKK = Continuation (equiv. to eval. context)

Don't need to re-partition program after every reduction

22

CK Machine

[Felleisen 1986]

(For by-value λ calculus)

• Separate program into two registers:

CCCC = Current subterm being evaluated

KKKK = Continuation (equiv. to eval. context)

Don't need to re-partition program after every reduction

[Garcia et al. 2009]: lazy CK machine

23

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

24

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

25

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

26

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

27

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

28

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

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

CCCC = (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5KKKK = mt

31

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4)KKKK = (arg M5) mt

32

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λy.(λz.(y M)) M0 M1 M2) M3 M4KKKK = (bind x M5) mt

33

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λy.(λz.(y M)) M0 M1 M2) M3KKKK = (arg M4) (bind x M5) mt

34

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λy.(λz.(y M)) M0 M1 M2)KKKK = (arg M3) (arg M4) (bind x M5) mt

35

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λz.(y M)) M0 M1 M2KKKK = (bind y M3) (arg M4) (bind x M5) mt

36

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λz.(y M)) M0 M1KKKK = (arg M2) (bind y M3) (arg M4) (bind x M5) mt

37

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = (λz.(y M)) M0KKKK = (arg M1) (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

CCCC = (λz.(y M))KKKK = (arg M0) (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

CCCC = (y M)KKKK = (bind z M0) (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

CCCC = yKKKK = (arg M) (bind z M0) (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

CCCC = yKKKK = (arg M)(arg M)(arg M)(arg M) (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

CCCC = yKKKK = (arg M) (bind z M0)(bind z M0)(bind z M0)(bind z M0) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt

43

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1)(arg M1)(arg M1)(arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt

44

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2)(arg M2)(arg M2)(arg M2) (bind y M3) (arg M4) (bind x M5) mt

45

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5) mt

46

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5) mt

47

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5) mt

48

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5) mt

49

Example (Garcia Machine)

(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5

CCCC = yKKKK = (arg M) (bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5) mt

• Linear search to find argument

50

CK+ Machine: Stack Structure

• Reorganize stack to be stack of stacks

bind continuations on top

51

CK+ Machine: Stack Structure

• Reorganize stack to be stack of stacks

bind continuations on top

(arg M) (bind z M0) (arg M1) (arg M2) (bind y M3) (arg M4) (bind x M5) mt

52

CK+ Machine: Stack Structure

• Reorganize stack to be stack of stacks

bind continuations on top

(arg M) (bind z M0)(bind z M0)(bind z M0)(bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5)(bind x M5)(bind x M5)(bind x M5) mt

53

CK+ Machine: Stack Structure

• Reorganize stack to be stack of stacks

bind continuations on top

(arg M) (bind z M0)(bind z M0)(bind z M0)(bind z M0) (arg M1) (arg M2) (bind y M3)(bind y M3)(bind y M3)(bind y M3) (arg M4) (bind x M5)(bind x M5)(bind x M5)(bind x M5) mt

(arg M)

mt

(bind M0)(bind M0)(bind M0)(bind M0)

(arg M1)

(arg M2)

mt

(bind M3)(bind M3)(bind M3)(bind M3)

(arg M4)

mt

(bind M5)(bind M5)(bind M5)(bind M5)

mt

54

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

55

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.M

56

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

57

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

K = mt | (arg M K) | (bind x M K) | (op x K K)

58

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

K = mt | (arg M K) | (bind x M K) | (op x K K)K = mt | (arg M K) | (bind M K) | (op K K)

59

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

K = mt | (arg M K) | (bind x M K) | (op x K K)K = mt | (arg M K) | (bind M K) | (op K K)

λx.(x λy.(x y))

60

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

K = mt | (arg M K) | (bind x M K) | (op x K K)K = mt | (arg M K) | (bind M K) | (op K K)

λx.(x λy.(x y))λ.(0 λ.(1 0))

61

CK+ Machine: Lexical Addresses

• Replace variables with lexical addresses

[De Bruijn 1972]

M = x | M M | λx.MM = n | M M | λ.M

K = mt | (arg M K) | (bind x M K) | (op x K K)K = mt | (arg M K) | (bind M K) | (op K K)

λx.(x λy.(x y))λ.(0 λ.(1 0))

62

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

63

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

64

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5KKKK = mt

65

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4)KKKK = (arg M5)

mt

66

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(λ.(1 M)) M0 M1 M2) M3 M4KKKK = mt (bind M5)

mt

67

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(λ.(1 M)) M0 M1 M2) M3KKKK = (arg M4)

mt

(bind M5)

mt

68

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(λ.(1 M)) M0 M1 M2)KKKK = (arg M3)

(arg M4)

mt

(bind M5)

mt

69

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(1 M)) M0 M1 M2KKKK = mt (bind M3)

(arg M4)

mt

(bind M5)

mt

70

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(1 M)) M0 M1KKKK = (arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

71

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(1 M)) M0KKKK = (arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

72

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (λ.(1 M))KKKK = (arg M0)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

73

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = (1 M)KKKK = mt (bind M0)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

74

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = 1KKKK = (arg M)

mt

(bind M0)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

75

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = 1KKKK = (arg M)

mt

(bind M0)

(arg M1)

(arg M2)

mt0

(bind M3)

(arg M4)

mt1

(bind M5)

mt2

76

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = 1KKKK = (arg M)

mt

(bind M0)

(arg M1)

(arg M2)

mt0

(bind M3)

(arg M4)

mt1

(bind M5)

mt2

77

CK+ Machine: Example

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5

CCCC = 1KKKK = (arg M)

mt

(bind M0)

(arg M1)

(arg M2)

mt0

(bind M3)

(arg M4)

mt1

(bind M5)

mt2

• Direct index instead of search

78

Stack Compaction

79

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

80

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

81

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

mt

(bind M0)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

82

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

mt

(bind M0)(bind M0)(bind M0)(bind M0)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

83

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)

mt

84

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

(arg M1)

(arg M2)

mt

(bind M3)(bind M3)(bind M3)(bind M3)

(arg M4)

mt

(bind M5)

mt

85

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

(bind M5)(bind M5)(bind M5)(bind M5)

mt

86

Stack Compaction

((λx.M) N) Mwhere x ∉ FV(M)

(λ.(λ.(λ.(1 M)) M0 M1 M2) M3 M4) M5where

No variables reference M0 or M5

CCCC = 1

KKKK = (arg M)

(arg M1)

(arg M2)

mt

(bind M3)

(arg M4)

mt

87

Thanks!

88