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Evaluating Call-by-need on the Control Stack Stephen Chang, David Van Horn, Matthias Felleisen Northeastern University 1
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
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Example (Garcia Machine)
(λx.(λy.(λz.(y M)) M0 M1 M2) M3 M4) M5
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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
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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
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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
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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
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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
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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
