THE CALL-BY-NEED LAMBDA CALCULUS, REVISITED
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1 THE CALL-BY-NEED LAMBDA CALCULUS, REVISITED Stephen Chang and Matthias Felleisen Northeastern University 26/3/2012 1
2 Church 12
3 13
4 24
5 35
6 46
7 57
8 λ call-by-name (.e) e{x:= } (β) 68-69
9 λ call-by-name E[(.e) ] E[e{x:= }] (β) "leftmost-outermost" e E 70
10 λ v call-by-value E[(.e) v x ] E[e{x:=v x }] (β v ) "leftmost-outermost" e E v x 71
11 call-by-need 72
12 call-by-need 1) Evaluate argument only when needed. 73
13 call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. 74
14 λ?? need call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. 75
15 λ need-af Ariola/Felleisen '94,'95,'97? = λ need? = λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. 76
16 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need 1) Evaluate argument only when needed. 2) Evaluate argument at most once. 77
17 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need 78
18 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need x 79
19 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need x 80
20 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need x 81
21 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need x x (reshuffle) 92
22 λ need-af Ariola/Felleisen '94,'95,'97 λ need λ need-mow Maraist/Odersky/Wadler '94,'95,'98 call-by-need v x v x x v x (like β) 103
23 our λ need call-by-need x 104
24 our λ need call-by-need x 105
25 our λ need call-by-need x "demand path" 106
26 our λ need call-by-need x (β need ) "demand path" 107
27 OLD λ need : OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. 108
28 OLD λ need : OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. 109
29 OLD λ need : DEMAND CONTEXTS D = [ ] D e 110
30 OLD λ need : DEMAND CONTEXTS D = [ ] D e 111
31 OLD λ need : DEMAND CONTEXTS D = [ ] D e (.D) e 112
32 OLD λ need : DEMAND CONTEXTS D = [ ] D e (.D) e 113
33 OLD λ need : DEMAND CONTEXTS D = [ ] D e (.D) e 114
34 OLD λ need : DEMAND CONTEXTS D = [ ] D e (.D) e binding structure 115
35 OLD λ need : DEMAND CONTEXTS D = [ ] D e B[D] binding structure B = [ ] (.B) e 116
36 OLD λ need : BINDING STRUCTURE B = [ ] (.B) e (.(λy.(λz....) e z ) e y ) 117
37 OLD λ need : BINDING STRUCTURE B = [ ] (.B) e (.(λy.(λz....) e z ) e y ) 118
38 OLD λ need : BINDING STRUCTURE B = [ ] (.B) e (.(λy.(λz....) e z ) e y ) 119
39 OLD λ need : BINDING STRUCTURE B = [ ] (.B) e (.(λy.(λz....) e z ) e y ) λy λz... e z e y 130
40 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (((.λy.λz....) ) e y ) e z λy λz... e y e z 131
41 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (((.λy.λz....) ) e y ) e z λy λz... e y e z 132
42 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (((.λy.λz....) ) e y ) e z λy λz... e y e z 133
43 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e ((.(λy.λz....) e y ) ) e z λy λz... e y e z 149
44 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e ((.(λy.λz....) e y ) ) e z λy λz... e y e z 150
45 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (.((λy.λz....) e y ) e z ) λy λz... e z e y 161
![OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.](/docs-images/81/84567951/images/46-0.jpg "B) e (.((λy.λz....) e y ) e z ) λy λz.")
46 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (.((λy.λz....) e y ) e z ) λy λz... e z e y 162
47 OLD λ need : RESHUFFLING OF BINDINGS B = [ ] (.B) e (.(λy.(λz....) e z ) e y ) λy λz... e z e y 173
48 PROBLEMS WITH OLD CALL-BY-NEED CALCULUS 1) Reshuffling rules. 174
49 OLD λ need : OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. 175
50 OLD λ need : OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. 176
51 OLD λ need : OPERATIONAL OVERVIEW 1) Find the next demanded variable. 2) Find its corresponding argument and evaluate it. 3) Substitute evaluated argument for demanded variable. 177
52 OLD λ need : DEREFERENCING (.(λy.(λz....) e z ) e y ) λy λz... e z e y 178
53 OLD λ need : DEREFERENCING (.(λy.(λz. y ) e z ) e y ) λz y λy e z e y 179
54 OLD λ need : DEREFERENCING (.(λy.(λz. y ) e z ) v y ) λz y λy e z v y 190
55 OLD λ need : DEREFERENCING (.(λy.(λz. y ) e z ) v y ) λz y λy e z v y (λy.d[y]) v (λy.d[v]) v (deref) 191
56 OLD λ need : DEREFERENCING (.(λy.(λz. v y ) e z ) v y ) λz v y λy e z v y (λy.d[y]) v (λy.d[v]) v (deref) 202
57 PROBLEMS WITH OLD CALL-BY-NEED CALCULUS 1) Reshuffling rules. 2) Arguments and applications never go away. 203
58 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e z e y
59 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e z e y B = [ ] (.B) e 206
60 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ]??? 207
61 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e z e y A = [ ]??? 208
62 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ]??? 209
63 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ] (.A) e 210
64 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ] (.A) e 211
65 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ] (.A) e 212
66 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ] A[(.A)] e 213
67 NEW λ need : HANDLING ARBITRARY BINDING STRUCTURE λy λz... e y e z A = [ ] A[(.A)] e D = [ ] D e A[D] 214
68 PROBLEMS WITH OLD CALL-BY-NEED CALCULUS 1) Reshuffling rules. 2) Arguments and applications never go away. 215
69 NEW λ need : SPLITTING CONTEXTS λy λz... e y e z 216
70 NEW λ need : SPLITTING CONTEXTS λy λz... e y e z A[...] 217
71 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z A[D[y]] 218
72 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z A[D[y]] 219
73 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z... (λy...d[y])...e y
74 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z... (λy...d[y])...e y
75 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z...a[λy...d[y] ] e y
76 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z...a[λy...d[y] ] e y
77 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z...a[λy. [D[y]]] e y... = [ ] A[. ] 224
78 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z...a[λy. [D[y]]] e y... = [ ] A[. ] 225
79 NEW λ need : SPLITTING CONTEXTS λy λz D[y] e y e z [A[λy. [D[y]]] e y ] = [ ] A[. ] = [ ] A[ ] e 226
80 NEW λ need : SPLITTING CONTEXTS λy λz D[y] v y e z [A[λy. [D[y]]] v y ] = [ ] A[. ] = [ ] A[ ] e 237
81 NEW λ need : β need λy λz D[y] v y e z [A[λy. [D[y]]] v y ] [A[ [D[y]]{y:=v y }]] (β need) 238
82 NEW λ need : β need e z v y e z λy λz λz D[y] {y:=v y} D[y] [A[λy. [D[y]]] v y ] [A[ [D[y]]{y:=v y }]] (β need) 249
83 PROBLEMS WITH PREVIOUS CALL-BY-NEED CALCULUS 1) Reassociation rules. 2) Function calls not resolved. 250
84 NEW λ need : EVALUATING ARGUMENTS D = [ ] D e A[D]
85 NEW λ need : EVALUATING ARGUMENTS D = [ ] D e A[D] [A[λy. [D[y]]] D] 253
86 OTHER INTERESTING THINGS IN THE PAPER... Correspondence to Launchbury's (1993) machine semantics
87 OTHER INTERESTING THINGS IN THE PAPER... Correspondence to Launchbury's (1993) machine semantics. Confluence, Standardization properties. 256
88 OTHER INTERESTING THINGS IN THE PAPER... Correspondence to Launchbury's (1993) machine semantics. Confluence, Standardization properties. Soundness with respect to observational equivalence. 257
89 Thanks! 258
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