1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
|
#lang racket
(require redex)
; An STG-like language as described in
; "Making a Fast Curry: Push/Enter vs. Eval/Apply for Higher-order Languages"
; Only lightly-tested.
; A list of differences from STG proper:
; * Right-hand sides of let-bindings have different syntax
; * Let-binding is not recursive
; * Missing semantics for top-level bindings
; * Missing semantics for primops, e.g.
; * Exceptions
; * Concurrency
; * STM
; * Missing let-no-escape
; * Missing IND (for heap indirections after thunk evaluation)
; Some other things that these semantics might want to capture
; * Selector thunks
; * Pointer tagging
; * Stack chunks / STACK_AP
; * CAFs
; Useful sanity checks:
; * Formalize Core (with typing), define translation to STG, test for
; progress given that Core type-checks
(define-language L
((x y z f g h) variable-not-otherwise-mentioned)
(C variable-not-otherwise-mentioned)
(n integer)
(lit integer
real)
((a v) x
lit)
(k • ; unknown arity
n)
(e a
(f k a ...)
(⊕ a ...) ; saturated
(let (x obj) e)
(case e alt ...)
)
(alt ((C x ...) e)
((x) e))
(obj val
(THUNK e)
BLACKHOLE)
(val (FUN (x ...) e)
(PAP f a ...)
(CON C a ...)) ; saturated
)
(define-extended-language Ev L
(p (e s H))
(H ((x_!_ obj) ...))
(κ (case • alt ...)
(upd x •)
(• a ...))
(s (κ ...))
)
; use the tutorial substitution
(require redex/tut-subst)
(define-metafunction Ev
subst : (x v) ... e -> e
[(subst (x v) ... e)
,(subst/proc x?
(term (x ...))
(term (v ...))
(term e))])
(define x? (redex-match Ev x))
; We need to do some negative matches, a metafunction will be easiest
(define-metafunction Ev [(lit? e) ,(redex-match? Ev lit (term e))])
(define-metafunction
Ev
[(heapval? e H)
,(redex-match Ev
(x_i ((x_0 obj_0) ... (x_i val_i) (x_i+1 obj_i+1) ...))
(term (e H)))
])
; eval/apply reduction semantics
(define red
(reduction-relation
Ev
#:domain p
(--> ((let (x obj) e) s ((x_1 obj_1) ...))
((subst (x x_0) e) s ((x_0 obj) (x_1 obj_1) ...))
(fresh x_0)
"Let")
(--> ((case x_i alt_0 ... ((C_i y_i ..._i_) e_i) alt_i+1 ...) s H)
((subst (y_i a_i) ... e_i) s H)
(where ((x_0 obj_0) ... (x_i (CON C_i a_i ..._i_)) (x_i+1 obj_i+1) ...) H)
"CaseCon")
(--> ((case x_i alt ... ((x) e)) s H)
((subst (x x_i) e) s H)
(where ((x_0 obj_0) ... (x_i val_i) (x_i+1 obj_i+1) ...) H)
(side-condition ; this terri-bad side condition
(not (redex-match ; NB: not redex-match? which is buggy
Ev
((CON C_i a_i ...) alt_0 ... ((C_i y_i ...) e_i) alt_i+1 ...)
(term (val_i alt ...)))))
"CaseAnyHeap")
(--> ((case lit alt ... ((x) e)) s H)
((subst (x lit) e) s H)
"CaseAnyLit")
; I kind of like the original presentation, where we have an execution
; code that tells us whether or not we need to enter the scrutinee
(--> ((case e alt ...) (κ ...) H)
(e ((case • alt ...) κ ...) H)
(where #f (lit? e))
(where #f (heapval? e H))
"Case")
(--> (lit ((case • alt ...) κ ...) H)
((case lit alt ...) (κ ...) H)
"RetLit")
(--> (x_i ((case • alt ...) κ ...) H)
((case x_i alt ...) (κ ...) H)
(where ((x_0 obj_0) ... (x_i val_i) (x_i+1 obj_i+1) ...) H)
"Ret")
(--> (x_i s
((x_0 obj_0) ... (x_i (THUNK e)) (x_i+1 obj_i+1) ...))
(e ((upd x_i •) ,@(term s)) ; nifty idiom for splicing in
((x_0 obj_0) ... (x_i BLACKHOLE) (x_i+1 obj_i+1) ...))
"Thunk")
(--> (y_j ((upd x_i •) κ ...) H)
(y_j (κ ...) ((x_0 obj_x0) ... (x_i val_j) (x_i+1 obj_i+1) ...))
(where ((x_0 obj_x0) ... (x_i BLACKHOLE) (x_i+1 obj_i+1) ...) H)
(where ((y_0 obj_y0) ... (y_j val_j) (y_j+1 obj_j+1) ...) H)
"Update")
(--> ((f_i n a ..._n_) s H)
((subst (x a) ... e) s H)
(where ((f_0 obj_0) ... (f_i (FUN (x ..._n_) e)) (f_i+1 obj_i+1) ...) H)
(side-condition (= (length (term (a ...))) (length (term (x ...))) (term n)))
"KnownCall")
; Primop rule is omitted
; n.b. named ellipses do not carry over
(--> ((f_i • a ...) s H)
((subst (x a) ... e) s H)
(where ((f_0 obj_0) ... (f_i (FUN (x ...) e)) (f_i+1 obj_i+1) ...) H)
(side-condition (= (length (term (a ...))) (length (term (x ...)))))
"Exact")
(--> ((f_i k a_1→n ..._1→n a_n+1→m ...)
(κ ...)
(name H ((f_0 obj_0) ... (f_i (FUN (x ..._1→n) e)) (f_i+1 obj_i+1) ...)))
((subst (x a_1→n) ... e) ((• a_n+1→m ...) κ ...) H)
(side-condition (> (length (term (a_n+1→m ...))) 0))
"CallK")
(--> ((f_i k a ..._1→m)
s
(name H ((f_1 obj_1) ... (f_i (FUN (x_1→m ..._1→m x_m+1→n ...) e)) (f_i+1 obj_i+1) ...)))
(f_0 s ((f_0 (PAP f_i a ...)) ,@(term H)))
(fresh f_0)
(side-condition (> (length (term (x_m+1→n ...))) 0))
"PAP")
(--> ((f_i • a ...) (κ ...) H)
(f_i ((• a ...) κ ...) H)
(where ((f_0 obj_0) ... (f_i (THUNK e)) (f_i+1 obj_i+1) ...) H)
"TCall")
(--> ((f_i k a_m ...) s H)
((g • a_n ... a_m ...) s H)
(where ((f_0 obj_0) ... (f_i (PAP g a_n ...)) (f_i+1 obj_i+1) ...) H)
"PCall")
(--> (f_i ((• a ...) κ ...) H)
((f_i • a ...) (κ ...) H)
; technically CON should not be allowed, but we'll get stuck one step later
(where ((f_0 obj_0) ... (f_i val) (f_i+1 obj_i+1) ...) H)
"RetFun")
))
(define dH (term ((f (FUN (x y z) (⊕ x y z))) (g (THUNK f)) (h (PAP f 0)) (y (THUNK z)) (z (CON C_I 0)))))
; XXX these tests are pretty fragile
(define dHc (term ((f (FUN (x y z) (⊕ x y z))) (g (THUNK f)) (h (PAP f 0)) (y (CON C_I 0)) (z (CON C_I 0)))))
; Case
(test-->> red
(term ((case z ((C_I x) (⊕ x 1)) ((x) x)) () ,dH))
(term ((⊕ 0 1) () ,dH)))
(test-->> red
(term ((case y ((C_I x) (⊕ x 1)) ((x) x)) () ,dH))
(term ((⊕ 0 1) () ,dHc)))
(test-->> red
(term ((case z ((C_J x) (⊕ x 1)) ((x) x)) () ,dH))
(term (z () ,dH)))
(test-->> red
(term ((case y ((C_J x) (⊕ x 1)) ((x) x)) () ,dH))
(term (z () ,dHc)))
(test-->> red
(term ((case 0 ((C_J x) (⊕ x 1)) ((x) x)) () ,dH))
(term (0 () ,dH)))
; KnownCall/Exact/CallK/PAP
(test-->> red
(term ((f • 0) () ,dH))
(term (f_0 () ((f_0 (PAP f 0)) ,@dH))))
(test-->> red
(term ((f 3 0) () ,dH))
(term (f_0 () ((f_0 (PAP f 0)) ,@dH))))
(test-->> red
(term ((f • 0 1 2 3) () ,dH))
(term ((⊕ 0 1 2) ((• 3)) ,dH)))
(test-->> red
(term ((f 3 0 1 2 3) () ,dH))
(term ((⊕ 0 1 2) ((• 3)) ,dH)))
(test-->> red
(term ((f • 0 1 2) () ,dH))
(term ((⊕ 0 1 2) () ,dH)))
(test-->> red
(term ((f 3 0 1 2) () ,dH))
(term ((⊕ 0 1 2) () ,dH)))
; TCall/Thunk/Update
(define dHe (term ((f (FUN (x y z) (⊕ x y z))) (g (FUN (x y z) (⊕ x y z))) (h (PAP f 0)) (y (THUNK z)) (z (CON C_I 0)))))
(test-->> red
(term ((g • 0) () ,dH))
(term (f_0 () ((f_0 (PAP f 0)) ,@dHe))))
(test-->> red
(term ((g • 0 1 2 3) () ,dH))
(term ((⊕ 0 1 2) ((• 3)) ,dHe)))
(test-->> red
(term ((g • 0 1 2) () ,dH))
(term ((⊕ 0 1 2) () ,dHe)))
; PCall
(test-->> red
(term ((h • 1) () ,dH))
(term (f_0 () ((f_0 (PAP f 0 1)) ,@dH))))
(test-->> red
(term ((h 2 1) () ,dH))
(term (f_0 () ((f_0 (PAP f 0 1)) ,@dH))))
(test-->> red
(term ((h • 1 2 3) () ,dH))
(term ((⊕ 0 1 2) ((• 3)) ,dH)))
(test-->> red
(term ((h 3 1 2 3) () ,dH))
(term ((⊕ 0 1 2) ((• 3)) ,dH)))
(test-->> red
(term ((h • 1 2) () ,dH))
(term ((⊕ 0 1 2) () ,dH)))
(test-->> red
(term ((h 3 1 2) () ,dH))
(term ((⊕ 0 1 2) () ,dH)))
|
