PS4

1
PS4 6 if you got points off because you didnt
do an intersect with accessible and the start
states, submit a re-grade. Today
let, let, letrec in the env. model
side-effecting lists (set-car!,
set-cdr!) Tomorrow more on
side-effecting data structures (e.g., chains)
2

• To evaluate a lambda in environment E, build a
  closure.
• To evaluate a (non-special form) combination (f
  a1 ... an) in E
• evaluate f in E and get a closure c
• evaluate each argument ai in E to get a value vi
• apply the closure c to the values v1 ... vn
• To apply a closure c to values v1 ... vn
• build a new frame, binding closure formal
  parameters to
• actual parameters v1 ... vn
• link the frame into the environment of the
  closure to yield a
• new environment
• evaluate the body of the closure in the new
  environment

3
To evaluate (let ((x1 e1) (x2 e2)
... (xn en)) e) in environment E 1.
evaluate e1, e2, ..., en in environment E
producing values v1,v2,...,vn. 2. build a new
frame that binds x1 to v1, x2 to v2, ..., xn to
vn. 3. link the new frame into E to yield a new
environment. 4. evaluate the body of the let, e,
in the new environment.
4
(let ((x 1) (y 2)) ( x y))
mult
1. evaluate 1 and 2 in the current env.
5
(let ((x 1) (y 2)) ( x y))
mult
x 1 y 2
2. build new frame with bindings 3. link it in to
the env.
6
(let ((x 1) (y 2)) ( x y))
mult
x 1 y 2
4. evaluate body of let in new env.
7
(let ((x 3) (y ( x x))) ( x y))
mult
1. evaluate 1 in environment to get 1...
8
(let ((x 3) (y ( x x))) ( x y))
mult
...but evaluating ( x x) fails! (no binding for
x in the environment!)
9
Alternative (let ((x1 e1) (x2 e2)
... (xn en)) e) in environment E,
evaluate ((lambda (x1 x2 ... xn) e) e1 e2 ...
en)
10
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
11
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
12
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
args (x y) body ( x y) env
13
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
args (x y) body ( x y) env
Now apply the closure to 1 and 2...
14
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
args (x y) body ( x y) env
x 1 y 2
...and we get 2.
15
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
args (x y) body ( x y) env
x 1 y 2
afterwards, we continue evaluating in the
top-level environment...
16
(let ((x 1) (y 2)) gt ((lambda (x y)
( x y)) 1 2) ( x y))
mult
args (x y) body ( x y) env
x 1 y 2
and the intermediate closure and frame are
garbage collected
17
To evaluate (let ((x1 e1) (x2 e2)
... (xn en)) e) in environment
E 1. evaluate e1 in E to v1 2. add a new frame
mapping x1 to v1, yielding E2 3. evaluate e2 in
E2 to v2 4. add a new frame mapping x2 to v2,
yielding E3 ... 2n-1. evaluate en in En to vn 2n.
add a new frame mapping xn to vn, yielding
E 2n1. evaluate the body e in E
18
(let ((x 3) (y ( x x))) ( x y))
mult
19
(let ((x 3) (y ( x x))) ( x y))
mult
x3
20
(let ((x 3) (y ( x x))) ( x y))
mult
x3
The ( x x) yields 9...
21
(let ((x 3) (y ( x x))) ( x y))
mult
x3
y9
22
(let ((x 3) (y ( x x))) ( x y))
mult
x3
y9
The ( x y) yields 39 27
23
Alternative (let ((x1 e1) (x2 e2)
... (xn en)) e) in environment E,
evaluate (let ((x1 e1)) (((...(lambda
(x1) (let ((x2 e2)) ((lambda
(x2) ... gt ...
(let ((xn en)) ((lambda (xn)
e)...)) e))) e1) e2)
... en)
24
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
mult
25
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
args x body env
mult
26
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
args x body env
mult
x3
27
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
args x body env
mult
x3
28
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
mult
x3
29
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
mult
x3
args y body ( x y) env
30
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
mult
x3
args y body ( x y) env
And then we apply the closure to 9...
31
(let ((x 3) (let ((x 3)) (y (
x x))) gt (let ((y ( x x)) gt ( x y))
( x y))) ((lambda (x) ((lambda
(y) ( x y)) ( x x))) 3)
mult
x3
args y body ( x y) env
y9
...and as before get 27 as the result.
32

• To evaluate
• (letrec ((x1 e1)
• (x2 e2)
• ...
• (xn en))
• e)
• in environment E
• 1. build a new frame mapping each xi to void
• looking up void should be an error...
• 2. link the frame into E yielding a new
  environment E
• 3. evaluate each ei in the new environment E to
  a value vi
• 4. set! the bindings in the frame so that xi maps
  to vi
• 5. evaluate the body e in E

33
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
1. create a new frame mapping f to void
34
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
f void
1. create a new frame mapping f to void 2. link
it in to the environment
35
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
f void
3. evaluate the expression in the new
environment to get a value v.
36
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f void
3. evaluate the expression in the new
environment to get a value v.
37
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
4. set! the binding in the frame so that f maps
to v
38
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
5. now evaluate the body of the letrec in the
new environment.
39
(letrec ((f (lambda (x) (if ( 1
x) 1 ( x (f (- 1 x))))))) (f 3))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
40
(if ( 1 x) 1 ( x (f (- 1 x)))) gt (if ( 1 3) 1
( x (f (- 1 x)))) gt (if f 1 ( x (f (- 1
x)))) gt ( x (f (- 1 x))) gt ( 3 (f (- 1 x)))
gt ( 3 (f 2))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
41
(if ( 1 x) 1 ( x (f (- 1 x)))) gt (if ( 1 2) 1
( x (f (- 1 x)))) gt (if f 1 ( x (f (- 1
x)))) gt ( x (f (- 1 x))) gt ( 2 (f (- 1 x)))
gt ( 2 (f 1))
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
x 2
42
(if ( 1 x) 1 ( x (f (- 1 x)))) gt (if ( 1 1) 1
( x (f (- 1 x)))) gt (if t 1 ( x (f (- 1
x)))) gt 1
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
x 2
x 1
43
(if ( 1 x) 1 ( x (f (- 1 x)))) gt ( 2 (f 1))
gt ( 2 1) gt 2
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
x 2
44
(if ( 1 x) 1 ( x (f (- 1 x)))) gt ( 3 (f 2))
gt ( 3 2) gt 6
mult add equal - sub
args x body (if ( 1 x) 1 ( x (f (-
1 x)))) env
f
x 3
45
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
46
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
f void g void
47
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
f void g void
48
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f void g void
49
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f void g void
50
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f void g void
args y body (f y) env
51
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f g
args y body (f y) env
52
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f g
args y body (f y) env
53
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f g
x 0
args y body (f y) env
54
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0))
args x body (g x) env
f g
x 0
args y body (f y) env
y 0
55
Alternative to evaluate (letrec ((x1 e1)
... (xn en)) e) in
environment E, evaluate?
(let ((x1 void) define place-holders
... (xn void)) (let ((v1 e1)
evaluate e1,...,en ... and save
values as v1,...,vn (vn en)) (set! x1
v1) set x1,...,xn to v1,...,vn ...
(set! xn vn) e))
56
(letrec ((f (lambda (x) (g x))) (g
(lambda (y) (f y)))) (f 0)) gt (let ((f
void) (g void)) (let ((v1 (lambda (x)
(g x))) (v2 (lambda (y) (f y))))
(set! f v1) (set! g v2) (f 0)))
57
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
f void g void
58
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f void g void
59
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f void g void
args y body (f y) env
60
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f void g void
args y body (f y) env
v1 v2
61
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f g void
args y body (f y) env
v1 v2
62
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f g
args y body (f y) env
v1 v2
63
(let ((f void) (g void)) (let ((v1
(lambda (x) (g x))) (v2 (lambda (y) (f
y)))) (set! f v1) (set! g v2) (f 0)))
args x body (g x) env
f g
args y body (f y) env
v1 v2
64

• Mutable Data
• (define mylist (list 1 2 3))
• (let ((new (list 4)))
• (set! (tail new) (tail mylist))
• (set! (tail mylist) new))

mylist
65

• Mutable Data
• (define mylist (list 1 2 3))
• (let ((new (list 4)))
• (set! (tail new) (tail mylist))
• (set! (tail mylist) new))

mylist
new
66

• Mutable Data
• (define mylist (list 1 2 3))
• (let ((new (list 4)))
• (set! (tail new) (tail mylist))
• (set! (tail mylist) new))

mylist
new
67

• Mutable Data
• (define mylist (list 1 2 3))
• (let ((new (list 4)))
• (set! (tail new) (tail mylist))
• (set! (tail mylist) new))

X
mylist
new
68

• Mutable Data
• (define mylist (list 1 2 3))
• (let ((new (list 4)))
• (set! (tail new) (tail mylist))
• (set! (tail mylist) new))

mylist
new