p: (d e)b3: (f e d)

b2E1

GE

a:8b:5c:8

f:

p: (x y)b1: (+ x y)

a:8b:5c:3

GE53

p:

p: (a b c)b2: (let…)

b3E2

d:13e:40

E1

53 b1E3

x:40y:13

E2

53Practice Session 10

The Environment Model

• Frame: A substitution, a mapping between variables and values. Every variable in a frame has a single value.

• Environment: A finite sequence of frames, where the last frame is the global environment.

• The global environment: A single-frame environment, the only environment that statically exists.

• Enclosing environment: For an environment E, the enclosing environment is E, excluding its first frame.

GEI

x:3y:5

IIz:6x:7

IIIn:1y:2

E1 E2

E1=<II,I>, E2=<III,I>, GE=<I>

Enclosing-env (E1) = GEEnclosing-env (E2) = GE

Environments:

Environment Model: Introduction

GEI

x:3y:5

IIz:6x:7

IIIn:1y:2

E1 E2

E1=<II,I>, E2=<III,I>, GE=<I>

Enclosing-env (E1) = GEEnclosing-env (E2) = GE

• The value of variable in an environment:The value of x in the environment E is its values in the first frame in which it is define.

Environments:

Environment Model: Introduction

• A procedure value (closure) is a data structure with:

(1) Procedure parameters.

(2) Procedure body.

(3) The environment in which it has been created.

Procedures:

> (define square (lambda (x) (* x x)))

GEsquare:

p:(x(b:(* x x(

p:(y(b:y

> (lambda (y) y)

b1

b2

Environment Model: Introduction

• Procedure application:

(1) Create new frame, mapping the procedure params to application values.

(2) The new frame extends the environment associated with the procedure.

(3) Evaluate the body of the procedure in the new environment.

Procedures:

GEsquare:

p:(x(b:(* x x(

GEE1

Bx:5

> (define square (lambda (x) (* x x)))

> (square 5)

b1

Environment Model: Introduction

GE

sq:

sum-of-squares:

f:

p: (x)b1: (* x x)

p: (x y(b2: (+ (sq x) (sq y))

p: (a)b3: (sum-of-squares (+ a 1) (* a 2))

Environment Model: Definition and Application

B3E1

a:5

GE136

B2E2

x:6y:10

B1E3

x:6B1

x:10

E2

100

E1

136

GE

sq:

sum-of-squares:

f:

p: (x)b1: (* x x)

p: (x y(b2: (+ (sq x) (sq y))

p: (a)b3: (sum-of-squares (+ a 1) (* a 2))

Environment Model: Definition and Application

E2

36

E4

Environment Model: Definition and Let

GE

a:8b:5c:8f:

b1E1

x:8y:8

GE16

p: (x y)b1: (+ x y)

GE53b2E1

a:8b:5c:3

p: (a b c)b2: (let…)

E1

53b3E2

d:13e:40

E2

53b1E3

x:40y:13

f:

p: (x y)b1: (+ x y)

p:

p: (d e)b3: (f e d)

Environment Model: Definition and Let

GE

a:8b:5c:8

GE

p:(n)b:(if…)

fact:

bE1

n:3

GE6

bE2

n:2

E1

2bE3

n:1

E2

1bE4

n:0

E3

1

Environment Model: Recursion

GE

p:(x y)b1:(lambda(sel)…)

make-pair:P1:

p:(sel)b2:(sel x y)

b1E1

x:5y:10

GE

Environment Model: Pair ADT

GEb2E2

sel:

b3E3

a:5b:10

E2

5

GE

p:(x y)b1:(lambda(sel)…)

make-pair:P1:

p:(sel)b2:(sel x y)

b1E1

x:5y:10

GE

p:(a b)b3:a

Environment Model: Pair ADT

( >p1 (lambda (a b( a((

5

p:(x y)b3:(p1(lambda…)

b3E2

x:1y:2

GE

p:(first second)b4:first

b1E3 E2

sel:

b4E4

first:5second:10

GE

p:(x y)b1:(lambda(sel)…)

make-pair:P1:

p:(sel)b2:(sel x y)

b1E1

x:5y:10

GE

Environment Model: Pair ADT

> (let ((x 1( (y 2(( (p1 (lambda (first second( first(((

b4

b3

Environment Model: Lexical (static) vs Dynamic Scoping

Lexical Scoping:

• A closure “carries” the environment in which it has been created.

• In application of a closure, its carried environment is extended.

Dynamic Scoping:

• A closure does not correspond to any environment.

• In application of a closure, the calling environment is extended.

• Simpler implementation (no need to “store” environments).

GE

p:(x y)b1:(lambda(sel)…)

make-pair:p1:

p:(sel)b2:(sel x y)

b1E1

x:5y:10

p:(x y)b3:(p1(lambda…)b3E2

x:1y:2

p:(first second)b4:first

b2E3

sel:

b4E4

first:1second:2

1

11

Environment Model: Dynamic Scoping - Example

> (let ((x 1( (y 2(( (p1 (lambda (first second( first(((

b4

b3

p:(n c)b1:(if…)

GE

fact$:

b1E1

n:3

GE6

p(fact-3)b3:(fact-3)

c:

p:(fact-n-1)b2:(c…)

b1E2

n:2

E1

6

c:

p:(fact-n-1)b2:(c…)

b1E3

n:1

E2

6

c:

b2E4

fact-n-1:1

E3

6

b2E5

fact-n-1:2

E4

6

b3E6

fact-n-1:6

E5

6

Environment Model: CPS