Essentials of Programming Languages

1
Essentials of Programming Languages

• Section 1.3

2
Scoping and Binding of Variables

• Variables in Scheme appear in two different
  ways
  
• 1) References x in the expression (f x y)
• 2) Declarations x in the expressions
• (lambda (x) )
• (let (x ) )
• The value named by a variable is its denotation
• The denotation comes from a declaration
• The variable is bound by the declaration

3
Lexical Scoping in Scheme

• Declarations in Scheme have a limited scope (the
  area in which the variable is bound)
  
• For lambda expressions the scope of a variable is
  the body of the lambda expression
  
• For let expressions, the scope of a variable is
  the let body
  
• (let (( )
• ( )
• ...
• ( ))
• )

4
Static Binding

• (let ((x 3))
• (let ((y x)
• (x 5))
• (display x)))
• Scope can be determined by analyzing the text
• Lisp uses dynamic scoping binding is determined
  during execution

5
Lambda Calculus Expressions

• (lambda () )
• ( )
• Binding rule for Lambda Calculus expressions In
  (lambda () ), the
  occurrence of is a declaration that
  binds all occurrences of that variable in
  unless some intervening declaration
  of the same variable occurs

6
Free Variables

• A variable x occurs free in E if and only if
  there is some use of x in E that is not bound by
  any declaration of x in E
  
• In ((lambda (x) x) y), y occurs free
• The variable y occurs free in
• (define foo
• (lambda (x)
• ( x y)))

7
Bound Variables

• A variable x occurs bound in an expression E if
  and only if there is some use of x in E that is
  bound by a declaration of x in E
  
• The variable x is bound in
• ((lambda (x) x) y)
• The variable x is bound in
• (let ((x 3) (y 4)) ( x y z))

8
Free Variables in Lambda Calculus Expressions

• A variable x occurs free in a lambda calculus
  expression E if and only if
  
• 1) E is a variable reference and E is the
  same as x or
  
• 2) E is of the form (lambda (y) E), where y
  is different from x and x occurs free in E or
  
• 3) E is of the form (E1 E2) and x occurs
  bound in E1 or E2

9
Occurs-free?
(lambda (y) y) (cdr (lambda (y) x)) ? ((y) x) (
cadr (lambda (y) x))?(y) (caadr (lambda (y) x)
? y
(caddr (lambda (y) x) ? x

• (define occurs-free?
• (lambda (var exp)
• (cond
• ((symbol? exp) (eqv? exp var))
• ((eqv? (car exp) 'lambda)
• (and (not (eqv? (caadr exp) var))
• (occurs-free? var (caddr exp))))
• (else (or (occurs-free? var (car exp))
• (occurs-free? var (cadr
  exp)))))))
  
• (occurs-free? 'x 'x)
• (occurs-free? 'x '(lambda (x) x))
• (occurs-free? 'x '(lambda (y) x))

Rule 1 Rule 2 Rule 3
10
Bound Variables in Lambda Calculus Expressions

• A variable occurs bound in a lambda calculus
  expression E if and only if
  
• 1) E is of the form (lambda (y) E), where x
  occurs bound in E or x and y are the same
  variable and y occurs free in E or
  
• 2) E is of the form (E1,E2) and x occurs
  bound in E1 or E2

11
Occurs-bound?

• (define occurs-bound?
• (lambda (var exp)
• (cond
• ((symbol? exp) f)
• ((eqv? (car exp) 'lambda)
• (or (occurs-bound? var (caddr exp))
• (and (eqv? (caadr exp) var)
• (occurs-free? var (caddr
  exp)))))
  
• (else (or (occurs-bound? var (car exp))
• (occurs-bound? var (cadr
  exp)))))))
  
• (occurs-bound? 'x 'x)
• (occurs-bound? 'x '(lambda (x) x))
• (occurs-bound? 'x '(lambda (y) x))

12
Scope and Lexical Address

• Given a declaration, which variable references
  refer to it?
  
• The scope of x in (lambda (x) ) is the body of
  the lambda expression
  
• The scope of x in (define x ) is the whole
  program
  
• Nesting of blocks complicates the scope rules

13
Nesting of Blocks

• (define x call this x1
• (lambda (x) call this x2
• (map
• (lambda (x) call this x3
• ( x 1)) refers to x3
• x))) refers to x2
• (x '(1 2 3)) refers to x1

14
Scope and Lexical Address

• Scope of a variable declaration is the text
  within which references to the variable refer to
  the declaration
  
• Shadowing occurs when the inner declaration of a
  variable takes precedence over an outer
  declaration of the same variable
  
• The inner declaration of a variable creates a
  hole in the scope of an outer variable with the
  same name

15
Contour Diagrams

• Borders of regions that denote scope are called
  contours
  
• (lambda (z)
• ((lambda (a b c)
• (a (lambda (a) ( a c) ) b ) )
• (lambda (f x) (f (z x)) )) )

16
Lexical Addresses

• The declarations associated with a region may be
  numbered in the order of their appearance in the
  text. Each variable reference may then be
  associated with two numbers its lexical depth
  and the position of its declaration in the
  declaring contour (its declaration position)
• Taken together, these two numbers are the lexical
  address of the variable reference. Thus a
  variable reference v in an expression, may be
  replaced by (v d p), where d is its lexical
  depth and p is its declaration position, (using
  zero-base indexing for both d and p)
• Note that lexical depth is measured from the
  declaration which is depth 0

17
Lexical Addresses (vd p)

• (lambda (x y)
• (lambda (a)
• ( x y a)))
• (lambda (x y)
• (lambda (a)
• ( x y a) ) )

Position 0 Position 1
Position 0
Level 0
Level 1
Region